# De Sitter magnetic black hole dipole with a supersymmetric horizon

D.Astesiano and S.L. Cacciatori  Dipartimento di Scienza e Alta Tecnologia,
Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy, and INFN, Via
Celoria 16, 20133 Milano, Italy

###### Abstract.

We find a new non BPS solution in $N=2$ $D=4$ gauged supergravity coupled to
$U(1)$ gauge fields and matter. It consists in a closed universe with two
extremal black holes of equal size, surrounding two singularities. They have
opposite magnetic charges (and no electric charges), but stay in static
equilibrium thanks to the positive pressure of a cosmological constant. The
geometry is perfectly symmetric under the exchange of the black holes and the
flip of the sign of the charges. However the scalar field is non constant and
non symmetric, with different values at the horizons, which depend on a real
modulus. Remarkably we show that it satisfies the attractor mechanism and the
entropy indeed depends only on the magnetic charges. At one of the horizons
the solution becomes $\frac{1}{2}$-BPS supersymmetric, while at the other one
there is no supersymmetry, but the entropy remains independent from the scalar
modulus.

##  1\. Introduction

Black holes are of great relevance in theoretical physics, and nowadays
overwhelmingly entered in the realm of experimental physics, since the era of
gravitational way spectroscopy started up. Their thermodynamical properties,
like the microstate counting of the entropy, are the first trials by fire for
any quantum theory. In supergravity theories, extremal black holes may
represent supersymmetric solutions, [1] [2] [3], and, in gauged supergravities
they can be seen as interpolating between a partially supersymmetric
$AdS_{2}\times S^{d-2}$ topology at the horizon, the near horizon limit, and
the fully supersymmetric anti-de Sitter solution at infinity [4] [5] [6].
These are particularly important on the light of the conjectured duality
between conformal quantum field theories on flat backgrounds and classical and
quantum gravity in an anti-de Sitter space time, known under the name of
AdS/CFT correspondence, [7] [8] [9]. It has numerous applications to QCD,
condensed matter physics, hydrodynamics, superconductivity and in several
cases goes beyond conformal field theories, [10] [11] [12] [13] [14] [16] [17]
[18].
In particular, black hole solutions in gauged supergravities are dual of
condensed matter systems at finite temperature [14]. In [16] [15] it is shown
that black holes coupled with abelian gauge fields and neutral scalars can be
used to describe transitions from fermi liquids to non fermi liquids, while
when the scalar fields are charged the dual describes strongly coupled
superconductors [17]. In recent years there have been progresses in finding
BPS, non BPS and thermal black holes solutions in $N=2$ gauged supergravity in
four dimensions, coupled with matter, see for example [20], [21], [22] [23],
[24], [25], [26], [27], [28] and [29]. The work presented here belongs in this
precise context, and is inspired by [19], where a new class of near horizon
solutions, both stationary and static, in gauged supergravity coupled to
scalar fields have been found, but they resisted to any effort in determining
the black hole solutions of which they are the near horizon limit. In this
work we fill this gap for the static case, and will show that a number of
surprises arise. The stationary solution in [19] belongs to a class of BPS
solutions, which are everywhere regular with non constant scalar fields. More
precisely, we considered the Fayet-Iliopoulos gauged supergravity coupled to
three vector multiplets in the STU model with prepotential
$F(X^{0},X^{1},X^{2},X^{3})=-2i\sqrt{X^{0}X^{1}X^{2}X^{3}}$ and the solutions
were characterized by an $AdS_{2}\times S^{2}$ geometry, and preserved four
supersymmetry generators among the eight of the $N=2$, $d=4$ gauged
supergravity coupled with four abelian gauge groups and three complex scalar
fields. We will show that it is indeed the near horizon limit of a black hole
solution, which, however, is not supersymmetric at all: it consists in a pair
of extremal black holes of equal sizes, facing symmetrically each other in a
closed universe of positive curvature. It is a well known fact that a positive
cosmological constant forbids supersymmetry. Multi black holes solutions in a
de Sitter back ground are already known [30]; the Kastor-Traschen solution
generalizes the Majumdar-Papapetrou multi black hole solution in
asymptotically flat background [31] [32], but differ from the last, being non
static. In our case, instead, the two black holes resemble more the black hole
pairs of [33] (and reference therein), but are in static equilibrium, despite,
beyond gravitational attraction, they have to face also the electromagnetic
attraction since they have opposite magnetic charges and zero electric
charges, to form a sort of magnetic dipole. This attractive force is perfectly
balanced by the pressure determined by a positive cosmological constant. The
spatial section of such universe are represented by a finite space having the
shape of a three dimensional axially symmetric rugby ball at whose tips the
curvature becomes singular. The two singularities are hidden by two horizon
and the geometry is also symmetric under the reflection exchanging the tips.
At the center of symmetry, the geometry is the one of a de Sitter space
carrying a constant magnetic flux. However, while the same symmetry holds for
the gauge fields (up to a sign), it does not extend to the scalar fields,
which assume different values at the horizons. As a consequence, even though
the geometry is the same, only at one of the horizon the full supersymmetric
solution of [19] is reproduced: the second horizon results to be non
supersymmetric. It is worth to mention that the values of the scalar fields at
the horizons depend on a real free modulus. On the other hand, the symmetric
geometry is sufficient to ensure that both black holes have the same entropy,
and the supersymmetry of one of them protect it from the dependence on the
value of the scalar field at the horizon, t.i. on the scalar modulus. Indeed,
the solution turn out to satisfy the attractor mechanism at both horizons. The
whole spacetime metric depends on three parameters, which can be interpreted
as the positive cosmological constant, the value of the scalar modulus, and
the magnetostatic energy gap between the two horizons. All these properties
make evident the peculiarity of this new solution.
We can now pass to the proof of our assertions.

##  2\. Equations of motion

We consider $N=2$, $D=4$ gauged supergravity coupled to $n$ abelian vector
multiplets. Apart from the vierbein $e^{a}_{\mu}$, the bosonic field content
includes the vectors $A^{I}_{\mu}$ enumerated by $I=0,\ldots,n$, and the
complex scalars $z^{\alpha}$ where $\alpha=1,\ldots,n$. These scalars
parametrize a special Kähler manifold, i. e. , an $n$-dimensional Hodge-Kähler
manifold that is the base of a symplectic bundle, with the covariantly
holomorphic sections

${\mathcal{V}}=\left(\begin{array}[]{c}X^{I}\\\
F_{I}\end{array}\right)\,,\qquad{\mathcal{D}}_{\bar{\alpha}}{\mathcal{V}}=\partial_{\bar{\alpha}}{\mathcal{V}}-\frac{1}{2}(\partial_{\bar{\alpha}}{\mathcal{K}}){\mathcal{V}}=0\,,$
(2.1)

where ${\mathcal{K}}$ is the Kähler potential and ${\mathcal{D}}$ denotes the
Kähler-covariant derivative. ${\mathcal{V}}$ obeys the symplectic constraint

$\langle{\mathcal{V}},\bar{\mathcal{V}}\rangle=X^{I}\bar{F}_{I}-F_{I}\bar{X}^{I}=i\,.$
(2.2)

To solve this condition, one defines

${\mathcal{V}}=e^{{\mathcal{K}}(z,\bar{z})/2}v(z)\,,$ (2.3)

where $v(z)$ is a holomorphic symplectic vector,

$v(z)=\left(\begin{array}[]{c}Z^{I}(z)\\\ \frac{\partial}{\partial
Z^{I}}F(Z)\end{array}\right)\,.$ (2.4)

F is a homogeneous function of degree two, called the prepotential, whose
existence is assumed to obtain the last expression. The Kähler potential is
then

$e^{-{\mathcal{K}}(z,\bar{z})}=-i\langle v,\bar{v}\rangle\,.$ (2.5)

The matrix ${\mathcal{N}}_{IJ}$ determining the coupling between the scalars
$z^{\alpha}$ and the vectors $A^{I}_{\mu}$ is defined by the relations

$F_{I}={\mathcal{N}}_{IJ}X^{J}\,,\qquad{\mathcal{D}}_{\bar{\alpha}}\bar{F}_{I}={\mathcal{N}}_{IJ}{\mathcal{D}}_{\bar{\alpha}}\bar{X}^{J}\,.$
(2.6)

