††institutetext: International Centre for Theoretical Physics Asia-Pacific,
University of Chinese Academy of Sciences, 100190 Beijing, China

# Gravitational Waves of Non-Extremal Kerr Black Holes from Conformal Symmetry

Jun Nian  and Weijie Tian<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In the low-energy limit, the near region of a generic Kerr black hole has been
conjectured to be holographically dual to a two-dimensional conformal field
theory. In this paper, we consider a test object in a circular orbit near the
horizon of a non-extremal Kerr black hole. We first couple it to a massless
scalar field. The resulting scalar radiation at the horizon is computed from
the perspectives of gravity and dual conformal field theory. Considering the
influence of nonzero temperature, the agreement of both computations is found.
Then, we generalize the analysis to the gravitational radiation and find
agreement again. The agreement supports the conjectured holography and
provides a potential theoretical tool for gravitational wave computations.

##  1 Introduction

Black holes are so fascinating that a lot of effort has been put into
understanding them. One of the most significant breakthroughs is the discovery
of the Bekenstein-Hawking entropy, which was inspired by the analogy between
the laws of black hole dynamics and the laws of thermodynamics
Christodoulou:1970wf ; Christodoulou:1971pcn ; Penrose:1971uk ; Hawking:1971tu
; WOS:A1972N081300008 ; Bekenstein:1973ur ; Bardeen:1973gs ; Bekenstein:1974ax
; Hawking:1974rv ; Hawking:1975vcx . A direct way to explain the microscopic
origin of the black hole entropy is the quantum theory of gravity, such as
string theory, and it succeeded for some classes of black holes
Strominger:1996sh ; Callan:1996dv . However, a universal statistical
mechanical interpretation of the Bekenstein-Hawking entropy is still missing.
A promising way for this is the holographic principle tHooft:1993dmi ;
Susskind:1994vu ; Maldacena:1997re , of which the validity is independent of
the details of quantum gravity. For example, for black holes whose near-
horizon regions are locally three-dimensional anti-de-Sitter
($\mathrm{AdS_{3}}$) space, the derivation of the entropy through holographic
duality applies to any consistent, unitary quantum theory of gravity
Strominger:1997eq .

For black holes existing in the universe, such as Kerr black holes, things
become a little different. The simplest case might be the extreme Kerr black
hole whose mass $M$ and angular momentum $J$ satisfy $J=GM^{2}$. For an
extreme Kerr black hole, there is a near-horizon region, known as the near-
horizon extreme Kerr (NHEK) geometry Bardeen:1999px , in which the asymptotic
symmetry generators form the Virasoro algebra under some consistent boundary
conditions Guica:2008mu . This motivated the Kerr/CFT correspondence
Guica:2008mu , which posits that the NHEK geometry is dual to a two-
dimensional conformal field theory (CFT). The conjecture was supported by the
facts that the CFT microstate degeneracy inferred from the Cardy formula
reproduces the black hole entropy Guica:2008mu  and the CFT correlators agree
precisely with the scattering amplitudes of the Kerr black hole
Bredberg:2009pv . The Kerr/CFT correspondence can be exploited to compute
gravitational waves. For an extreme or near-extremal Kerr black hole, whether
the gravitational radiation is emitted by a star in a circular orbit near the
horizon or by a star plunging into the black hole, agreements were found
between the gravity and the CFT analyses Porfyriadis:2014fja ; Hadar:2014dpa ;
Hadar:2015xpa . In fact, through conformal symmetry, all corotating equatorial
orbits near the horizon are related, which provides convenience for analytical
computations of gravitational waves from plunges into supermassive near-
extremal Kerr black holes Compere:2017hsi .

However, for non-extremal Kerr black holes, the NHEK geometry disappears, and
the near-horizon conformal symmetry is not manifest. Fortunately, the solution
space of the wave equation still has a conformal symmetry called the hidden
conformal symmetry Castro:2010fd . In the low-energy limit $\omega M\ll 1$,
where $\omega$ is the frequency of the propagating field, the spacetime can be
divided into a near region ($r\ll 1/\omega$) and a far region ($r\gg M$),
which have an overlap. The hidden conformal symmetry acts on the near-region
wave equation at low frequencies. Note that the near region defined above is
not the near-horizon region in the definition of the NHEK geometry. With the
hidden conformal symmetry, the Bekenstein-Hawking entropy and the scattering
amplitudes of the non-extremal Kerr black hole can be reproduced Castro:2010fd
, which corroborates the generalized conjecture that the near region of a
generic non-extremal Kerr black hole is dual to a two-dimensional CFT.
Complementarily, the hidden conformal symmetry of a non-extremal Kerr black
hole can be explicitly realized using the covariant phase space formalism
Haco:2018ske .

