OU-HET-1191

Flows of Rotating Extremal Attractor

Black Holes

Norihiro Iizuka1, Akihiro Ishibashi2 and Kengo Maeda3

1Department of Physics, Osaka University

Toyonaka, Osaka 560-0043, JAPAN

2Department of Physics and Research Institute for Science and Technology,

Kindai University, Higashi-Osaka 577-8502, JAPAN

3Faculty of Engineering, Shibaura Institute of Technology,

Saitama 330-8570, JAPAN

iizuka at phys.sci.osaka-u.ac.jp, akihiro at phys.kindai.ac.jp,
maeda302 at sic.shibaura-it.ac.jp

###### Contents

  1. 1 Introduction
  2. 2 Attractor mechanism for the near horizon geometry in extremal rotating black holes
    1. 2.1 The setup
    2. 2.2 Attractor conditions
    3. 2.3 The local no-hair near the horizon
  3. 3 Asymptotically flat case
  4. 4 The attractor black hole solutions in AdS4
  5. 5 Summary and discussions
  6. A The field equations for $f$ and $g$
  7. B Expansions of the field equations
  8. C The first law of the extremal Myers-Perry AdS black hole

##  1 Introduction

An attractor mechanism appearing in an extremal charged black hole is a
beautiful phenomenon. A scalar field, that has no bare potential and therefore
can take vacuum expectation value (VEV) freely at spatial infinity, can only
take a specific value at the black hole horizon. Furthermore, the specific
value at the horizon is determined by the charge of a black hole only.
Thinking of the VEV of a scalar field as a function of radial coordinate, the
scalar VEV can take any value at infinity, but it is attracted to the specific
value at the horizon. This attractor mechanism was found originally in
${\cal{N}}=2$ supersymmetric black holes [1], and developed furthermore in [2,
3, 4, 5, 6, 7, 8, 9, 10] . Later it was shown in [11] to hold more generally
at zero temperature without supersymmetry.

The fact that a scalar field can only take a specific value at the horizon for
zero temperature black hole is deeply tied to the fact that near horizon
geometry determines the entropy of the black hole. The entropy of the extremal
charged black hole is determined by the charge only. Since the charge is
quantized, its entropy should not allow any free parameter to tune. On the
other hand, if there are moduli at the horizon, the existence of the free
parameter at the horizon is in contradiction with the fact that the ground
state entropy of the system is fixed by the charge only.

By thinking in this way, it is easy to imagine that the attractor phenomena
works not only for an extremal charged black hole but also for a rotating
extreme black hole. In fact, the attractor mechanism for rotating black holes
has also been investigated [12] where the attractor value is determined by
extremizing the entropy function [13] and thus it is determined locally near
the horizon. In many situations of rotating attractors, the extremality
condition for black holes is achieved with nonzero charges. In this paper, we
investigate the attractor mechanism of a rotating black hole without any
charges.

More concretely, we consider Einstein gravity minimally coupled to multi
complex scalars, which is the model studied recently in [14]. In this case,
the analytic nature of the horizon places strong restrictions on the scalar,
forcing it to be zero under certain conditions which we clarify. Under this,
in this paper, we investigate the rotating attractor mechanism and its
deformation. This paper can be regarded as the parallel work done here [15],
replacing the charge of a charged black hole with angular momentum.

In general, it is much more difficult to analyze stationary rotating black
holes than static ones since the geometry is no longer spherically symmetric,
and the field equations become highly involved in partial differential
equations. For the five-(more generally odd-)dimensions, however, there are
exceptional cases in the sense that the Einstein equations coupled to a class
of multi complex scalar fields reduce to ordinary differential equations when
we apply the cohomogeneity-$1$ metric ansatz. The rotating boson star [16] and
rotating AdS black hole solutions [17] were constructed for complex doublet
scalar models, on this ansatz. The complex scalar model was generalized to a
multiplet complex scalar for both the rotating AdS black hole and rotating
boson stars [14]. In this paper, we investigate the attractor mechanism of
extremal rotating black holes under the cohomogeneity-$1$ ansatz in the
Einstein gravity minimally coupled with a multiplet complex scalar. We find
that the only allowed attractor value is zero under certain conditions, due to
the existence of angular momentum of the black hole. This means that the local
geometry of the regular extremal AdS black hole is determined by the Myers-
Perry black hole, as the scalar field rapidly decays to zero near the horizon.

Introducing a relevant deformation to the complex scalar system through a bare
potential, we find various extremal rotating AdS black hole solutions
interpolating near the horizon to the AdS boundary. In particular, we find
that the mass and the angular momentum of the black hole solutions are
discretized under the usual Dirichlet boundary condition at the AdS boundary.
When the mass of the scalar field is negative, the scalar field admits two
normalizable modes near the boundary. In this case, we find that the energy of
the extremal rotating black hole with a complex multiplet scalar hair is less
than that of the extremal Myers-Perry AdS black hole [18] for the same angular
momentum.

The organization of the paper is as follows. In section 2, we briefly review
the multiplet complex scalar model [14] and investigate the attractor
conditions for the extremal rotating black holes, where the near horizon
geometry is analyzed. In section 3, we analyze asymptotically flat spacetime
and in section 4, we numerically find the extremal hairy black hole solutions
interpolating the near horizon to the AdS boundary under the various boundary
conditions. Section 5 is devoted to summary and discussions.

##  2 Attractor mechanism for the near horizon geometry in extremal rotating
black holes

In this section, we investigate the near horizon geometry of five-dimensional
extremal rotating black holes. Our model contains multiple complex scalar
fields. We see that the attractor mechanism to work and the value at the
horizon is strictly restricted to be zero. This shows that the local geometry
of the extremal black holes is determined by the Myers-Perry black hole. Note
that the argument in this section 2 is local and independent of the asymptotic
geometry.

###  2.1 The setup

We consider stationary extremal black hole solutions in the five-dimensional
Einstein gravity minimally coupled to multi complex scalar fields $\vec{\Psi}$
with a potential $V(\vec{\Psi})$. The action is given by

$\displaystyle S=\frac{1}{2}\int
d^{5}x\sqrt{-g}\left[R-2|\nabla\vec{\Psi}|^{2}-2V(\vec{\Psi})\right],$ (2.1)

where here and in the following we set $8\pi G=1$. We take the following
metric ansatz:

$\displaystyle ds_{5}^{2}$
$\displaystyle=-f(u)g(u)dt^{2}+\frac{r_{+}^{2}}{4u^{3}f(u)}du^{2}+\frac{r_{+}^{2}}{4u}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})$
$\displaystyle\qquad+\frac{r_{+}^{2}}{u}h(u)\left(d\psi+\frac{\cos\theta}{2}d\varphi-\Omega(u)dt\right)^{2},$
(2.2)

which is a cohomogeneity-1 spacetime. We choose the gauge that the horizon is
located at $u=1$ ($f(1)=0$). Each surface on $t=\mbox{const.}$ and
$u=\mbox{const.}$ is homogenously squashed $S^{3}$, which is composed of
$S^{1}$ fibers over $S^{2}$ (the angular coordinate ranges are $0\leq\psi\leq
2\pi$, $0\leq\theta\leq\pi$, and $0\leq\varphi\leq 2\pi$). Here, $r_{+}$ is
the horizon raidus on $S^{2}$. When $h=1$, it reduces to a sphere $S^{3}$ with
$SO(4)$ symmetry. This spacetime admits asymptotic timelike Killing vector
$(\partial_{t})^{\mu}$ and spacelike vectors $\xi_{i}^{\mu}$ on $S^{3}$ given
by Bianchi type IX,

$\displaystyle\xi_{1}=\cos\varphi\,\partial_{\theta}+\frac{\sin\varphi}{2\sin\theta}\partial_{\psi}-\cot\theta\sin\varphi\,\partial_{\varphi},$
$\displaystyle\xi_{2}=-\sin\varphi\,\partial_{\theta}+\frac{\cos\varphi}{2\sin\theta}\partial_{\psi}-\cot\theta\cos\varphi\,\partial_{\varphi},$
$\displaystyle\xi_{3}=\partial_{\varphi}.$ (2.3)

The null Killing vector on the horizon is given by the linear combination of
$\partial_{t}$ and $\partial_{\psi}$ as

$\displaystyle l=\partial_{t}+\Omega|_{u=1}\partial_{\psi},$ (2.4)

where $\Omega|_{u=1}$ is the angular velocity of the black hole.