The bosonic action reads

$\displaystyle e^{-1}{\mathcal{L}}_{\text{bos}}=$
$\displaystyle\frac{1}{2}R+\frac{1}{4}(\text{Im}\,{\mathcal{N}})_{IJ}G^{I}_{\mu\nu}G^{J\mu\nu}-\frac{1}{8}(\text{Re}\,{\mathcal{N}})_{IJ}\,e^{-1}\epsilon^{\mu\nu\rho\sigma}G^{I}_{\mu\nu}G^{J}_{\rho\sigma}$
$\displaystyle-
g_{\alpha\bar{\beta}}\partial_{\mu}z^{\alpha}\partial^{\mu}\bar{z}^{\bar{\beta}}-V\,,$
(2.7)

with the scalar potential

$\displaystyle
V=-2g^{2}\xi_{I}\xi_{J}[(\text{Im}\,{\mathcal{N}})^{-1|IJ}+8\bar{X}^{I}X^{J}]\,,$
(2.8)

that results from U$(1)$ Fayet-Iliopoulos gauging. Here, $g$ denotes the gauge
coupling and the $\xi_{I}$ are FI constants. In what follows, we define
$g_{I}\equiv g\xi_{I}$. The Einstein’s equations of motion from (2.7) are
given by

$\displaystyle G_{\mu\nu}=$ $\displaystyle
T_{\mu\nu}=^{(0)}T_{\mu\nu}+^{(1)}T_{\mu\nu}-g_{\mu\nu}V$ (2.9)
${}^{(0)}T_{\mu\nu}=$ $\displaystyle
2g_{\alpha\bar{\beta}}\partial_{(\mu}z^{\alpha}\partial_{\nu)}\bar{z}^{\bar{\beta}}-g_{\mu\nu}g_{\alpha\bar{\beta}}\partial_{\sigma}z^{\alpha}\partial^{\sigma}\bar{z}^{\bar{\beta}}$
(2.10) ${}^{(1)}T_{\mu\nu}=$
$\displaystyle-(Im\mathcal{N})_{IJ}G^{I}_{\mu\sigma}G_{\nu}^{J\sigma}+g_{\mu\nu}\frac{1}{4}(Im\mathcal{N})_{IJ}G^{I}_{\sigma\rho}G^{J\sigma\rho}$
(2.11)

we make explicit the contribution form the spin $0$ and the spin $1$ parts.
These equation give the curvature of the space-time

$\displaystyle
R=2g_{\alpha\bar{\beta}}\partial_{\sigma}z^{\alpha}\partial^{\sigma}\bar{z}^{\bar{\beta}}+4V,$
(2.12)

which can be used to rewrite the full system as

$\displaystyle R_{\mu\nu}$
$\displaystyle=-(\text{Im}\,{\mathcal{N}})_{IJ}G^{I}_{\mu\lambda}G^{J\
\lambda}_{\
\nu}+2g_{\alpha\bar{\beta}}\partial_{(\mu}z^{\alpha}\partial_{\nu)}\bar{z}^{\bar{\beta}}+g_{\mu\nu}\left[\frac{1}{4}(\text{Im}\,{\mathcal{N}})_{IJ}G^{I}_{\rho\sigma}G^{J\rho\sigma}+V\right]\,,$
(2.13)
$\displaystyle\nabla_{\mu}\left[(\text{Im}\,{\mathcal{N}})_{IJ}G^{J\mu\nu}-\frac{1}{2}(\text{Re}\,{\mathcal{N}})_{IJ}\,e^{-1}\epsilon^{\mu\nu\rho\sigma}G^{J}_{\rho\sigma}\right]=0,\,$
(2.14)
$\displaystyle\begin{split}\frac{1}{4}\frac{\delta(\text{Im}\,{\mathcal{N}})_{IJ}}{\delta
z^{\alpha}}&G^{I}_{\mu\nu}G^{J\mu\nu}-\frac{1}{8}\frac{\delta(\text{Re}\,{\mathcal{N}})_{IJ}}{\delta
z^{\alpha}}\,e^{-1}\epsilon^{\mu\nu\rho\sigma}G^{I}_{\mu\nu}G^{J}_{\rho\sigma}+\frac{\delta
g_{\alpha\bar{\beta}}}{\delta\bar{z}^{\bar{\gamma}}}\partial_{\lambda}\bar{z}^{\bar{\gamma}}\partial^{\lambda}\bar{z}^{\bar{\beta}}\\\
&+g_{\alpha\bar{\beta}}\nabla_{\lambda}\nabla^{\lambda}\bar{z}^{\bar{\beta}}-\frac{\delta
V}{\delta z^{\alpha}}=0\,,\end{split}$ (2.15)

which hold independently from the existence and the choice of a prepotential
$F(X)$.
Defining the tensor

$\displaystyle
G_{I\mu\nu}=R_{IJ}G^{I}_{\mu\nu}+I_{IJ}\tilde{G}_{\mu\nu}\quad,\quad\tilde{G}^{J}_{\mu\nu}=\frac{1}{2}\sqrt{-g}\epsilon_{\mu\nu\rho\sigma}G^{J\rho\sigma},$
(2.16)

then eq.(2.14), the Bianchi identities and the charges can be written as

$\displaystyle\epsilon^{\mu\nu\rho\sigma}\partial_{\mu}\begin{pmatrix}G^{I}_{\rho\sigma}\\\
G_{I\rho\sigma}\\\
\end{pmatrix}=0\quad,\quad\frac{1}{4\pi}\int_{\Sigma_{\infty}}\begin{pmatrix}G^{I}\\\
G_{I}\end{pmatrix}=\begin{pmatrix}p^{I}\\\ q_{I}\\\ \end{pmatrix}.$ (2.17)

We now specialize to the “magnetic” $STU$ model, for which the prepotential
can be written as

$\displaystyle F=-2i\sqrt{(X^{0}X^{1}X^{2}X^{3})}.$ (2.18)

The symplectic section can be parametrised in terms of three complex scalar
fields $z^{\alpha}=\tau_{\alpha}$, $\alpha=1,2,3$, so that

$\displaystyle v^{T}=e^{-\mathcal{K}/2}(X^{J},\partial F/\partial X^{J})$
(2.19)
$\displaystyle=(1,\tau_{2}\tau_{3},\tau_{1}\tau_{3},\tau_{1}\tau_{2},-i\tau_{1}\tau_{2}\tau_{3},-i\tau_{1},-i\tau_{2},-i\tau_{3}).$
(2.20)

$\tau_{\alpha}$ and $\bar{\tau}_{\bar{\alpha}}$ are the complex coordinates on
the scalar manifold. The Kähler potential and the non vanishing components of
the metric on the scalar manifold are respectively

$\displaystyle e^{-\mathcal{K}}$ $\displaystyle=8{\rm Re}\tau^{1}{\rm
Re}\tau^{2}{\rm Re}\tau^{3},$ (2.21) $\displaystyle g_{\alpha\bar{\alpha}}$
$\displaystyle=g_{\bar{\alpha}\alpha}=\partial_{\alpha}\partial_{\bar{\alpha}}\mathcal{K}=(\tau_{\alpha}+\bar{\tau}_{\bar{\alpha}})^{-2}.$
(2.22)

The scalars $z^{\alpha}$ are coupled to the gauge fields via the period matrix
$\mathcal{N}_{IJ}$ defined by

$\displaystyle F_{I}$ $\displaystyle=\mathcal{N}_{IJ}X^{J},$ (2.23)
$\displaystyle\partial_{\bar{\alpha}}\frac{\bar{F}}{\bar{X}^{I}}-\frac{1}{2}(\partial_{\bar{\alpha}}\mathcal{K})\frac{\bar{F}}{\bar{X}^{I}}$
$\displaystyle=\mathcal{N}_{IJ}\left(\partial_{\bar{\alpha}}\bar{X}^{J}-\frac{1}{2}(\partial_{\bar{\alpha}}\mathcal{K})\bar{X}^{J}\right).$
(2.24)

In looking for a static solution, we now propose the ansatz

$\displaystyle\tau_{1}$
$\displaystyle=\sqrt{\frac{g_{0}g_{1}}{g_{2}g_{3}}}\tau(r),\quad\tau_{2}=\sqrt{\frac{g_{0}g_{2}}{g_{1}g_{3}}}k_{2},\quad\tau_{3}=\sqrt{\frac{g_{0}g_{3}}{g_{1}g_{2}}}k_{3},$
(2.25)

where $k_{2}$ and $k_{3}$ are real constants. The $X^{I}$ components of the
symplectic section are

$\displaystyle
X^{I}=\frac{1}{8}\sqrt{\frac{G}{8}}\left(\begin{array}[]{c}\frac{1}{|g_{0}|}\frac{1}{\sqrt{{\rm
Re}\tau k_{2}k_{3}}}\\\ \frac{1}{|g_{1}|}\sqrt{\frac{k_{2}k_{3}}{{\rm
Re}\tau}}\\\ \frac{1}{|g_{2}|}\sqrt{\frac{k_{3}}{{\rm Re}\tau k_{2}}}\tau\\\
\frac{1}{|g_{3}|}\sqrt{\frac{k_{2}}{{\rm Re}\tau
k_{3}}}\tau\end{array}\right)$ (2.30)

where $G\equiv 64\sqrt{g_{0}g_{1}g_{2}g_{3}}$. Since
$\mathcal{N}_{IJ}=\frac{F}{2(X^{I})^{2}}\delta_{IJ}$, we obtain

$\displaystyle\mathcal{N}_{IJ}=-i\frac{64}{G}\begin{pmatrix}g_{0}^{2}k_{2}k_{3}\tau&0&0&0\\\
0&g_{1}^{2}\frac{\tau}{k_{2}k_{3}}&0&0\\\ 0&0&g_{2}^{2}\frac{k_{2}}{\tau
k_{3}}&0\\\ 0&0&0&g_{3}^{2}\frac{k_{3}}{\tau k_{2}}\\\ \end{pmatrix},$ (2.31)

the choices $k_{2}k_{3}>0$, together with Re$\tau>0$ imposed by Kähler
geometry, guarantee

$\displaystyle{\rm Im}(\mathcal{N}_{IJ})<0.$ (2.32)

For the potential we have

$\displaystyle V=$
$\displaystyle-\frac{G}{16}\left[\frac{g_{0}}{|g_{0}|}\frac{g_{1}}{|g_{1}|}\frac{1}{{\rm
Re}\tau}+\frac{g_{0}}{|g_{0}|}\frac{g_{2}}{|g_{2}|}\frac{1}{k_{2}}+\frac{g_{0}}{|g_{0}|}\frac{g_{3}}{|g_{3}|}\frac{1}{k_{3}}\right.$
(2.33)
$\displaystyle\left.+\frac{g_{1}}{|g_{1}|}\frac{g_{2}}{|g_{2}|}k_{3}+\frac{g_{1}}{|g_{1}|}\frac{g_{3}}{|g_{3}|}k_{2}+\frac{g_{2}}{|g_{2}|}\frac{g_{3}}{|g_{3}|}\frac{|\tau|^{2}}{{\rm
Re}\tau}\right].$ (2.34)

Now, from the equations of motion we get

$\displaystyle
R-4V=\frac{1}{2}g^{rr}\frac{(\partial_{r}\tau)(\partial_{r}\bar{\tau})}{({\rm
Re}\tau)^{2}}.$ (2.35)

Usually, at infinity one expects the scalars to approach a constant value, so
the r.h.s of eq.(2.35) vanishes there, and spacetime should have constant
curvature, $R_{\infty}=4V_{\infty}$, $V_{\infty}$ being determined by the
asymptotic value of $\tau$. For example, like in [5], or in [4] (see (4.42),
(4.43) and (4.44) therein), we can assume $g_{I}>0$. Assuming that at infinity
the fields reach the extremum of the potential (at which
$\tau=k_{2}=k_{3}=1$), we set $k_{2}=k_{3}=1$ and get

$\displaystyle\tau_{ex}=1,\qquad V_{ex}=-\frac{3G}{8}.$ (2.36)

for the minimum of the potential $V$. Thus, at infinity

$\displaystyle R_{ex}=-\frac{3}{2}G<0,$ (2.37)

so we expect an anti-de Sitter space. This is indeed what happens for the
static version of the solutions in [5] (see [4]), which interpolate between a
$AdS_{2}\times\Sigma$ $\frac{1}{2}$-supersymmetric geometry on the horizon,
and a full supersymmetric $AdS_{4}$. But things go very different for the new
solutions in [19], where $g_{0},g_{1}<0$ and $g_{2},g_{3}>0$. In that case

$\displaystyle V=\frac{G}{16}\left[-\frac{1+|\tau|^{2}}{{\rm
Re}\tau}+4\right],$ (2.38)

so, the minimum of the potential and the corresponding curvature are

$\displaystyle\tau$ $\displaystyle=1,\qquad k_{2}=1,\qquad k_{3}=1,$ (2.39)
$\displaystyle V_{ex}$ $\displaystyle=\frac{G}{8}\qquad R_{ex}=\frac{G}{2},$
(2.40)

leading then to a de Sitter space! To look for the explicit solution having as
near horizon limit the static solution in [19], we make the ansatz

$\displaystyle(ds)^{2}=-A(r)dt^{2}+A(r)^{-1}dr^{2}+B(r)d\Omega_{\kappa}^{2},$
(2.41) $\displaystyle
G^{I}_{tr}=\frac{(\text{Im}\,{\mathcal{N}})^{-1|IJ}}{B(r)}\left((\text{Re}\mathcal{N})_{JS}\,p^{S}-q_{J}\right).$
(2.42)

where $d\Omega_{k}^{2}=d\theta^{2}+f_{\kappa}^{2}(\theta)d\phi^{2}$ is the
metric on the two-dimensional surfaces
$\Sigma=\\{\mathrm{S}^{2},\mathrm{H}^{2}\\}$ of constant scalar curvature
$R=2\kappa$, with $\kappa=\\{1,-1\\}$, and

$f_{\kappa}(\theta)=\frac{1}{\sqrt{\kappa}}\sin(\sqrt{\kappa}\theta)=\left\\{\begin{array}[]{c@{\quad}l}\sin\theta&\kappa=1\,,\\\
\sinh\theta&\kappa=-1\,.\end{array}\right.$ (2.43)

The stress tensor for the spin-1 part ${}^{(1)}T_{\mu\nu}$ can be computed as

${}^{(1)}T^{t}_{\,t}=^{(1)}T^{r}_{\,r}=-^{(1)}T^{\theta}_{\,\theta}=-^{(1)}T^{\phi}_{\,\phi}=-\frac{1}{B^{2}}V_{BH}.$
(2.44)

One can also check that

$\frac{1}{4}\frac{\partial I_{IJ}}{\partial
z^{a}}G_{\mu\nu}^{I}G^{J|\,\mu\nu}-\frac{1}{8}\frac{\partial R_{IJ}}{\partial
z^{a}}G_{\mu\nu}^{I}{{}^{\ast}{G}}^{J|\,\mu\nu}=-\frac{1}{B^{2}}\frac{\partial
V_{BH}}{\partial z^{a}},$ (2.45)

where $\mathcal{N}_{IJ}\equiv R_{IJ}+iI_{IJ}$. We define the so-called Black
Hole potential( BH potential)

$V_{BH}\equiv-\frac{1}{2}(p^{I},\,q_{I})\left(\begin{array}[]{cc}I_{IJ}+R_{IS}(I)^{-1|\,ST}R_{TJ}&\;-R_{IS}(I)^{-1|\,SJ}\\\
-(I)^{-1|\,IS}R_{SJ}&\;(I)^{-1|\,IJ}\end{array}\right)\left(\begin{array}[]{c}p^{J}\\\
q_{J}\end{array}\right).$ (2.46)

The field Maxwell and Bianchi field equations (2.17) are satisfied, while the
Einstein’s equations of motion (2.13) and the scalar field equation