For the case

$\displaystyle V(\vec{\Psi})=\Lambda=-\frac{6}{\ell^{2}}$ (2.5)

i.e., with negative cosmological constant, and in the absence of the scalar
field, the action (2.1) admits the extremal Myers-Perry AdS black hole metric
[18]

$\displaystyle
f(u)=\frac{r_{+}^{2}}{\ell^{2}u}(1-u)^{2}\left(1+2u+\frac{\ell^{2}}{r_{+}^{2}}u\right),$
$\displaystyle h(u)=1+\left(1+\frac{2r_{+}^{2}}{\ell^{2}}\right)u^{2},\qquad
g(u)=\frac{1}{h(u)},$
$\displaystyle\Omega(u)=\frac{2r_{+}^{2}}{\ell^{3}}\left(1+\frac{\ell^{2}}{r_{+}^{2}}\right)\sqrt{1+\frac{\ell^{2}}{2r_{+}^{2}}}\frac{u^{2}}{h(u)}$
(2.6)

as a solution. Here $u=1$ is a horizon and $u=0$ is AdS boundary.

From our ansatz for the metric eq. (2.1) and its Killing symmetry, all the
equations reduce to a set of ordinary differential equations (ODEs) if one
considers $2j+1$ components of the scalar field in the form [14]

$\displaystyle\vec{\Psi}$
$\displaystyle=\\{\Psi_{j},\,\Psi_{j-1},\,\cdots,\,\Psi_{-j}\\}$
$\displaystyle=\phi(u)\,e^{-i\omega
t}\left(\Theta^{j}_{j,k},\,\Theta^{j}_{j-1,k},\,\Theta^{j}_{j-2,k},\,\cdots,\,\Theta^{j}_{-j,k}\right)\,.$
(2.7)

Here $\phi(u)$ is a real-valued function and
$\Theta^{j}_{m,k}~{}(m=-j,-j+1,\cdots,j-1,j)$ are Wigner D-matrices, which are
eigenfunctions of the Laplace equations on the homogeneously squashed $S^{3}$,
and thus satisfy the following equations

$\displaystyle\left(\sum_{i=1}^{3}L_{i}^{2}\right)\Theta^{j}_{m,k}=j(j+1)\Theta^{j}_{m,k},\quad
j=0,\,\frac{1}{2},\,1,\,\frac{3}{2},\cdots,$ $\displaystyle
L_{3}\Theta^{j}_{m,k}=m\Theta^{j}_{m,k},\quad m=-j,\,-j+1,\cdots,\,j.$
$\displaystyle\frac{i}{2}\partial_{\psi}\Theta^{j}_{m,k}=k\Theta^{j}_{m,k},\quad
k=-j,\,-j+1,\cdots,\,j,$ (2.8)

where $L_{i}~{}(i=1,2,3)=i\xi_{i}$ is the angular momentum operator satisfying
the commutation relation

$\displaystyle[L_{i},\,L_{j}]=i\epsilon_{ijk}L_{k}.$ (2.9)

Let us define the scalar potential $V(\vec{\Psi})$ as

$\displaystyle
V(\vec{\Psi})=V(|\vec{\Psi}|)=-\frac{6}{\ell^{2}}+\frac{1}{2}M^{2}|\phi|^{2}+\cdots,$
$\displaystyle|\vec{\Psi}|\equiv\sqrt{\sum_{m=-j}^{j}\Psi_{m}\Psi^{\ast}_{m}}=|\phi|\sqrt{\sum_{m=-j}^{j}\Theta^{j}_{m,k}\,{\Theta^{\ast}}^{j}_{m,k}}=|\phi|.$
(2.10)

Here $\cdots$ corresponds to terms of higher power in $|\phi|^{2}$ and we used
the formula

$\displaystyle\sum_{m=-j}^{j}{\Theta^{\ast}}^{j}_{m,k}\Theta^{j}_{m,k}=1$
(2.11)

in the last equality. So, the scalar potential $V(\vec{\Psi})$ becomes the
function of only $u$ via $|\phi(u)|$ in eq. (2.1). Combining the properties of
the Wigner D-matrix (2.1), one can check that the stress-energy momentum
tensor

$\displaystyle
T_{\mu\nu}=\nabla_{\mu}\vec{\Psi}\cdot\nabla_{\nu}\vec{\Psi^{\ast}}+\nabla_{\mu}\vec{\Psi^{\ast}}\cdot\nabla_{\nu}\vec{\Psi}-g_{\mu\nu}(|\nabla\vec{\Psi}|^{2}+V(\vec{\Psi}))$
(2.12)

is consistent with the metric ansatz eq. (2.1) [14]. Although the complex
scalar fields, eq. (2.1), are functions of all the coordinates,
$(t,\,u,\,\psi,\,\theta,\,\varphi)$, the backreaction of the complex $(2j+1)$
scalars to the metric is determined only through $\phi$.

From the Einstein equations $R_{\mu\nu}-(g_{\mu\nu}/2)R=T_{\mu\nu}$, we derive
two coupled second order ODEs for $h(u)$ and $\Omega(u)$, and as

$\displaystyle
h^{\prime\prime}+\frac{3u\\{f+h+(\epsilon_{k}^{2}+\epsilon_{k+1}^{2})\phi^{2}\\}+r_{+}^{2}V(|\phi|)-6u}{3u^{2}f}h^{\prime}-\frac{h^{\prime
2}}{h}$
$\displaystyle\qquad+\frac{2gh(1-h)+g\\{4k^{2}-h(\epsilon_{k}^{2}+\epsilon_{k+1}^{2})\\}\phi^{2}+r_{+}^{2}uh^{2}\Omega^{\prime
2}}{u^{2}fg}=0\,,$ (2.13)
$\displaystyle\Omega^{\prime\prime}-\left(\frac{1}{u}+\frac{g^{\prime}}{2g}-\frac{3h^{\prime}}{2h}\right)\Omega^{\prime}-\frac{2k(\omega+2k\Omega)}{u^{2}fh}\phi^{2}=0\,,$
(2.14)

where the coefficient $\epsilon_{k}$ is defined by

$\displaystyle\epsilon_{k}:=\sqrt{(j+k)(j-k+1)}$ (2.15)

and a dash denotes the derivative with respect to $u$. We also have two
coupled first order ODEs for $f(u)$ and $g(u)$, which are given in Appendix A.

Due to the properties eq. (2.1), the scalar field equation for $\Psi_{m}$
reduces to a single scalar equation,

$\displaystyle
f\phi^{\prime\prime}+\left(f^{\prime}+\frac{g^{\prime}f}{2g}+\frac{h^{\prime}f}{2h}\right)\phi^{\prime}+\left(r_{+}^{2}\frac{(\omega+2k\Omega)^{2}}{4u^{3}fg}-\frac{k^{2}(1-h)}{u^{2}h}-\frac{j(j+1)}{u^{2}}\right)\phi$
$\displaystyle\qquad-\frac{r_{+}^{2}\phi}{8u^{3}|\phi|}\frac{\partial
V}{\partial|\phi|}=0.$ (2.16)

###  2.2 Attractor conditions

Let us investigate the attractor value of the scalar field on the horizon.
Hereafter, we assume that the metric is $C^{2}$ outside the horizon and the
scalar field $\phi$ is $C^{1}$. Therefore the functions near the horizon can
be expanded as

$\displaystyle\Omega=\Omega_{0}+\Omega_{1}(1-u)+\Omega_{2}(1-u)^{2}+\cdots,$
$\displaystyle h=h_{0}+h_{1}(1-u)+h_{2}(1-u)^{2}+\cdots,$ $\displaystyle
f=f_{0}(1-u)^{2}+\cdots,$ $\displaystyle g=g_{0}+g_{1}(1-u)+\cdots,$
$\displaystyle\phi=\phi_{0}+\phi_{1}(1-u)^{\lambda},\quad\lambda\geq 1,$
(2.17)

where the coefficients $h_{0}$, $f_{0}$, $g_{0}$ are positive, and the
frequency $\omega$ must satisfy the condition

$\displaystyle\omega=-2k\Omega_{0}$ (2.18)

so that the complex scalar field is stationary in a co-rotating frame near the
extremal rotating black hole horizon. As shown in Ref. [15], in the presence
of the bare potential $V(\phi)$, one can take any attractor values $\phi_{0}$
for the case of static charged extremal black hole by tuning the bare
potential $V(\phi)$111Without bare potential, the attractor value $\phi_{0}$
is determined by the charges of the black hole only. See §3 in [15]. In our
rotating case, however, one cannot take arbitrary value $\phi_{0}$, and indeed
$\phi_{0}=0$ is the only attractor value, as shown below.