$\displaystyle R_{tt}$
$\displaystyle=\frac{A}{2B}\left(A^{\prime\prime}B+A^{\prime}B^{\prime}\right)=\frac{A}{B^{2}}V_{BH}-AV,$
(2.47) $\displaystyle R_{rr}$
$\displaystyle=-\frac{1}{2AB^{2}}\left(A^{\prime\prime}B^{2}+A^{\prime}B^{\prime}B+2AB^{\prime\prime}B-AB^{\prime
2}\right)=-\frac{1}{AB^{2}}V_{BH}+\frac{1}{A}V+\frac{\tau^{\prime}{\bar{\tau}^{\prime}}}{2(Re\tau)^{2}},$
(2.48) $\displaystyle R_{\theta\theta}$
$\displaystyle=-\frac{1}{2}\left(A^{\prime}B^{\prime}+AB^{\prime\prime}\right)+\kappa=\frac{1}{B}V_{BH}+BV,$
(2.49) $\displaystyle 0$
$\displaystyle=\frac{1}{B^{2}}\partial_{\tau}V_{BH}+\partial_{\tau}V-\frac{1}{B}\frac{\left(BA\bar{\tau}^{\prime}\right)^{\prime}}{4(Re\tau)^{2}}+A\frac{({\bar{\tau}}^{\prime})^{2}}{4(Re\tau)^{3}}.$
(2.50)

Summing and subtracting eq.(2.47) and (2.48) we can rewrite the system as

$\displaystyle\frac{\left(2B^{\prime\prime}B-B^{\prime 2}\right)}{B^{2}}$
$\displaystyle=4\frac{(\sqrt{B})^{\prime\prime}}{\sqrt{B}}=-\frac{\tau^{\prime}{\bar{\tau}^{\prime}}}{(Re\tau)^{2}},$
(2.51) $\displaystyle
A^{\prime}B^{\prime}+A^{\prime\prime}B=(A^{\prime}B)^{\prime}$
$\displaystyle=2\frac{V_{BH}}{B}-2BV,$ (2.52)
$\displaystyle-\frac{1}{2}\left(A^{\prime}B^{\prime}+AB^{\prime\prime}\right)+\kappa=-\frac{1}{2}\left(AB^{\prime}\right)^{\prime}+\kappa$
$\displaystyle=\frac{V_{BH}}{B}+BV,$ (2.53) $\displaystyle 0$
$\displaystyle=\frac{1}{B^{2}}\partial_{\tau}V_{BH}+\partial_{\tau}V-\frac{1}{B}\frac{\left(BA\bar{\tau}^{\prime}\right)^{\prime}}{4(Re\tau)^{2}}+A\frac{({\bar{\tau}}^{\prime})^{2}}{4(Re\tau)^{3}}.$
(2.54)

The system can be rewritten as

$\displaystyle\frac{\left(2B^{\prime\prime}B-B^{\prime
2}\right)}{B^{2}}=-\frac{\tau^{\prime}{\bar{\tau}^{\prime}}}{(Re\tau)^{2}},$
(2.55) $\displaystyle
A^{\prime\prime}B-AB^{\prime\prime}=4\frac{V_{BH}}{B}-2\kappa,$ (2.56)
$\displaystyle(AB)^{\prime\prime}=-4BV+2\kappa,$ (2.57) $\displaystyle
0=\frac{1}{B^{2}}\partial_{\tau}V_{BH}+\partial_{\tau}V-\frac{1}{B}\frac{\left(BA\bar{\tau}^{\prime}\right)^{\prime}}{4(Re\tau)^{2}}+A\frac{({\bar{\tau}}^{\prime})^{2}}{4(Re\tau)^{3}}.$
(2.58)

For the solution in [19], which we want to extend, we have

$\displaystyle p^{I}=\frac{1}{8g^{I}},\qquad q_{I}=0,\qquad
V_{BH}=\frac{1}{G}\left(\frac{1+|\tau|^{2}}{Re\tau}\right),\qquad\partial_{\tau}V_{BH}=\frac{1}{2G}\left(\frac{\bar{\tau}^{2}-1}{(Re\tau)^{2}}\right),\qquad\kappa=1,$
(2.59)

so we specify the equations to this case. Eq.(2.58) becomes

$\displaystyle
0=\frac{1}{2G}\left[\bar{\tau}^{2}-1\right]-B^{2}\frac{G}{32}\left[\bar{\tau}^{2}-1\right]-B\frac{\left(AB\bar{\tau}^{\prime}\right)^{\prime}}{4}+AB^{2}\frac{({\bar{\tau}}^{\prime})^{2}}{4(Re\tau)},$
(2.60)

and using $\tau=f(r)+ig(r)$, we get

$\displaystyle-\frac{1}{4}B\left(BAf^{\prime}\right)^{\prime}+\frac{AB^{2}\left(f^{\prime
2}-g^{\prime
2}\right)}{4f}+\left(\frac{1}{2G}-\frac{1}{32}GB^{2}\right)\left(f^{2}-g^{2}-1\right)=0,$
(2.61)
$\displaystyle\frac{B}{4}\left(BA^{\prime}g^{\prime}\right)^{\prime}-\frac{AB^{2}f^{\prime}g^{\prime}}{2f}-2\left(\frac{1}{2G}-\frac{1}{32}GB^{2}\right)fg=0.$
(2.62)

After imposing

$\displaystyle f(r)=\sqrt{1+g(r)(g_{H}-g(r))},$ (2.63)

where $g_{H}$ is the value at the horizon, eq.(2.61) and (2.62) becomes
linearly dependent.

##  3\. The solution: Black hole dipole

Let us now present the full solution. The Fayet-Iliopoulos potential is given
by (2.38) and the electromagnetic fields and their duals are

$\displaystyle G^{I}=$ $\displaystyle\frac{1}{8B}\frac{{\rm Im}\tau}{{\rm
Re}\tau}\begin{pmatrix}-g_{0}^{-1}\\\ -g_{1}^{-1}\\\ g_{2}^{-1}\\\
g_{3}^{-1}\end{pmatrix}dt\wedge
dr+\frac{\sin{\theta}}{8}\begin{pmatrix}g_{0}^{-1}\\\ g_{1}^{-1}\\\
g_{2}^{-1}\\\ g_{3}^{-1}\end{pmatrix}d\theta\wedge d\phi,$ (3.1)
$\displaystyle G_{I}$ $\displaystyle=-\frac{8B^{-1}}{G{\rm
Re}\tau}\begin{pmatrix}|\tau|^{2}g_{0}\\\ |\tau|^{2}g_{1}\\\ g_{2}\\\
g_{3}\end{pmatrix}dt\wedge dr.$ (3.2)

The Maxwell equations and Bianchi identities for the gauge fields are already
satisfied, while the Einstein’s equations and the equations of motion for the
scalars give the general solution

$\displaystyle ds^{2}$
$\displaystyle=-\frac{G}{c^{2}}\frac{(1-c^{2}r^{2})^{2}}{2(1-c^{2}r^{2})+Gr_{A}^{2}}dt^{2}+\frac{4c^{2}}{G}\frac{2(1-c^{2}r^{2})+Gr_{A}^{2}}{(1-c^{2}r^{2})^{2}}dr^{2}$
(3.3)
$\displaystyle\quad+4\frac{2(1-c^{2}r^{2})+Gr_{A}^{2}}{G^{2}r_{A}^{2}}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right),$
(3.4) $\displaystyle\tau_{1}$
$\displaystyle=\sqrt{\frac{g_{0}g_{1}}{g_{2}g_{3}}}\frac{Gr_{A}^{2}+2(1-c^{2}r^{2})-i2\sqrt{2G}r_{A}cr}{2(1+cr)^{2}+Gr_{A}^{2}},\qquad\tau_{2}=\sqrt{\frac{g_{0}g_{2}}{g_{1}g_{3}}},\qquad\tau_{3}=\sqrt{\frac{g_{0}g_{3}}{g_{1}g_{3}}},$
(3.5) $\displaystyle G^{I}$
$\displaystyle=\frac{G^{5/2}r_{A}^{3}}{\sqrt{2}\,8}\frac{cr}{\left(2(1-c^{2}r^{2})+Gr_{A}^{2}\right)^{2}}\begin{pmatrix}g_{0}^{-1}\\\
g_{1}^{-1}\\\ -g_{2}^{-1}\\\ -g_{3}^{-1}\end{pmatrix}dt\wedge
dr+\frac{\sin{\theta}}{8}\begin{pmatrix}g_{0}^{-1}\\\ g_{1}^{-1}\\\
g_{2}^{-1}\\\ g_{3}^{-1}\end{pmatrix}d\theta\wedge d\phi,$ (3.6)

where $c$ is a constant assumed positive, and $r$ ranges in

$\displaystyle
r_{\leftarrow}\equiv-\frac{1}{c}\sqrt{\frac{2+Gr_{A}^{2}}{2}}<r<\frac{1}{c}\sqrt{\frac{2+Gr_{A}^{2}}{2}}\equiv
r_{\rightarrow}.$ (3.7)

Notice that we rescaled the time coordinate $t\to t/2$ in order to get a more
direct comparison with the near horizon limit in [19]. The curvature is
everywhere regular with the exception of these two points, where it diverges
at $+\infty$. The spatial section (at $\theta=\frac{\pi}{2}$) is therefore not
embeddable in $\mathbb{R}^{3}$, but we can roughly think at it as a sort of
rugby ball with the singular points at the tips, where also the Kähler
geometry becomes singular, since Re $\tau$ goes to zero there. The components
$g_{tt}=-4g^{rr}$ have two double roots at

$\displaystyle r_{\pm}=\pm\frac{1}{c},$ (3.8)

which represent two extremal event horizons.

Figure 1. Representation at $t,\phi$ constants of the geometry of the “Black
hole dipole” with parameters $G=2,c=0.1,r_{A}=1$. The circles are the lines of
constant $r$ and the longitudinal lines are the magnetic field lines. The
horizons are at $r_{\pm}=\pm 10$ and are represented in blue, while the
singularities are the tips at $r_{s\pm}=\pm 10\sqrt{2}$. Here
$f(r,\theta)=g_{\theta\theta}\cos{\theta}$ and
$g(r,\theta)=g_{\theta\theta}\sin{\theta}$.

Both the horizons share the same geometry, since the limit at
$r=\\{r_{-},r_{+}\\}$ gives

$\displaystyle
ds^{2}=-4\frac{r^{2}}{r^{2}_{A}}dt^{2}+\frac{r_{A}^{2}}{r^{2}}dr^{2}+\frac{4}{G}d\Omega^{2},$
(3.9)

which is exactly the static near-horizon solution found in [19], after
identifying

$\displaystyle r_{A}=\sqrt{\frac{2}{G}}\frac{a}{\alpha}.$ (3.10)

In particular, the horizons have the topology of $AdS_{2}\times S^{2}$, with
radii $|r_{A}|$ and $r_{H}=2/\sqrt{G}$ respectively. Therefore, we have two
black holes facing each other in a symmetric way, see Figure 1. However, this
symmetry under the parity transformation

$\displaystyle r\rightarrow-r,$ (3.11)

is not shared by the scalar $\tau_{1}$ and only up to a sign by the
electromagnetic field, see Figure 2. The fluxes have opposite signs, which
means that the two black holes have opposite (magnetic) charges

$\displaystyle p^{I}_{\mp}=\pm\frac{1}{8g_{I}},\qquad q^{\mp}_{I}=0.$ (3.12)

Thus, they attract each other both gravitationally and electromagnetically.

Figure 2. Electric field interpreted as the $t,r$ component of $G^{I}$ for
$G=2,c=0.1,r_{A}=1,g_{1}<0.$

Why then they remain in static equilibrium? This can be understood noticing
that the extremum (2.40) of the potential is reached at $r=0$, and analyzing
the metric around such region one checks that it approximate a de Sitter
metric with cosmological constant $\Lambda=\frac{G}{8}$, and a magnetic flux
$Q=\sqrt{\frac{2}{G}}$, see Figure 3.

Figure 3. Ricci scalar curvature for $G=2,c=0.1,r_{A}=1$.

It is interesting to compare the values of the non constant scalar at the
horizons. One gets

$\displaystyle\tau_{1}^{+}$
$\displaystyle=\frac{Gr_{A}^{2}}{8+Gr_{A}^{2}}\bar{\tau}_{1}^{-},\quad\tau_{1}^{-}=\sqrt{\frac{g_{0}g_{1}}{g_{2}g_{3}}}\left(1+2i\frac{\alpha}{a}\right),$
(3.13)

see Figure 4. In particular, $\tau_{1}^{-}$ is exactly the scalar field of the
supersymmetric solution in [19] and we see that the horizon $r=r_{-}$ is
$\frac{1}{2}$-BPS supersymmetric. Instead, the limit geometry at $r=r_{+}$
does not solve the BPS equations and is not supersymmetric.

Figure 4. Real (blue line) and imaginary part (orange line) of the scalar
field with parameters $G=2,c=0.1,r_{A}=1.$

However, since the spacetime geometry is the same, the two black hole have the
same entropy, which inherits some properties of supersymmetric black holes.
Indeed,

$\displaystyle S_{\pm}=\frac{4\pi}{G}=4\pi\sqrt{\prod_{I=1}^{4}p^{I}_{\pm}},$
(3.14)

which is independent on the values of the scalar at the horizons, which depend
on the real modulus $\alpha/a$.
The metric has four Killing vector fields, generating the four dimensional
isometry group of rotations and time translations. This group is enhanced to
$SO(2,1)\times SO(3)$ at the horizons. Since the solution is a closed
universe, with no asymptotic regions, the problem of correctly assigning a
value to the masses of the black holes is a nontrivial task that needs a
rather lengthy analysis, and that we demand to a separated work. However, it
is worth to anticipate a couple of comments. First, from (3.2) one easily gets
the total electromagnetic energy separating the two horizons. For
$G_{I}\equiv-d(\Upsilon_{I}\ dt)$ and
$\Delta\Upsilon_{I}=\Upsilon_{I}(r_{+})-\Upsilon_{I}(r_{-})$, one gets

$\displaystyle\Delta\mathcal{E}_{\Upsilon}=\frac{1}{2}\sum_{I}\Delta\Upsilon_{I}p^{I}=\frac{1}{c}.$
(3.15)

This gives a physical meaning to $c$, as a measure of the magnetostatic energy
between the two black holes. Notice that it looks like a sort of (magnetic)
capacitor whose walls are the horizons. The same value of the electromagnetic
energy can be found directly by integrating $-^{(1)}T_{t}^{t}$ on the spatial
region between the two horizons. Second, despite we recognized a
supersymmetric like characterization of the entropy, we expect a nontrivial
contribution of the scalars to the first law of black holes thermodynamics
[34]. The computation of this contribution, however, is not trivial and
requires an improved version of the method in [37] [38].
For extremal solutions the attractor mechanism implies for the scalars to be
attracted toward the configuration in which, see [39],

$\displaystyle\partial_{\tau}V_{E}|_{(r_{H})}=0,$ (3.16)

where $V_{E}$ is the effective potential,

$\displaystyle V_{E}=(1-\sqrt{1-4V_{BH}V})/(2V).$ (3.17)

Remarkably, a direct check shows that in our case this is what happens at both
horizons, and the entropy takes exactly the form

$\displaystyle S=\pi V_{E}(\tau_{rH})=\frac{4\pi}{G}.$ (3.18)

It suggests that our solution behaves like a flow connecting a supersymmetric
solution to a non supersymmetric one. Making this flow more explicit, combined
with AdS2/CFT1, would allow us to dualize a non-supersymmetric conformal field
theory CFT${}^{ns}_{1}$ to a supersymmetric conformal field theory
CFT${}^{s}_{1}$.