Suppose that $\phi_{0}\neq 0$. Then, substituting eqs. (2.2) into eq. (2.14),
one finds

$\displaystyle\Omega_{1}=0.$ (2.19)

Let us consider a $t,\,u=const.$ spacelike surface $S_{0}$ on the future event
horizon and a past-directed null geodesic congruence orthogonal to $S_{0}$ and
transverse to the horizon. The tangent vector $k^{\mu}$ of affinely
parametrized such null geodesics and the expansion $\theta$ thereof are given
by

$\displaystyle
k^{\mu}=-E\left(\frac{1}{fg}\partial_{t}+\frac{\Omega}{fg}\partial_{\psi}+\frac{2u^{\frac{3}{2}}}{r_{+}\sqrt{g}}\partial_{u}\right)^{\mu},$
$\displaystyle\theta:=\frac{{\cal L}_{k}{\cal A}}{{\cal
A}}=E\frac{3h-uh^{\prime}}{r_{+}h}\sqrt{\frac{u}{g}},$ (2.20)

where $E$ is a positive constant and ${\cal A}$ is the area element of the
null geodesic congruence. In the usual black hole case, the expansion $\theta$
on $S_{0}$ needs to be positive in the outgoing direction. Thus, this leads to

$\displaystyle\theta(u=1)>0\,\,\,\Longleftrightarrow\,\,\,h_{1}+3h_{0}>0\,.$
(2.21)

Under the condition (2.18) and the expansion (2.2), eqs. (A) and (A) are
rewritten near the horizon as

$\displaystyle 3u^{2}(uh^{\prime}-3h)f^{\prime}=F_{0}+O(1-u),$ $\displaystyle
3u^{2}(uh^{\prime}-3h)g^{\prime}=\frac{g_{0}G_{0}}{f_{0}(1-u)^{2}}+O((1-u)^{-1}),$
(2.22)

where the coefficients $F_{0}$ and $G_{0}$ are functions of $h_{0}$, $h_{1}$,
$\phi_{0}^{2}$, and $V(|\phi_{0}|)$ (in details, see Appendix B). Since the
metric functions are $C^{2}$ and $f^{\prime}=0$ on the horizon, each
coefficient $F_{0}$ and $G_{0}$ must be zero. This yields

$\displaystyle\phi_{0}^{2}=\frac{h_{0}(4-h_{0}-r_{+}^{2}V(|\phi_{0}|))}{2\\{2k^{2}+h_{0}(\epsilon_{k}^{2}+\epsilon_{k+1}^{2})\\}},\qquad
h_{1}+3h_{0}=0.$ (2.23)

This, however, contradicts eq. (2.21). Thus, $\phi_{0}$ must be zero in the
extremal rotating black holes.

###  2.3 The local no-hair near the horizon

As shown in the previous subsection, the attractor value of the scalar field
$\phi_{0}$ must be zero for the extremal black holes. So we can expect that
the back reaction of the scalar field on the metric is negligible near the
horizon. To illustrate this, let us first consider the solution of the scalar
field eq. (2.1) near the horizon on the background of the extremal Myers-Perry
black hole metric (2.1). Expanding the scalar potential $V(|\phi|)$ as

$\displaystyle
V(|\phi|)=-\frac{6}{\ell^{2}}+\frac{1}{2}M^{2}|\phi|^{2}+O(|\phi|^{3}),$
(2.24)

eq. (2.1) near the horizon reduces to the following equation,

$\displaystyle\qquad(1-u)^{2}\phi^{\prime\prime}-2(1-u)\phi^{\prime}-\zeta\phi\simeq
0,$
$\displaystyle\zeta=\frac{1}{1+3\hat{r}_{+}^{2}}\left[j(j+1)+\frac{\hat{r}_{+}^{2}\hat{M}^{2}}{8}-\frac{3k^{2}(1+2\hat{r}_{+}^{2})}{2(1+3\hat{r}_{+}^{2})}\right],$
(2.25)

and the solution is approximately given by

$\displaystyle\phi\simeq\alpha_{+}(1-u)^{\lambda_{+}}+\alpha_{-}(1-u)^{\lambda_{-}},\qquad\lambda_{\pm}=\frac{-1\pm\sqrt{1+4\zeta}}{2},$
(2.26)

where here and hereafter

$\hat{r}_{+}:=\dfrac{r_{+}}{\ell}\,,\quad\hat{M}:=\dfrac{M}{\ell}\,.$ (2.27)

By imposing that $\phi$ is $C^{1}$, the second term with index $\lambda_{-}$
should be discarded and $\zeta$ must satisfy the inequality

$\displaystyle\lambda_{+}\geq 1\,\,\Longrightarrow\,\,\zeta\geq 2.$ (2.28)

This is equivalent to the inequality

$\displaystyle
j(j+1)\geq\frac{4(1+3\hat{r}_{+})^{2}+3k^{2}(1+2\hat{r}_{+}^{2})}{2(1+3\hat{r}_{+}^{2})}-\frac{\hat{r}_{+}^{2}\hat{M}^{2}}{8},$
(2.29)

which gives a lower bound of the index $j$ for each black hole horizon radius
$r_{+}=\hat{r}_{+}\ell$, the eigen value $k$, and the mass $M=\hat{M}\ell$ of
the complex scalar field.

Let us first look at the the back reaction on the metric function $\Omega$ in
eq. (2.14). The formal solution of eq. (2.14) is given by

$\displaystyle\Omega^{\prime}=u\sqrt{\frac{g}{h^{3}}}\left[C+2k\int^{u}_{1}\sqrt{\frac{h}{g}}\,\frac{\omega+2k\Omega}{u^{3}f}\,\phi^{2}du\right],$
(2.30)

where $C$ is an integration constant. Since $\lambda\geq 1$ in eqs. (2.2), the
scalar field appears in $\Omega^{\prime}$ in the following subleading terms as

$\displaystyle\Omega^{\prime}=u\sqrt{\frac{g}{h^{3}}}\left(C+B(1-u)^{2\lambda}\right),$
(2.31)

where $B$ is some coefficient, whose explicit expression is not relevant for
the rest of our argument. Similarly, it is expected that the back reaction
effect of the scalar field on the other metric functions are also negligible
near the horizon. In other words, the near horizon geometry of the extremal
black hole locally approaches AdS2 geometry. Hereafter, we will show that the
geometry near the horizon is given by the Myers-Perry extremal black hole
metric (2.1) apart from local gauge freedom.

Substituting eqs. (2.2) and (2.31) into eq. (2.1), we obtain

$\displaystyle\dfrac{\\{uh-2(\hat{r}_{+}^{2}+u)\\}hh^{\prime}+2h^{2}(1-h)+{\ell^{2}C^{2}\hat{r}_{+}^{2}u^{3}}}{u^{2}fh}+h^{\prime\prime}-\frac{h^{\prime
2}}{h}+\frac{h^{\prime}}{u}=O((1-u)^{2\lambda-2}).$ (2.32)

Since the metric is $C^{2}$, we obtain the numerator of the first term in eq.
(2.33) should be $O((1-u)^{2})$, i. e. ,

$\displaystyle\\{uh-2(\hat{r}_{+}^{2}+u)\\}hh^{\prime}+2h^{2}(1-h)+\ell^{2}C^{2}\hat{r}_{+}^{2}u^{3}=O((1-u)^{2}).$
(2.33)

Similarly, by requiring $f^{\prime}=O(1-u)$, we obtain the other constraint
from eq. (A) as

$\displaystyle\ell^{2}C^{2}\hat{r}_{+}^{2}u^{4}+h^{2}(6\hat{r}_{+}^{2}+8u-5uh)+2uhh^{\prime}(uh-2\hat{r}_{+}^{2}-2u)=O(1-u).$
(2.34)

Substituting the expansion (2.2) into eqs. (2.33) and (2.34), we obtain the
coefficients $h_{0}$, $h_{1}$, and $C^{2}$ under the condition (2.21) as

$\displaystyle h_{0}=2\left(1+\hat{r}_{+}^{2}\right),\qquad
h_{1}=-2(1+2\hat{r}_{+}^{2}),$ $\displaystyle
C^{2}=\frac{8(1+2\hat{r}_{+}^{2})(1+\hat{r}_{+}^{2})^{2}}{\ell^{2}\hat{r}_{+}^{2}}.$
(2.35)

This is nothing but the coefficient of the metric expansion of the Myers-Perry
black hole (2.1). By calculating the next subleading order in eq. (A), we also
obtain $f_{0}=1+3\hat{r}_{+}^{2}$, agreeing with the metric expansion of the
Myers-Perry black hole (2.1). It is noteworthy that the metric (2.1) and the
field equations (2.1), (2.14), (A), (A), and (2.1) are invariant under the
transformation:

$\displaystyle\psi\to\psi+\xi\,t,\qquad\Omega_{0}\to\Omega_{0}+\xi,\qquad\omega\to\omega-2k\xi.$
(2.36)

So, apart from this gauge freedom and the scaling of $t$, any regular solution
under our ansatz for the metric and the complex scalar field reduces to the
Myers-Perry black hole (2.1) locally near the horizon.