###  3.1. Comments on the mass and conserved quantities

Since the dipole solution is closed and does not contain any asymptotically
flat region, finding a good definition of mass for the contained black hoes is
a quite hard problem.
A first idea to tackle this problem could be to notice that the region between
the two horizons behaves as the throat of a wormhole. Using cut and sewing
techniques one could get a true wormhole solution, as looked for in [40]. In
our case we do not need to add a massless Dirac field, it is the stationary
point of the potential that allows to sustain the wormhole. This
interpretation can allow us to infer the mass of the two holes composing the
wormhole. In that case one could be tempted to define the mass of each hole,
as seen by an “external” observer, as the quantity measured by
$M=\sqrt{V}_{BH}|_{Hor}$, the black hole potential at the horizon. Using the
explicit values it would give

$\displaystyle M=\frac{r_{H}}{r_{A}}\sqrt{\frac{r_{H}^{2}+r_{A}^{2}}{2}}.$
(3.19)

However, such a proposal is too naive and no further hint can help us in
confirming such a result. Indeed, formula (3.19) for the mass is usually true
for solutions that admit an asymptotically flat or maximally symmetric
solution, which is what would give rise a meaning to the word “external”
referred to the observer. But the proposed mass formula does not result to
satisfy any kind of Smarr relation, for example. Interestingly, the dipole
solution has been found by us while looking for asymptotically flat or AdS
solutions having the near horizon limit described in [19], but we failed in
finding such a solution. So it is not at all evident that the above cut and
glue operation can be really done in order to provide a meaning to our
proposal.
One could try to define masses by looking at Komar integrals. Again, we have
not an asymptotically flat region so that the usual results cannot be directly
applied. Nevertheless, we could try with the following reasoning:
Let us review shortly the case of an asymptotically flat, for simplicity
static, black hole, and let $K$ a timelike Killing vector field such that in
the flat region, say $S_{\infty}$, $K^{\mu}K_{\mu}=-1$. Then, the mass of the
black hole is

$\displaystyle M=2\int_{S_{\infty}}\nabla^{\mu}K^{\nu}dS_{\mu\nu}.$ (3.20)

On the other hand, using that $\nabla_{\nu}K^{\nu}=0$ and
$[\nabla_{\nu},\nabla^{\mu}]K^{\nu}=R^{\mu}_{\nu}K^{\nu}$, assuming that the
black hole has an external event horizon at $S_{+}$, we can write

$\displaystyle
M=2\int_{S_{+}}\nabla^{\mu}K^{\nu}dS_{\mu\nu}+\int_{\Sigma}(2T^{t}_{\mu}-\delta^{t}_{\mu}T)d\Sigma_{t},$
(3.21)

where $\Sigma$ is the spatial region external to the black hole. This formula
is used in [43] to interpret the total mass of the black hole as the sum of
the pure black hole mass

$\displaystyle M_{BH}=2\int_{S_{+}}\nabla^{\mu}K^{\nu}dS_{\mu\nu}$ (3.22)

plus the mass contribution from the matter outside the black hole

$\displaystyle
M_{matt}=\int_{\Sigma}(2T^{t}_{\mu}-\delta^{t}_{\mu}T)d\Sigma_{t}.$ (3.23)

For example, for a Reissner-Nordstrom black hole one would get111The $8\pi$
factor here arises from the convention $G_{N}=1/8\pi$ in place of $G_{N}=1$ as
usual for the Newton constant.

$\displaystyle M_{BH}=8\pi\sqrt{M^{2}-Q^{2}},\qquad
M_{matt}=8\pi(M-\sqrt{M^{2}-Q^{2}}).$ (3.24)

Notice that it is the total mass that enters the first law of thermodynamics.
For an extremal black hole $M_{BH}=0$ and the total mass is completely given
by the matter contribution. This may suggest us that, since our horizons are
both extremal, the total mass of each black hole could be given by the matter
mass contribution of each half of the dipole external region (from $r_{-}$ to
$r=0$ for the left horizon and from $r=0$ to $r_{+}$ for the right horizon).
Indeed, a direct computation shows that for each of the horizons of the dipole

$\displaystyle\int_{S_{\pm}}\nabla^{\mu}K^{\nu}dS_{\mu\nu}=0.$ (3.25)

Unfortunately, also

$\displaystyle\int_{\\{r=0\\}}\nabla^{\mu}K^{\nu}dS_{\mu\nu}=0,\qquad\int_{r_{-}<r<0}(2T^{t}_{\mu}-\delta^{t}_{\mu}T)d\Sigma_{t}=\int_{0<r<r_{+}}(2T^{t}_{\mu}-\delta^{t}_{\mu}T)d\Sigma_{t}=0,$

so that this strategy is not able to give us any reasonable non vanishing mass
for the black holes in the dipole.
The only interesting quantities we can associate to the dipole solution are
the conserved ones associated to the Killing vectors describing the symmetries
of the solutions. We have a timelike Killing vector $K_{(0)}=\partial_{t}$,
associated to staticity, and the generators of the spherical symmetry. A
direct computation show immediately that rotations give rise to vanishing
integrals, so that the only non trivial conserved quantity is the total
energy222Here we have included a factor $1/8\pi$ to easily compare the results
with the expressions given in standard units (for which the Newton constant is
1)

$\displaystyle E:=$
$\displaystyle-\frac{1}{8\pi}\int_{\Sigma}T^{t}_{t}d\Sigma_{t}=-\frac{1}{8\pi}\int_{\Sigma}{}^{(0)}T^{t}_{t}d\Sigma_{t}-\frac{1}{8\pi}\int_{\Sigma}{}^{(1)}T^{t}_{t}d\Sigma_{t}+\frac{1}{8\pi}\int_{\Sigma}Vd\Sigma_{t}$
(3.26) $\displaystyle=:$ $\displaystyle E^{(0)}+E^{(1)}+E_{P}.$ (3.27)

Here $E^{(0)}$ is the kinetic energy of the scalar fields, $E^{(1)}$
represents the electromagnetic energy, and $E_{P}$ is the potential energy of
the scalars. Explicitly:

$\displaystyle E^{(0)}=$
$\displaystyle\frac{\left[4\sqrt{Gr_{A}^{2}+2}\left(7Gr_{A}^{2}+4\right)+\sqrt{2}Gr_{A}^{2}\left(3Gr_{A}^{2}+8\right)\log\left(\frac{Gr_{A}^{2}+4-2\sqrt{2Gr_{A}^{2}+4}}{Gr_{A}^{2}}\right)\right]}{4cGr_{A}^{2}\sqrt{Gr_{A}^{2}+2}}$
(3.28) $\displaystyle E^{(1)}=$ $\displaystyle\frac{1}{c},$ (3.29)
$\displaystyle E_{P}=$ $\displaystyle E^{(1)}=\frac{1}{c}.$ (3.30)

###  3.2. Geodesic motions

Let us now study the geodesic equations in the black dipole background:

$\displaystyle\ddot{r}$
$\displaystyle-\frac{G^{2}r}{2c^{2}}\frac{(1-c^{2}r^{2})(1-c^{2}r^{2}+Gr_{A}^{2})}{(2(1-c^{2}r^{2})+Gr_{A}^{2})^{3}}\dot{t}^{2}+2c^{2}r\left(\frac{1}{1-c^{2}r^{2}}-\frac{1}{2(1-c^{2}r^{2})+Gr_{A}^{2}}\right)\dot{r}^{2}+$
$\displaystyle+\frac{2r}{Gr_{A}^{2}}\frac{(1-c^{2}r^{2})^{2}}{2(1-c^{2}r^{2})+Gr_{A}^{2}}\left(\dot{\theta}^{2}+\sin(\theta)^{2}\dot{\phi}^{2}\right)=0,$
(3.31) $\displaystyle\ddot{t}$
$\displaystyle+4c^{2}r\left(\frac{1}{1-c^{2}r^{2}}+\frac{1}{2(1-c^{2}r^{2})+Gr_{A}^{2}}\right)\dot{t}\,\dot{r}=0,$
(3.32) $\displaystyle\ddot{\theta}$
$\displaystyle-\frac{4c^{2}r}{2(1-c^{2}r^{2})+Gr_{A}^{2}}\dot{\theta}\,\dot{r}-\cos(\theta)\sin(\theta)\dot{\phi}^{2}=0,$
(3.33) $\displaystyle\ddot{\phi}$
$\displaystyle-\frac{4c^{2}r}{2(1-c^{2}r^{2})+Gr_{A}^{2}}\dot{\phi}\,\dot{r}+\frac{2}{\tan(\theta)}\dot{\theta}\dot{\phi}=0,$
(3.34)

where $\dot{f}$ is the derivative respect to the affine parameter $\tau$. In
the middle at $r=0$ there is a geodesics