##  3 Asymptotically flat case

Before considering the asymptotically AdS spacetime, it is worth noting that
for generic $\omega$, the solution cannot be asymptotically flat. Suppose the
asymptotically flat solution would be realized by taking the limit
$\ell\to\infty$. Then,

$\displaystyle f=g=h=1,\qquad\Omega=0\,\,\,\mbox{for}\,\,\,u=0,$ (3.1)

and the asymptotic form of eq. (2.1) becomes

$\displaystyle\phi^{\prime\prime}\simeq-\frac{a^{2}}{4u^{3}}\phi\qquad
a^{2}:=r_{+}^{2}\left(\omega^{2}-\frac{M^{2}}{2}\right)$ (3.2)

in the asymptotic region, $u\to 0$. The asymptotic solution is given by the
(modified) Bessel functions $J_{\nu}(x)~{}(I_{\nu}(x))$ and
$N_{\nu}(x)~{}(K_{\nu}(x))$, depending on the sign of $a^{2}$ as

$\displaystyle\phi\simeq\sqrt{u}\left[c_{1}J_{1}\left(\frac{a}{\sqrt{u}}\right)+c_{2}N_{1}\left(\frac{a}{\sqrt{u}}\right)\right],\qquad
a^{2}>0,$
$\displaystyle\phi\simeq\sqrt{u}\left[{\tilde{c}}_{1}I_{1}\left(\frac{\tilde{a}}{\sqrt{u}}\right)+{\tilde{c}}_{2}K_{1}\left(\frac{\tilde{a}}{\sqrt{u}}\right)\right],\qquad\tilde{a}^{2}:=-a^{2}>0.$
(3.3)

Let us analyze the following three cases separately.

  1. 1.

$\omega^{2}<M^{2}/2$ case: the solution is given by the linear combination of
the Bessel functions. By imposing the regularity condition on the horizon,
generically the solution includes the asymptotically diverging mode,
$I_{1}(\tilde{a}/\sqrt{u})\sim e^{\tilde{a}/\sqrt{u}}\to\infty$. Thus in the
case of $\omega^{2}<M^{2}/2$, for the generic value of $\omega$, there is no
asymptotically flat solution for the $\omega$ satisfying the co-rotating
condition eq. (2.18)222However, for some quantized number $\omega$ (or
$\Omega_{0}$ in Eq. (2.18)), the regularity condition both on the horizon and
at the asymptotic infinity can be satisfied. Thus there may possibly be an
exceptional case in which one can adjust the co-rotating condition, (2.18),
for that quantized number $\omega$, so that the asymptotically flat solution
can be the case. We will not investigate such a spacial case here anymore..

  2. 2.

$\omega^{2}=M^{2}/2$ case: the solution is given from eq. (3.2) as,

$\displaystyle\phi\simeq{c}^{\prime}_{1}+\frac{{c}^{\prime}_{2}}{r^{2}}\,,\quad\mbox{near
$u=0$}$ (3.4)

where we used $u=r_{+}^{2}/r^{2}$. The regularity condition imposed at the
horizon requires specific linear combinations between $c_{1}^{\prime}$ and
$c_{2}^{\prime}$, and generically the constant $c_{1}^{\prime}\neq 0$.
However, when $\omega\neq 0$ due to the co-rotating condition eq. (2.18), this
implies that the gravitational energy diverges due to its oscillation at
infinity with nonzero magnitude, and thus, an asymptotically flat solution is
not allowed.

  3. 3.

$\omega^{2}>M^{2}/2$ case: the asymptotic behavior of $\phi$ is given by

$\displaystyle\phi\sim
u^{3/4}\cos\left(\frac{a}{\sqrt{u}}+\delta\right)\sim{O}\left({r^{-3/2}}\right)\,.$
(3.5)

However, this scalar field does not decay fast enough asymptotically as
$\phi\sim r^{-2}$ which follows the Gauss law. In other words, by substituting
Eq. (3.5) into eq. (A)

$\displaystyle f\simeq 1+O(\sqrt{u})\simeq 1+O\left(\frac{1}{r}\right),$ (3.6)

Thus, we conclude that for the generic value of $\omega$, the gravitational
energy is divergent for such slow decay of the scalar field, and thus one
cannot have an asymptotically flat solution as $f\simeq 1+O(1/r^{2})$.

Thus unlike the static extremal attractor black holes, in the rotating
extremal black hole case, due to the co-rotating condition eq. (2.18), the
construction of asymptotically flat black holes in our setup is highly
restricted and impossible generically. Therefore in the next section, we
consider the case with asymptotically AdS case.

##  4 The attractor black hole solutions in AdS4

In this section, we numerically construct extremal AdS black hole solutions
with complex scalars interpolating between the near horizon solution (2.3) and
asymptotically AdS solution. In our setup, we consider the case that the mass-
squared, $M^{2}$, of the scalar field is negative, and that the metric
asymptotically approaches AdS spacetime 333If one introduces a radial
coordinate $r$ by $u=r_{+}^{2}/r^{2}$, the metric becomes usual asymptotic
form,
$ds^{2}\simeq-r^{2}/\ell^{2}dt^{2}+\ell^{2}/r^{2}dr^{2}+r^{2}d\Omega_{3}^{2}$.,

$\displaystyle f\simeq\frac{r_{+}^{2}}{\ell^{2}u},\qquad h\simeq g\simeq
1,\qquad\Omega\simeq 0.$ (4.1)

Then, the asymptotic solution of Eq. (2.1) is

$\displaystyle\phi=k_{+}u^{\rho_{+}}+k_{-}u^{\rho_{-}},\qquad\rho_{\pm}=1\pm\sqrt{1+\frac{\hat{M}^{2}}{8}}.$
(4.2)

Since both of the modes are normalizable, i. e., $\rho_{\pm}>0$, $\phi$
approaches zero toward the AdS boundary, $u=0$.

As shown in the previous section, the attractor value of the scalar field is
zero and the near horizon geometry reduces to that of the Myers-Perry black
hole via eq. (2.3), and $f_{0}=1+3\hat{r}_{+}^{2}$, except the gauge freedom
(2.36). This gauge freedom (2.36) is used to satisfy one of the asymptotic
boundary conditions (4.1), $\Omega(0)=0$. Another trivial gauge freedom is the
choice of $g_{0}$ in eqs. (2.2), corresponding to the scaling of $t$ as $t\to
s_{0}t$ with some constant $s_{0}\neq 0$. This freedom is used to satisfy
another asymptotic boundary condition, $g(0)=1$ in the boundary conditions
(4.1). So, once the coefficients $h_{0}$, $h_{1}$, $\Omega_{1}$, and $f_{0}$
in the expansion (2.2) are determined, the ODEs (2.1), (2.14), (A), (A) are
uniquely determined except the gauge freedoms mentioned above. So, the
remaining free parameters on the horizon are

$\displaystyle\alpha_{+},\qquad\hat{r}_{+}$ (4.3)

in Eqs. (2.26) and (2.27). For simplicity, we consider the following potential
with two free parameters, $b_{1}$ and $b_{2}$ as

$\displaystyle
V(|\phi|)=-\frac{6}{\ell^{2}}-\frac{15}{2\ell^{2}}|\phi|^{2}+\frac{b_{1}}{\ell^{2}}|\phi|^{3}+\frac{b_{2}}{\ell^{2}}|\phi|^{4},$
(4.4)

where $\hat{M}^{2}=-15/2$. In this case, the asymptotic solution (4.2) reduces
to

$\phi\simeq k_{-}u^{3/4}+k_{+}u^{5/4},$ (4.5)

where each mode is normalizable.

First, we numerically find the regular extremal black hole solutions under the
usual boundary condition

$k_{-}=0.$ (4.6)

Figure 1: $\phi$ with $\hat{r}_{+}\simeq 2.91$, $j=9$, $k=-1$ in the odd
potential.

Figure 2: $h$ with $\hat{r}_{+}\simeq 2.91$, $j=9$, and $k=-1$ in the odd
potential. The dotted (blue) curve represents Myers-Perry extremal BH.

Figure 3: $fu$ with $\hat{r}_{+}\simeq 2.91$, $j=9$, and $k=-1$ in the odd
potential. The dotted (blue) curve represents Myers-Perry extremal BH.

Figure 4: $(\Omega,g)$ with $\hat{r}_{+}\simeq 2.91$, $j=9$, and $k=-1$ in the
odd potential. The solid (blue) and dashed (green) curves are $\Omega$ and
$g$, respectively.

Among the free parameters $\Omega_{0}$, $g_{0}$, $\alpha_{+}$, and
$\hat{r}_{+}$ on the horizon (4.3), the first two parameters are adjusted to
satisfy the asymptotic boundary conditions, $g(0)=1$ and $\Omega(0)=0$ in
(4.1). Then, the remaining boundary conditions are Eq. (4.6) and

$\displaystyle h(0)=1$ (4.7)

for the only two free parameters, $\alpha_{+}$ and $\hat{r}_{+}$ 444$f\simeq
r_{+}^{2}/u$ in (4.1) is automatically satisfied under the boundary condition
(4.5).. This implies that the solutions are discretized for each “quantum”
numbers, $j$ and $k$. In other words, the horizon radius and the entropy are
characterized by such quantum numbers of the complex scalar field. Since it is
in general difficult to find numerical solutions interpolating between the
horizon and the AdS boundary in the full non-linear regime $|\phi|\gg 1$, we
shall pay attention to the perturbed region in which $|\phi|\lesssim 1/2$ is
satisfied and the deviation from the extremal Myers-Perry black hole solution
is not so large.

In the perturbation region, $|\alpha_{+}|\lesssim 1/2$, we find two discrete
solutions for each $(j,\,k)=(7,-1),\,(9,-1)$ in the odd potential case,
$b_{1}=-23$ and $b_{2}=0$. The numerical data for $(j,\,k)=(9,-1)$ are given
in Figs. 4, 4, 4, 4. We find two discrete solutions with horizon radii,
$\hat{r}_{+}=2.91,\,3.07$ for $(j,\,k)=(9,-1)$. Similarly, we also find two
discrete solutions with $\hat{r}_{+}=2.5,\,2.7$ for $(j,\,k)=(7,-1)$.

Secondly, we investigate regular solutions for generalized boundary conditions
with $k_{\pm}\neq 0$. In this case, the total energy generically diverges
according to the slower fall off term with coefficient $k_{-}$in Eq. (4.5). To
avoid such a divergence, we follow the approach adopted by Ref. [19].

To calculate the energy, we use the usual radial coordinate $r=r_{+}/\sqrt{u}$
instead of dimensionless coordinate $u$. By Eqs. (2.1), (2.14), (2.1), (A),
and (A), one obtains the asymptotic solution near the AdS boundary as

$\displaystyle
f=\frac{r^{2}}{\ell^{2}}+1+\frac{k_{-}^{2}r_{+}^{3}}{\ell^{2}r}+\frac{C_{1}}{r^{2}}+\cdots,$
$\displaystyle g=1-\frac{k_{-}^{2}r_{+}^{3}}{r^{3}}+\cdots$ $\displaystyle
h=1+\frac{C_{2}}{r^{4}}+\cdots,$
$\displaystyle\Omega=\frac{C_{3}}{r^{4}}+\cdots,$ (4.8)

with $C_{1},C_{2},C_{3}$ being some constants. Due to the existence of the
third term for $f$ in the asymptotic expansion (4), usual gravitational energy
diverges. However, as shown below, the total Hamiltonian of the gravitational
energy and the scalar field energy is finite.

According to the Hamiltonian approach, let us consider a timelike vector field
$\xi=\partial_{t}$. Then, the variation of the gravitational energy is given
by

$\displaystyle\delta Q_{G}[\xi]=\frac{1}{2}\int d\psi d\theta
d\varphi\,\delta^{r}_{i}\,\overline{G}^{ijkl}(\xi_{\perp}\bar{D}_{j}\delta
h_{kl}-\delta h_{kl}\bar{D}_{j}\xi^{\perp}),$ $\displaystyle
G^{ijkl}=\frac{1}{2}\sqrt{g}(g^{ik}g^{jl}+g^{il}g^{jk}-2g^{ij}g^{kl}),\qquad
h_{ij}=g_{ij}-\bar{g}_{ij}.$ (4.9)

Here, $\bar{g}_{ij}~{}(i,\,j=r,\psi,\theta,\varphi)$ is the spatial metric of
pure AdS and any geometric quantity $A$ with respect to $\bar{g}_{ij}$ is
denoted by $\bar{A}$, and $\xi_{\perp}=\xi^{\mu}n_{\mu}$, with $n^{\mu}$ being
the (past-directed) unit normal to the Cauchy surface $\Sigma$. The variation
of the surface term by the scalar field is also given by

$\delta Q_{\phi}=-2\int\xi^{\perp}\delta\phi
D_{i}\phi\,dS^{i}=-\frac{1}{2}\int\xi^{\perp}\delta\phi
D_{i}\phi\,\delta^{i}_{r}\,r^{3}\sqrt{fh}\sin\theta\,d\psi d\theta d\varphi.$
(4.10)

In general, both eqs. (4) and (4.10) diverge, due to the slower fall-off of
the scalar field with (4.5). Under the asymptotic expansion (4), one can
evaluate $\delta Q_{G}[\xi]$ and $\delta Q_{\phi}[\xi]$ as

$\displaystyle\delta
Q_{G}=\pi^{2}\left(-\frac{3\delta(\alpha^{2})}{\ell^{2}}r-3\delta
C_{1}+\frac{4\delta C_{2}}{\ell^{2}}\right),$ $\displaystyle\delta
Q_{\phi}=\frac{\pi^{2}}{\ell^{2}}\left(3\delta(\alpha^{2})r+6\delta(\alpha\beta)+4\beta\delta\alpha\right),$
(4.11)

where $\xi_{\perp}=\sqrt{fg}$, and $\alpha$, $\beta$ are defined by

$\displaystyle\alpha=k_{-}r_{+}^{3/2},\qquad\beta=k_{+}r_{+}^{5/2}.$ (4.12)

Since the total variation of the gravitational and scalar charge is finite, we
can define the total energy by choosing a function $\beta(\alpha)$ and
introducing an effective potential $W(\alpha)$ following [20, 19] as

$\displaystyle W(\alpha)=\int^{\alpha}_{0}\beta(\alpha)d\alpha.$ (4.13)

So, the total charge is finite and it is given by

$\displaystyle
Q_{tot}=\pi^{2}\left(\frac{4C_{2}}{\ell^{2}}-3C_{1}+\frac{6\alpha\beta}{\ell^{2}}+\frac{4W(\alpha)}{\ell^{2}}\right).$
(4.14)

Figure 5: $\alpha$-$\beta$ plot for $(j,\,k)=(7,-1)$ (green diamonds),
$(9,-1)$ (red triangles) for various radii $2.48\leq\hat{r}_{+}\leq 2.82$
($j=7$), $3.0\leq\hat{r}_{+}\leq 3.42$ ($j=9$) in the even potential with
$b_{1}=0$, $b_{2}=-63$. The blue curve $\beta\simeq 2.25\alpha(\alpha-13.4)$
intersects two solutions in $j=9$. Figure 6: $(Q_{tot},\,J)$ plot. The curve
corresponds to extremal Myers-Perry black holes. The (orange) boxes and (blue)
circles represent $(j,\,k)=(7,\,-1)$ and $(9,\,-1)$ solutions satisfying the
boundary condition (4.6) for the odd potential $(b_{1},\,b_{2})=(-23,0)$. The
(green) diamonds and (red) triangles represent $(j,\,k)=(7,\,-1)$ and
$(9,\,-1)$ solutions satisfying the generalized boundary conditions (4.16)
with $(k,\,\alpha_{0})\simeq(2.34,\,9.90)$ and $(2.25,\,13.43)$.

The angular momentum can be read off from the coefficient $C_{3}$ in the
expansion (4). Since the falloff of $\Omega$ is the same as that of the Myers-
Perry black hole (2.1), it is calculated by

$\displaystyle J=\frac{1}{2}\int_{\cal
S}\epsilon_{\mu\nu\alpha\beta\gamma}\nabla^{\beta}\,\psi^{\gamma}=4\pi^{2}C_{3},$
(4.15)

where ${\cal S}$ is the three-dimensional sphere at the AdS boundary and
$\psi^{\mu}=(\partial_{\psi})^{\mu}$ is the axial Killing vector 555Eq. (4.15)
is normalized so that extremal Myers-Perry black hole satisfies the first law
of the black hole for the gravitational energy (4.14) and the angular momentum
(4.15). See Appendix C in detail..

Before evaluating the energy and the angular momentum of the rotating black
holes with a complex multiplet scalar hair, let us look at the solutions in
$(\alpha,\,\beta)$ space in the even potential case, $b_{1}=0$. The figure 5
shows $(j,\,k)=(7,\,-1)$, $(9,-1)$ solutions for $\ell=1$ in $\alpha-\beta$
plane for various radii $\hat{r}_{+}$ in the even potential with $b_{2}=-63$
in the perturbative region $|\alpha_{+}|\lesssim 1/2$. Now, let us choose our
boundary condition as

$\displaystyle\beta=k\alpha(\alpha-\alpha_{0}),$ (4.16)

where $k$ and $\alpha_{0}$ are some positive constants. When $\alpha$ is small
enough, this corresponds to the usual Robin (mixed) boundary condition, and
for large $\alpha$ it reduces to $\beta\simeq k\alpha^{2}$, preserving all the
asymptotic AdS symmetries [22]. The solid curve in Figure 5 corresponds to
$k\simeq 2.25$ and $\alpha_{0}\simeq 13.4$. So, there are three solutions for
the theory; two hairly extremal black hole solutions with small $\alpha$ and
large $\alpha$, and $\alpha=0$ extremal Myers-Perry AdS solution (2.1).