$\displaystyle\theta,\phi=\text{constant},t=\tau,$ (3.35)

which is an unstable equilibrium configuration. From the second and the fourth
equations we directly get two constants of motion

$\displaystyle
E=\dot{t}\frac{(1-c^{2}r^{2})^{2}}{2(1-c^{2}r^{2})+Gr_{A}^{2}},$ (3.36)
$\displaystyle
L_{z}=\dot{\phi}\sin^{2}\theta\left[2(1-c^{2}r^{2})+Gr_{A}^{2}\right],$ (3.37)

which can be interpreted as the energy (eventually per unit mass) and the $z$
component of the angular momentum.
After multiplying (3.33) times $2\dot{\theta}(2(1-c^{2}r^{2})+Gr_{A}^{2})^{2}$
and (3.34) times $2\dot{\phi}\sin^{2}\theta(2(1-c^{2}r^{2})+Gr_{A}^{2})^{2}$
and summing up the results we get

$\displaystyle\frac{d}{dt}[(\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2})(2(1-c^{2}r^{2})+Gr_{A}^{2})^{2}]=0$
(3.38)

so we get the constant of motion

$\displaystyle
L^{2}=(\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2})(2(1-c^{2}r^{2})+Gr_{A}^{2})^{2},$
(3.39)

representing the square modulus of the total angular momentum. Replacing from
this last equation into (3.31) we get

$\displaystyle\ddot{r}$
$\displaystyle-\frac{G^{2}E^{2}r}{2c^{2}}\frac{1-c^{2}r^{2}+Gr_{A}^{2}}{(1-c^{2}r^{2})\left[2(1-c^{2}r^{2})+Gr_{A}^{2}\right]}+\frac{2r}{Gr_{A}^{2}}L^{2}\frac{(1-c^{2}r^{2})^{2}}{\left[2\left(1-c^{2}r^{2}\right)+Gr_{A}^{2}\right]^{3}}+$
$\displaystyle+2c^{2}r\dot{r}^{2}\frac{1-c^{2}r^{2}+Gr_{A}^{2}}{(1-c^{2}r^{2})\left[2(1-c^{2}r^{2})+Gr_{A}^{2}\right]}=0.$
(3.40)

This can be integrated giving a fourth integral of motion

$\displaystyle-\frac{G}{c^{2}}\frac{(1-c^{2}r^{2})^{2}}{2(1-c^{2}r^{2})+Gr_{A}^{2}}\dot{t}^{2}+\frac{4c^{2}}{G}\frac{2(1-c^{2}r^{2})+Gr_{A}^{2}}{(1-c^{2}r^{2})^{2}}\dot{r}^{2}$
(3.41)
$\displaystyle+\frac{4}{G^{2}r_{A}^{2}}\left[2(1-c^{2}r^{2})+Gr_{A}^{2}\right](\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2})=-\epsilon,$
(3.42)

where $\epsilon$ is 0 for massless particles and 1 in the massive case. This
is of course the relation $g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-\epsilon$.
After substituting from the two integrals of motion (3.36) and (3.39) we get

$\displaystyle-\frac{G}{c^{2}}E^{2}\frac{2(1-c^{2}r^{2})+Gr_{A}^{2}}{(1-c^{2}r^{2})^{2}}+\frac{4c^{2}}{G}\frac{2(1-c^{2}r^{2})+Gr_{A}^{2}}{(1-c^{2}r^{2})^{2}}\dot{r}^{2}+\frac{4}{G^{2}r_{A}^{2}}\frac{L^{2}}{2(1-c^{2}r^{2})+Gr_{A}^{2}}=-\epsilon.$
(3.43)

This equation can be equivalently rewritten as

$\displaystyle\frac{1}{2}\dot{r}^{2}+\tilde{V}(r)=\tilde{E},$ (3.44)
$\displaystyle\tilde{V}(r)=\frac{(1-c^{2}r^{2})^{2}}{2(1-c^{2}r^{2})+Gr_{A}^{2}}\left[\frac{L^{2}}{2c^{2}Gr_{A}^{2}}\frac{1}{2(1-c^{2}r^{2})+Gr_{A}^{2}}+\frac{G}{8c^{2}}\epsilon\right],$
(3.45) $\displaystyle\tilde{E}=\frac{G^{2}}{8c^{4}}E^{2}.$ (3.46)

Figure 5. $\tilde{V}$ with parameters $G=2,c=0.1,r_{A}=1,M=1,\epsilon=0.$

Thus, the problem of geodesic motions is completely reduced to quadratures.
For circular orbits we impose $r=r_{c}\neq 0$ in (3.40) and obtain

$\displaystyle
L^{2}=\frac{G^{3}r_{A}^{2}E^{2}}{4c^{2}}\frac{\left[2(1-c^{2}r_{c}^{2})+Gr_{A}^{2}\right]^{2}\left(1-c^{2}r_{c}^{2}+Gr_{A}^{2}\right)}{(1-c^{2}r_{c}^{2})^{3}},$
(3.47)

and if we then trade $E$ for $\epsilon$ using eq.(3.42) we get

$\displaystyle\epsilon=\frac{-4L^{2}}{G}\frac{1}{\left[2(1-c^{2}r_{c}^{2})+Gr_{A}^{2}\right]\left(1-c^{2}r_{c}^{2}+Gr_{A}^{2}\right)},$
(3.48)

which is not compatible with $\epsilon\geq 0$, showing the non existence of
circular orbits for $r\neq 0$, as one can expect from the form of
$\tilde{V}(r)$ in fig.5. The conservation of the angular momentum implies that
motions are planar. Indeed, let us set
$\rho^{2}(r)\equiv\frac{4}{G^{2}r_{A}^{2}}2(1-c^{2}r_{c}^{2})+Gr_{A}^{2}$ and
let $u_{r}$, $u_{\theta}$, $u_{\phi}$ the spatial vierbein vectors so that the
spatial position of a test particle is given by the vector

$\displaystyle\vec{x}=\rho u_{r}.$ (3.49)

The spatial velocity, with respect to a parameter $\tau$, is
$\vec{v}=\dot{\rho}u_{r}+\rho\dot{u}_{r}$. If we define the vector

$\displaystyle\vec{\Lambda}=\vec{x}\times\vec{v},$ (3.50)

we get immediately that

$\displaystyle\vec{\Lambda}\cdot\vec{\Lambda}=\frac{16}{G^{4}r_{A}^{4}}L^{2}.$
(3.51)

Moreover, let us define

$\displaystyle u_{z}=\cos\theta u_{r}-\sin\theta u_{\theta}.$ (3.52)

Then, it easily follows that

$\displaystyle\Lambda_{z}:=\vec{\Lambda}\cdot
u_{z}=\frac{4}{G^{2}r_{A}^{2}}L_{z}.$ (3.53)

We can then get an idea of the orbits by looking at the equations of motion on
the plane $\theta=\pi/2$. In this case we easily get

$\displaystyle\phi(r)-\phi_{0}=\int\frac{dr}{\sqrt{\frac{2\tilde{E}}{L^{2}}\left[2(1-c^{2}r^{2})+Gr_{A}^{2}\right]^{2}-\frac{(1-c^{2}r^{2})^{2}}{c^{2}Gr_{A}^{2}}+\frac{G}{4c^{2}L^{2}}(1-c^{2}r^{2})(2(1-c^{2}r^{2})+Gr_{A}^{2})\epsilon}},$
(3.54)

where $\epsilon=1$ for massive particles and $\epsilon=0$ for massless
particles. This can be explicitly expressed in terms of elliptic function, but
the explicit expressions would not give us major hints.
Notice that, as happens for Reissner-Nordstrom black holes, the singularities
are repulsive. No geodesics fall in the singular region, but they are expelled
from the future white hole.