We calculate the total energy (4.14) and the angular momentum (4.15) and
compare them with those of the extremal Myers-Perry AdS black hole (2.1). The
figure 6 shows $(Q_{tot},\,J)$ plot for the several hairly extremal black hole
solutions. The solid curve corresponds to the one-parameter family of extremal
Myers-Perry AdS black hole solutions. The left two boxes above the curve
correspond to the two solutions with $(j,\,k)=(7,\,-1)$ for the usual boundary
condition (4.6), while the right two circles above the curve do to the ones
with $(j,\,k)=(9,\,-1)$ for the same boundary condition (4.6). These imply
that the energy of the hairly extremal black hole solution is larger than that
of the extremal Myers-Perry black hole with the same angular momenta. The
diamonds below the curve correspond to the two solutions with
$(j,\,k)=(7,\,-1)$ for the generalized boundary condition (4.16) with
$(k,\,\alpha_{0})\simeq(2.34,\,9.90)$, while the right two triangles below the
curve do to the ones with $(j,\,k)=(9,\,-1)$ for the boundary condition
$(k,\,\alpha_{0})\simeq(2.25,\,13.43)$. The energies of these hairly extremal
black hole solutions are less than those of the extremal Myers-Perry black
holes with the same angular momenta.

##  5 Summary and discussions

We have investigated the attractor mechanism of the extremal rotating black
holes in the Einstein gravity minimally coupled with a multiplet complex
scalar field. In the non-rotating Einstein-Maxwell-dilaton system [15], one
can take any attractor values by adjusting the bare potential. On the other
hand, the only attractor value is zero in the extremal rotating case. This is
because the non-zero value is incompatible with the regularity of the extremal
horizon, as shown in section 2. So, the scalar field must decay to zero
rapidly near the horizon, and the near horizon geometry coincides with the
Myers-Perry AdS black hole [18], as described in subsection 2.3. Note that the
argument in section 2 is local and independent of the asymptotic region. This
is similar to the local argument of the entropy function extremization [13].

Then we discussed that this extremal rotating black hole does not admit
asymptotic flat spacetime. The crucial difference from the static extremal
black hole case is that due to the co-rotating condition (2.18), the scalar
field is oscillating and it does not admit finite energy configuration.

We have also investigated the extremal rotating AdS black hole solution
interpolating the near horizon geometry of the Myers-Perry black hole to the
AdS boundary. Since the near horizon geometry is restricted to the Myers-Perry
metric, the free parameters near the horizon are the magnitude $\alpha_{+}$ of
the scalar field in Eq. (2.26) and the horizon radius $\hat{r}_{+}$ only. If
one imposes usual Dirichlet boundary condition for the scalar field in
addition to the asymptotic boundary condition (4.1), two boundary conditions
must be imposed at the AdS boundary. This implies that the extremal hairly
black hole solution exists for a discrete set of the parameters
$(\alpha_{+},\,\hat{r}_{+})$. In other words, the energy and the angular
momentum are discretized, according to the quantum numbers of the scalar
field, $(j,\,k)$ and the parameters $(b_{1},\,b_{2})$ of the bare potential
(4.4). In this sense, all the properties of the extremal black hole are
determined by the “quantum” state of the scalar field. These are quite
different from the Myers-Perry AdS black hole in which the solution exists
continuously by varying black hole radius $r_{+}$.

For more general boundary condition (4.5) with $k_{\pm}\neq 0$, we can
evaluate the energy and the angular momentum of the extremal rotating black
hole. As shown in section 4, there are hairly rotating extremal AdS black hole
solutions whose energy is lower than that of the extremal Myers-Perry AdS
black hole for the same angular momentum. This implies that the extremal
Myers-Perry AdS black hole is unstable against the general boundary condition,
as already observed in the static AdS black hole case [21]. As shown in [17,
14], the Myers-Perry AdS black hole is unstable due to superradiance when the
black hole horizon is small compared with the AdS scale $\ell$. In our case,
the horizon radius is much larger than the AdS scale, i. e. , $\hat{r}_{+}\gg
1$, the instability of the extremal Myers-Perry AdS black hole would indicate
that the vaccum state with zero expectation value of the scalar operator
${\cal O}$ dual to the complex scalar fields $\vec{\Psi}$ is unstable for the
dual deformed boundary field theory. It would be interesting to pursue further
the issue for a wide range of quantum numbers, $(j,\,k)$, the parameters of
the potential, and the black hole radius.

Acknowledgments

This work was supported in part by JSPS KAKENHI Grant No. 18K03619 (N.I.),
15K05092, 20K03938 (A.I.), 20K03975 (K.M.) and also supported by MEXT KAKENHI
Grant-in-Aid for Transformative Research Areas A Extreme Universe No.21H05184
(N.I.), No.21H05186 (A.I. and K.M.), and 21H05182 (N.I. and A.I.).

##  Appendix A The field equations for $f$ and $g$

$\displaystyle
3u^{2}fgh(uh^{\prime}-3h)f^{\prime}+3h[r_{+}^{2}h(\omega+2k\Omega)^{2}-2ufg\\{2k^{2}-(\epsilon_{k}^{2}+\epsilon_{k+1}^{2})(2h-uh^{\prime})\\}]\phi^{2}$
$\displaystyle+12u^{3}f^{2}gh^{2}\phi^{\prime
2}-r_{+}^{2}fgh(2uh^{\prime}-3h)V(|\phi|)-3r_{+}^{2}u^{2}fh^{3}\Omega^{\prime
2}$
$\displaystyle+3ufg\\{h^{2}(-8+3f+5h)+u(4-3f-2h)hh^{\prime}+u^{2}fh^{\prime
2}\\}=0,$ (A.1) $\displaystyle
3u^{2}f^{2}h(uh^{\prime}-3h)g^{\prime}-6h[r_{+}^{2}h(\omega+2k\Omega)^{2}+ufg\\{-4k^{2}+(\epsilon_{k}^{2}+\epsilon_{k+1}^{2})(h-uh^{\prime})\\}]\phi^{2}$
$\displaystyle-24u^{3}f^{2}gh^{2}\phi^{\prime
2}+2r_{+}^{2}ufghh^{\prime}V(|\phi|)+6r_{+}^{2}u^{2}fh^{3}\Omega^{\prime 2}$
$\displaystyle-3ufg\\{4h^{2}(h-1)-uh(-4+f+2h)h^{\prime}+u^{2}fh^{\prime
2}\\}=0.$ (A.2)

##  Appendix B Expansions of the field equations

Under the conditions, (2.18) and (2.19), the field Eqs. (A) and (A) are
expanded around the horizon $u=1$ as

$\displaystyle
3u^{2}(uh^{\prime}-3h)f^{\prime}=3\\{h_{0}(8-5h_{0}-2h_{1})-2(2h_{0}+h_{1})(\epsilon_{k}^{2}+\epsilon_{k+1}^{2})\phi_{0}^{2}$
$\displaystyle+4(h_{1}+k^{2}\phi_{0}^{2})\\}-(3h_{0}+2h_{1})r_{+}^{2}V(|\phi_{0}|)+O(1-u)$
$\displaystyle=F_{0}+O(1-u),$
$\displaystyle\frac{3u^{2}f(uh^{\prime}-3h)g^{\prime}}{g}=6\\{h_{0}(-2+2h_{0}+h_{1})+(h_{0}+h_{1})(\epsilon_{k}^{2}+\epsilon_{k+1}^{2})\phi_{0}^{2}-2(h_{1}+2k^{2}\phi_{0}^{2})\\}$
$\displaystyle+2h_{1}r_{+}^{2}V(|\phi_{0}|)+O((1-u))$
$\displaystyle=G_{0}+O(1-u).$ (B.1)

The r. h. s. of Eqs (B) should be zero if the metric is $C^{2}$. This yields
the solution (2.23).

##  Appendix C The first law of the extremal Myers-Perry AdS black hole

In the extremal Myers-Perry AdS black hole, the energy (4.14) and the angular
momentum $J$ (4.15) are evaluated from the metric (2.1) under the condition
$\alpha=0$ as

$\displaystyle
Q_{tot}=\pi^{2}\ell^{2}\hat{r}_{+}^{2}(8\hat{r}_{+}^{4}+13\hat{r}_{+}^{2}+6),$
$\displaystyle
J=8\pi^{2}\ell^{3}\sqrt{1+\frac{1}{2\hat{r}_{+}^{2}}}\left(1+\frac{1}{\hat{r}_{+}^{2}}\right)\hat{r}_{+}^{6}.$
(C.1)

By taking the deviation $\hat{r}_{+}\to\hat{r}_{+}+\delta$, we obtain

$\displaystyle\delta
Q_{tot}=\frac{\sqrt{1+{1}/{2\hat{r}_{+}^{2}}}}{\ell}\delta
J=\Omega_{h}\,\delta J,$ (C.2)

where $\Omega_{h}=\Omega(1)$ is the angular velocity of the rotating Myers-
Perry black hole. Since the temperature of the extremal Myers-Perry AdS black
hole is zero, this is the first law of the extremal Myers-Perry AdS black
hole.