###  3.3. Causal structure

We want now to analyze the causal structure of the dipole black hole manifold.
Let us define new coordinates such that the $(t,r)$ part of the metric reads

$\displaystyle
ds^{2}_{|t,r}=-\frac{G}{4c^{2}}\frac{(1-c^{2}r^{2})^{2}}{2(1-c^{2}r^{2})+Gr_{A}^{2}}\left(-dt^{2}+dr_{*}^{2}\right),$
(3.55)

where, compared to the metric $(\ref{FS})$, we defined a new time
$t\rightarrow t/2$ and a new coordinate tortoise $r_{*}$

$\displaystyle
r_{*}=\frac{2c^{2}r_{A}^{2}r}{1-c^{2}r^{2}}+c\left(\frac{4}{G}+r_{A}^{2}\right)\log\left(\left|\frac{1+cr}{1-cr}\right|\right),$
(3.56)

which takes values in $(-\infty,+\infty)$ as $r\in(-\frac{1}{c},\frac{1}{c})$.
While the metric $(\ref{FS})$ was well defined in the range
$-\frac{1}{c}<r<\frac{1}{c}$, if we introduce the radial null coordinates as

$\displaystyle v=t+r_{*},\qquad u=t-r_{*}.$ (3.57)

the metric in the new coordinates $(v,r,\theta,\phi)$

$\displaystyle
ds^{2}=-\frac{G}{4c^{2}}\frac{(1-c^{2}r^{2})^{2}}{2(1-c^{2}r^{2})+Gr_{A}^{2}}dv^{2}+2dvdr+\frac{4}{G^{2}r_{A}^{2}}d\Omega^{2},$
(3.58)

is smooth until the singularities $r_{s\pm}=\pm
c\sqrt{\frac{2+Gr_{A}^{2}}{2}}$. To study the causal structure we introduce a
first set of Kruskal coordinates

$\displaystyle-e^{-\frac{u}{2c\left(\frac{4}{G}+r_{A}^{2}\right)}}\,:=$
$\displaystyle\begin{cases}U^{-}_{out}&\text{if}\quad
r\in(-\frac{1}{c},\frac{1}{c})\\\ U^{+}_{in}&\text{if}\quad
r\in(\frac{1}{c},r_{s+})\end{cases},$ (3.59) $\displaystyle
e^{\frac{v}{2c\left(\frac{4}{G}+r_{A}^{2}\right)}}\,:=$
$\displaystyle\begin{cases}V^{-}_{out}&\text{if}\quad
r\in(-\frac{1}{c},\frac{1}{c})\\\ V^{+}_{in}&\text{if}\quad
r\in(\frac{1}{c},r_{s+})\end{cases}.$ (3.60)

The products of these functions is

$\displaystyle
UV=-e^{\frac{cr_{A}^{2}}{\left(\frac{4}{G}+r_{A}^{2}\right)}\frac{r}{1-c^{2}r^{2}}}\left(\frac{1+cr}{1-cr}\right),$
(3.61)

with positivity defined by

$\displaystyle\frac{1+cr}{1-cr}>0\rightarrow\quad
r\in(-\frac{1}{c},\frac{1}{c}).$ (3.62)

Even though the Kruskal coordinates are defined for $U<0$ and $V>0$ we can
extend them to $U,V\in\mathbb{R}$ as usual, thinking $r=r(U,V)$. We have then
4 regions, two of them covered by the coordinates $(U^{-}_{out},V^{-}_{out})$
and $(U^{+}_{in},V^{+}_{in})$ respectively. The remaining two regions are
covered by $(U^{-}_{in},V^{-}_{in})$ and $(U^{+}_{out},V^{+}_{out})$ defined
as

$\displaystyle e^{-\frac{v}{2c\left(\frac{4}{G}+r_{A}^{2}\right)}}\,:=$
$\displaystyle\begin{cases}U^{-}_{in}&\text{if}\quad
r\in(-\frac{1}{c},\frac{1}{c})\\\ U^{+}_{out}&\text{if}\quad
r\in(\frac{1}{c},r_{s+})\end{cases},$ (3.63)
$\displaystyle-e^{\frac{u}{2c\left(\frac{4}{G}+r_{A}^{2}\right)}}\,:=$
$\displaystyle\begin{cases}V^{-}_{in}&\text{if}\quad
r\in(-\frac{1}{c},\frac{1}{c})\\\ V^{+}_{out}&\text{if}\quad
r\in(\frac{1}{c},r_{s+})\end{cases}.$ (3.64)

After compactifications the resulting Penrose diagram is illustrated in Figure
6.

$(U^{-}_{out},V^{-}_{out})$$(U^{+}_{out},V^{+}_{out})$$(U^{-}_{in},V^{-}_{in})$$(U^{+}_{in},V^{+}_{in})$$H^{f}_{-}$$H^{f}_{+}$$H^{p}_{-}$$H^{p}_{+}$$r=r_{s-}$$r=r_{s+}$
Figure 6. The red dotted lines are constant $r$, the blue dotted lines are
constant time. The central part is the exterior of the black holes.
$H^{f}_{\pm}$ are the future horizons of the left ($-$) and right ($+$) black
holes. $H^{p}_{\pm}$ are the past horizons. The darker sectors are the
internal regions of the two black holes, the red waved lines are the timelike
singularities. The yellow regions are periodic prolongations.

##  4\. Conclusions

In [19] we have found new supersymmetric backgrounds in $\mathcal{N}=2$, $D=4$
gauged supergravity coupled to vector multiplets for the STU model with
prepotential $F(X^{0},\ldots,X^{3})=-2i\sqrt{X^{0}X^{1}X^{2}X^{3}}$. These are
everywhere regular solutions with one nonconstant complex scalar field and
both magnetic and electric fields with vanishing fluxes. A parameter $\alpha$
can be set to zero to switch off the electric fields. In this limit the
topology of the solutions is the one of a fibration $\mathbb{R}^{1,1}$ over a
surface of genus $0$, while for $\alpha\neq 0$ the solutions have the same
topology as the Kerr-Newman throat: a fibration of $AdS_{2}$ fibres on the
base of conformal spheres with non constant scalar curvature. These are
stationary solutions, with a rotation parameter along a Killing spacelike
direction on the conformal spheres. We recognized this solutions as near-
horizon geometries of black holes. In this work we filled the gap in the
static case and we found the alleged black hole. Surprisingly, the full
solution presents a new interesting unusual black hole dipole. Among other
properties, it is worth to mention that the scalars in this system satisfy the
attractor mechanism. Indeed the entropy depends only on the charges. In [19]
is considered also the case $\alpha=0$. In this case, the near horizon
solution is a type II ultracold Nariai spacetime, which results to be the near
horizon limit of a triply degenerate de Sitter spacetime

$\displaystyle F^{I}=$ $\displaystyle\frac{\sin{\theta}}{8g_{I}}d\theta\wedge
d\phi,\quad\tau=1,$ (4.1) $\displaystyle ds^{2}=$
$\displaystyle-e^{2U(r)}dt^{2}+e^{-2U(r)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}\right),$
(4.2) $\displaystyle e^{2U}=$
$\displaystyle-\frac{(\sqrt{G}r-2)^{3}(\sqrt{G}r+6)}{24Gr^{2}},$ (4.3)

with horizon in $r=\frac{2}{\sqrt{G}}$, and magnetic charge
$Q^{2}=\frac{2}{G}$, see e.g. [33]. It would be interesting to generalize
these results to the rotating case and then analyze holography and the
attractor mechanism along the lines of [41] and [42]. First of all, in the
rotating case we expect more non vanishing conserved quantities that could
allow us to find nontrivial relations helping us in determining the mass of
the black holes. The second reason is that, as shown in [19], the shape of the
horizon does not have constant curvature, so it looks like a deformed sphere.
This is a consequence of the presence of non trivial scalar configurations
along the horizon. Therefore, in the rotating case we expect two possible
scenarios: first, a dipole-like solution generalizing the above static
solution, whose shape of the surfaces at constant $r$ is always the same when
$r$ varies; second, an asymptotically AdS solution, whose shape of constant
$r$ surfaces changes from a deformed sphere to a round sphere when moving from
the horizon to the spatial infinity. This could help us in understanding if
such deformations are allowed by the equations of motion and, then, if open
solutions, in place of the closed ones, exist or not according with the given
family of near horizon solutions. In the case of affirmative answer, this
should also provide a good definition of mass through the wormhole
prolongation procedure we described in section 3.

## Acknowledgments

We thank an anonymous referee whose comments helped us to improve our
manuscript. D.A. is thankful to Professor M. Trigiante and Professor A.
Anabalon for useful discussions.

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