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# Computational behavior recognition in child and adolescent psychiatry : A
statistical and machine learning analysis plan

Nicole N. Lønfeldt * Child and Adolescent Mental Health Center, Copenhagen
University Hospital – Mental Health Services CPH, Hellerup, Denmark  Flavia D.
Frumosu * Department of Applied Mathematics and Computer Science, Technical
University of Denmark, Kongens Lyngby, Denmark A.-R. Cecilie Mora-Jensen
11footnotemark: 1 Nicklas Leander Lund 22footnotemark: 2 Sneha Das
22footnotemark: 2 A. Katrine Pagsberg 11footnotemark: 1 , , Department of
Clinical Medicine, Faculty of Health and Medical Sciences, University of
Copenhagen, Copenhagen, DenmarkDepartment of Clinical Biochemistry, Hospital
Glostrup - University Hospital, Glostrup, Denmark * Equal
contributor,Correspondence<EMAIL_ADDRESS>Line K. H.
Clemmensen 22footnotemark: 2

###### Abstract

Motivation
Behavioral observations are an important resource in the study and evaluation
of psychological phenomena, but it is costly, time-consuming, and susceptible
to bias. Thus, we aim to automate coding of human behavior for use in
psychotherapy and research with the help of artificial intelligence (AI)
tools. Here, we present an analysis plan.
Methods
Videos of a gold-standard semi-structured diagnostic interview of 25 youth
with obsessive-compulsive disorder (OCD) and 12 youth without a psychiatric
diagnosis (no-OCD) will be analyzed. Youth were between 8 and 17 years old.
Features from the videos will be extracted and used to compute ratings of
behavior, which will be compared to ratings of behavior produced by mental
health professionals trained to use a specific behavioral coding manual. We
will test the effect of OCD diagnosis on the computationally-derived behavior
ratings using multivariate analysis of variance (MANOVA). Using the generated
features, a binary classification model will be built and used to classify
OCD/no-OCD classes.
Discussion
Here, we present a pre-defined plan for how data will be pre-processed,
analyzed and presented in the publication of results and their interpretation.
A challenge for the proposed study is that the AI approach will attempt to
derive behavioral ratings based solely on vision, whereas humans use visual,
paralinguistic and linguistic cues to rate behavior. Another challenge will be
using machine learning models for body and facial movement detection trained
primarily on adults and not on children. If the AI tools show promising
results, this pre-registered analysis plan may help reduce interpretation
bias. Trial registration: ClinicalTrials.gov - H-18010607

_Keywords_ machine learning $\cdot$ visual signals from video data $\cdot$
human behavioral coding $\cdot$ children and adolescents $\cdot$ obsessive-
compulsive disorder

## Introduction

Observations of human behavior comprise an important clinical and research
tool within psychology and psychiatry. For example, motor unrest and slowness
are symptoms of psychiatric disorders [2] [13]. Facial expressions, eye
contact, proximity, touch, and gaze provide information about mental states
such as emotions, attention, and cognitive effort, as well as interpersonal
processes such as alliance, affection, and reciprocity [4][8][9][12]. While
behavioral coding bypasses the problem of self-report bias, disadvantages also
exist. Human behavioral coding is a costly, time-consuming task, and it is
susceptible to other types of bias. Thus, we aim to use AI tools that will
reduce human labor as well as increase the speed and reliability of coding
human behavior using landmark detection i.e., extracting features from videos.

## Methods

### Participants and setting

A total of 37 participants between the ages of 8 and 17 years are included in
this study. Participants include 25 youth with OCD and 12 youth without a
psychiatric diagnosis from an larger case-control study and randomized
clinical trial – TECTO [14]. All participants were screened for psychiatric
disorders by mental health professionals (psychologists, medical doctors with
psychiatric training and graduate-level psychology students). Diagnoses of
patients were established by psychiatrists or specialized clinical
psychologists.

### Measures

The Kiddie Schedule for Affective Disorders and Schizophrenia (K-SADS) is a
semi-structured interview used to screen for previous and current psychiatric
symptoms and diagnoses in children and adolescents between the ages of 6 and
18 [15]. Diagnostic categories have a subscale of items. Items are scored on a
3-point scale (1 = symptom not present, 2 = subclinical, 3 = clinical). If an
item is rated 3, guidelines instruct the interviewer to complete the relevant
disorder supplement to establish whether the full criteria of a diagnosis has
been met. Items can also be scored with zero to indicate that no information
is available. Items are rated for parent and child responses. The total item
score takes the parent and child response as well as the clinician’s expertise
into account. Before inclusion in TECTO, all participants and their parents
were interviewed with the K-SADS interview. Here, we used the clinician’s
total score for current symptoms. The scores from the depression and mania
chapters were used to establish whether the video footage is of participants
discussing current psychiatric symptoms. Some participants scored 2 or 3 for
the depression chapter, while all the participants scored 1 for the mania
chapter.

### Video data source

We will extract visual/motor features from videos of K-SADS interviews of
children. Figure 1 shows how videos were selected for this study. TECTO aims
to include about 30% fewer controls than patients and started including
patients before controls. Therefore, there are more patient videos available
than control videos. Videos containing a depression and mania chapter were
available for 64 participants. The videos were initially recorded with the
purpose of quality assurance and training of mental health professionals. As
they were not filmed for video processing purposes, the data quality is not
optimal. The videos are shot from different angles and contain a varying
number of people in the frame. Many videos are missing for several reasons.
First, some patients did not consent to being filmed before being included in
TECTO. Second, many videos were discarded to follow data protection policies
in the clinic before they could be moved to research folders. In total, 7
participants or 14 videos were removed due to missing data on clinical
measures of interest, overexposure of the video or other technical problems.
Videos of children who received a K-SADS score over one on the motor unrest
item of the attention deficit hyperactivity chapter, the
stereotyped/repetitive behavior item in the autism spectrum disorder chapter
or the motor tic item were excluded to avoid videos that may include increased
movements unrelated to outcomes of interest. In this initial work, we decided
not to exclude children with vocal tics, attention deficits not problems with
social reciprocity. These symptoms may be rated correctly by humans and the
computational models. Finally, it could be argued that we were not
conservative enough and that other symptoms should also be removed. For
example, vocal tics may increase the vocalization scores, but vocalization
should capture socially motivated speech and laughter. Other symptoms should
be captured by behavioral codes. A lack of social reciprocity may result in
decreased gaze, attention and positive affect, and attention deficits are
expected to be reflected in lower attention codes. Nonetheless, this study is
a first step and in future work we will test successful models in more
symptom-diverse populations. The second author (CM) with expertise in
psychopathology and conducting K-SADS interviews, identified the start and
stop times of the depression and mania chapters of the K-SADS interview
videos.

Figure 1: Video selection for the study

### Coding Interactive Behavior (CIB) adolescent version

The CIB is a global rating system for coding interactions between two partners
on a scale from one to five, in which half-points also can be assigned (e.g.,
1.5 or 3.5) [3]. One indicates that the behavior is low in intensity,
frequency, and duration or not present. Five indicates that the behavior is
high in intensity, frequency, and duration. Separate codes are assigned to the
parent, child and the dyad as a unit. While the original coding system was
developed for parent-child interactions, it has also been used for
interactions between children and adults other than the children’s parents
[10]. The adolescent version was designed for youth between the ages of 12 and
18, but many of the codes appear in versions developed for infancy to
adolescence for continuity [8]. Several individual codes can be averaged to
represent theoretically meaningful constructs such as social engagement [8].
The manual recommends a minimum of three minutes of interaction for global
scores and 10 second slices for micro-codes. In this work, we sought to
decrease human labor and machine processing time to test our method and thus,
applied global scores to 30-second clips. We were careful to choose codes
which involve behaviors or mental processes that could be observed in 30
second intervals and not complex codes that require more context. This study
will focus on child behavior and thus, will only use child codes. We chose
codes that are relevant for evaluating child engagement and emotional states.
These constructs are valuable in and of themselves, but we also believe they
will contribute to assessing therapeutic alliance, parent-child synchrony, and
child distress during exposure-based therapy in future work using other videos
(Clinicaltrials.gov - H-18010607). We will use the following seven child
codes, as they are relevant to interactions between children and mental health
professionals:

  1. 1.

Gaze \- child’s eyes look in direction of the interviewer.

  2. 2.

Attention \- child listens to the interviewer and responds to the interviewer
in relevant ways.

  3. 3.

Positive affect \- verbal and nonverbal expressions that indicate the child is
feeling relaxed and comfortable, including smiling, laughter, warm tone of
voice and relaxed body posture.

  4. 4.

Negative emotionality \- verbal and nonverbal expressions of negative emotions
(e.g. sadness and anger) including frowning, closed body posture (crossed
arms), yelling, and crying.

  5. 5.

Vocalization \- speech and other vocal noise produced with the goal of social
interaction. Frequent, consistent, and clear speech is scored high.

  6. 6.

Anxiety \- overt signs of nervousness such as frequent looks to the camera or
interviewer, enthusiasm that is incongruent with the context, long silences,
actions such as pulling hair, fidgeting, nail biting, emotional lability, or
statements expressing anxiety, fear or nervousness.

  7. 7.

Activity-level/arousal \- level of expressed energy in the way the child talks
and acts. Calm speech and movements receive a low score, whereas passionate,
frantic or excited speech and activity receives a high score. Energetic speech
can be marked by rapidity, loudness and excess.

Gaze, attention, positive affect, and vocalization can be combined into a
composite score representative of child social engagement [8].

### Human behavioral coding procedure

One of the first authors and the second author (NL and CM), a psychologist and
medical doctor specializing in psychiatry, respectively, will code the videos.
The first author was certified by the developer of the CIB and subsequently
trained the second author. They reached a rater agreement of Kappa = 89% for a
separate set of videos coded for training. They will hold regular meetings to
prevent coder drift. Coders will not code videos in which they are the
interviewers. Blinding human coders to diagnostic status cannot be achieved as
the discussion of symptoms is inherent to the interview. However, we attempt
to blind coders to the extent possible. Thus, participants received a new ID
number that does not reveal whether they are cases or controls and video clips
received a code that does not reveal whether they contain the depression or
mania chapter. CM has had contact with many of the participants in the TECTO
study and is not blind to much of their clinical information, which may bias
CM to score patients more negatively than controls or patients with higher
clinical severity more negatively than patients with lower clinical severity.
Thus, we estimated the sample size needed to test for a difference in means
between CM’s codes and NL’s codes at a significance level of 0.05 (two-
tailed), a power of 80%, a mean difference of 1 (the minimum allowed
difference for agreement) and a standard deviation (SD) of 0.9 [11] using
G*Power [6]. Results indicate that we would need a sample of 14 per rater to
detect a mean difference of one. If we assume a less conservative SD of 0.11
[7] and a mean difference of one, a sample size of four would be needed to
achieve a power of 80%. Thus, 14 videos (33%) will be coded by both human
coders to test bias in the unblinded coder and inter-rater reliability. For
the entire analysis, in total, 44 videos will be coded by each of the human
coders.

## Statistical Analysis Plan

### Processing pipeline

#### Pre-processing video and annotations

The two video clips from the depression and mania chapters from each
participant varied in length. The shortest video was approximately 40 seconds
in duration. We extracted 30 seconds from each video from the middle section
and coded this. The middle section was deemed the most representative of the
diagnostic chapter as we expect this to be where the child is more active,
talkative and expressive. The diagnostic chapters start with the interviewer
posing a question and by the end of the chapter the potential emotion-
provoking stimuli (a question about a psychiatric symptom) may be distant or
no longer novel and so children may be more relaxed.

#### Video features

To extract features from the videos, pre-trained machine learning models
focusing on landmark detection and facial expression recognition are used.
These models automatically track the position and movement of the head, hands,
fingers, face, and torso. A computational facial expression recognition model
is used to classify facial expressions from images of human faces into emotion
categories: anger, disgust, fear, happiness, sadness, surprise and neutral.
With such models, we can automatically track specific movements or positions
as well as emotions that humans use to assign behavioral codes. Table 1
presents an a priori description of the computationally-tracked actions that
we believe will contribute to specific CIB codes.

Table 1: Models used for body movement tracking and facial expression recognition CIB codes/constructs | Tracked actions | Class machine learning models |  |
---|---|---|---|---
1 Gaze |  | Head and eyes
---
positioning/angles
Gaze tracking |  |
2 Attention |  | Nodding head/turning head
---
from side to side (yes or no),
Limited movement (focus)
| Body landmark detection,
---
Face landmark detection
|
3 Positive affect | Facial expression for happiness | Facial Expression Recognition |  |
4 Negative emotionality |  | Facial expression
---
for sadness or anger
Facial Expression Recognition |  |
5 Vocalization |  | Openness of mouth,
---
Ratio between mouth corners
and lower, upper lips
Face landmark detection |  |
6 Anxiety |  | Activity frequency (high),
---
Facial expression for fear,
Facial expression for disgust
| Body landmark detection,
---
Face landmark detection
Facial Expression Recognition
|
7 Activity |  | Hands: away from the torso,
---
touching head, touching arm,
crossing arms
Torso: frequent movements
Head: up, down,
left, right movements
| Body landmark detection,
---
Face landmark detection
|

As the data were not acquired for the purpose of video analysis, the videos
vary in terms of characteristics since different cameras were used to film the
interviews. Thus, the computationally-derived ratings of behavior will
initially be represented in percentages, e.g., how many times a certain event
has been observed, divided by the number of frames from the video.
Subsequently, the percentages are transformed into scores ranging from 1 to 5
to match the human-rated CIB scores.

## Statistical analyses

### Statistical analysis I

First, we will estimate the agreement between the computationally-derived
scores and the corresponding human-derived, CIB scores using a Cohen’s kappa,
for easy comparison to human agreement, and a correlation plot for
visualization. In other words, the reliability between the human coders and
the features extracted by the machine learning models are checked. The seven
scores can be treated as composite scores and reduced to a smaller number of
scores: social engagement, negative emotionality, anxiety and activity.

### Statistical analysis II

Furthermore, we will test for significant differences at a level of $\alpha$ =
0.05 in the computationally-derived scores between the diagnostic groups (OCD
and no-OCD). For this analysis, the composite scores are used in a MANOVA. The
composite scores will also be used to test differences between the depression
and mania chapters of the OCD and no-OCD groups. If necessary, a correction of
the p-values will be performed using the Benjamini-Hochberg procedure controls
[5] with a false discovery rate (FDR) of 20%.

### Statistical analysis III

Using the generated features data, a binary classification model can be
trained in which the classes are OCD and no-OCD classes. Examples include tree
based models (e.g., random forest, extreme gradient boosting) and logistic
regression. These models can be used for predicting OCD and no-OCD classes on
unseen video data. A tree approach is preferred in this context as it also
produces feature importance as output, which can help to identify the best
features for capturing the OCD/no-OCD classes.
Regarding model validation strategies, depending on the selected model
different statistics will be reported, e.g., inspection of qq-plots of the
residuals, Out-of-Bag (OOB) error. The data is imbalanced as OCD patients
outnumber no-OCD participants. Thus, we consider using either oversampling or
undersampling strategies to deal with the class imbalance. As performance
metrics we will report the F1-Score and area under the curve (AUC) computed
from the receiver operator characteristic curve (ROC).
For estimating the generalization error of the models, we plan to use either
leave one out cross validation (LOOCV) or 5-fold cross validation, as deemed
necessary. If relevant, stratified cross validation will be used.

## Discussion

### Strengths

Here, we present a pre-defined plan for how data will be pre-processed,
analyzed and presented in the publication of results and their interpretation.
If the AI tools show promising results, this pre-registered analysis plan may
help reduce interpretation bias. Automation of coding human behavior will save
clinicians and researchers of human behavior time and resources. An additional
advantage is that coding will become more consistent and less prone to human
error and bias. A challenge for the proposed study is that the AI approach
will attempt to derive behavioral ratings based solely on vision, whereas
humans use visual, paralinguistic and linguistic cues to rate behavior. If
successful, these AI tools would transcend human abilities. In future work,
visual AI tools can be supplemented with audio tools to improve automatic
ratings. Another challenge will be using machine learning models for body and
facial movement detection trained primarily on adults and not on children.
Children and adults differ in the expression of mental states, i.e., attention
and emotions, and thus, more work is needed on younger populations [1].

### Limitations

In this initial study, we chose to remove subclinical symptoms of motor
unrest, stereotyped/repetitive behaviors and motor tics and motor unrest. The
reason behind this choice is to ease the comparison between the OCD/no-OCD
groups, as we have few subjects, we want to minimize unnecessary noise. This
decision may decrease the influence of bias of the human coder with knowledge
of the patients’ diagnoses as the other human coder without this knowledge may
code more similarly to the machine that also lacks this knowledge. In this
case, agreement between the human coders would decease and we may falsely
conclude that machine-human agreement is at least as high as human-human
agreement. This decision also resulted in removing many children from the
sample and will limit the generalizability of our findings.

## Conclusion

The current predefined analysis plan will help to limit bias for reported
results in a future publication. If the obtained results from this analysis
plan are promising, we will be one step closer to automated human behavioural
coding within the context of child psychiatry.

## Competing interests

The authors declare that there are no competing interests for this work.

## Acknowledgements

The current work is funded by the Novo Nordisk Foundation (grant number:
NNF19OC0056795). The authors would also like to thank research assistant
Anders Buch Thuesen for his work while temporarily supporting the project.

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