In this paper, we attempt to apply the hidden conformal symmetry to
gravitational wave computations of a non-extremal Kerr black hole. For
simplicity, we consider only the case of a star in a circular orbit near the
horizon. In the low-energy and near-region limits, the wave equation can be
solved by hypergeometric functions, and the radiation flux at the horizon is
computed. In the dual CFT, the orbiting star leads to a deformation of the CFT
and transitions out of the thermal state. We use Fermi’s golden rule to
compute the transition rate, which is dual to the radiation flux at the
horizon. Through explicit computations, we find agreement between results from
the gravity and the CFT analyses, which provides new evidence for the dual
conformal symmetry.

This work also paves the way for the analytical study of gravitational waves
through the hidden conformal symmetry. Since LIGO first detected a
gravitational wave signal from a binary black hole merger
LIGOScientific:2016aoc , gravitational wave observations have developed
greatly. Astronomical observations show that many Kerr black holes spinning
rapidly or slowly exist in our universe Reynolds:2020jwt . If there is a star
orbiting near the horizon of a supermassive Kerr black hole ($\geq 10^{5}$
solar masses), the emitted gravitational waves are potentially observable and
are the primary goal of space-based gravitational wave detections. The results
of this paper suggest that the hidden conformal symmetry might benefit
gravitational wave computations.

This paper is organized as follows. Sec. 2 contains a brief review of the Kerr
black hole metric and the hidden conformal symmetry. In Sec. 3, we couple a
massless scalar field to the orbiting star as a warmup. The scalar radiation
is computed from the gravity and the dual CFT perspectives, and the results
are compared. We also reattach the far region and compute the scalar radiation
at the future null infinity. In Sec. 4, we adopt the Newman-Penrose formalism
and generalize the results of the scalar field to the gravitational field
case. A summary and some discussions are included in Sec. 5.

Throughout this paper we use units such that $G=c=\hbar=1$.

##  2 Review of Kerr black hole and hidden conformal symmetry

In this section, we briefly review the Kerr black hole and its dual CFT. For a
four-dimensional asymptotically flat Kerr black hole, the metric in Boyer-
Lindquist coordinates is

$ds^{2}=-\frac{\Delta}{\rho^{2}}(dt-a\sin^{2}\theta
d\phi)^{2}+\frac{\sin^{2}\theta}{\rho^{2}}\left((r^{2}+a^{2})d\phi-
adt\right)^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2},$ (1)

with

$\Delta=r^{2}-2Mr+a^{2},\ \rho^{2}=r^{2}+a^{2}\cos^{2}\theta,$

where $M$ is the mass, and $J=aM$ is the angular momentum. The inner and the
outer horizons of the Kerr Black hole are located at

$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}.$ (2)

The Klein-Gordon equation for a massless scalar field $\Psi$ is

$\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Psi)=0.$
(3)

In the Kerr metric, expanding $\Psi$ in modes

$\Psi(t,r,\theta,\phi)=e^{-i\omega t+im\phi}R(r)S(\theta),$ (4)

Eq. (3) can be separated into

$\left[\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta})-\frac{m^{2}}{\sin^{2}\theta}+\omega^{2}a^{2}\cos^{2}\theta\right]S(\theta)=-K_{l}\,S(\theta),$
(5)

and

$\left[\partial_{r}(\Delta\partial_{r})+\frac{(2Mr_{+}\omega-
am)^{2}}{(r-r_{+})(r_{+}-r_{-})}-\frac{(2Mr_{-}\omega-
am)^{2}}{(r-r_{-})(r_{+}-r_{-})}+\left(r^{2}+2M(r+2M)\right)\omega^{2}\right]R(r)=K_{l}\,R(r),$
(6)

where $K_{l}$ is the separation constant.

To show the hidden conformal symmetry of the non-extremal Kerr black hole
Castro:2010fd , we consider the low-energy limit $\omega M\ll 1$, which in
practice is satisfied for the low-frequency gravitational waves ($\lesssim$
mHz) emitted by supermassive black holes ($\geq 10^{5}$ solar masses). In this
case, Eq. (5) reduces to

$\left[\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta})-\frac{m^{2}}{\sin^{2}\theta}\right]S(\theta)=-K_{l}\,S(\theta),$
(7)

which is the standard Laplace equation on the 2-sphere with $K_{l}=l(l+1)$,
and Eq. (6) in the near region $r\ll 1/\omega$ becomes

$\left[\partial_{r}(\Delta\partial_{r})+\frac{(2Mr_{+}\omega-
am)^{2}}{(r-r_{+})(r_{+}-r_{-})}-\frac{(2Mr_{-}\omega-
am)^{2}}{(r-r_{-})(r_{+}-r_{-})}\right]R(r)=K_{l}\,R(r),$ (8)

which can be viewed as the eigenvalue equation of the Casimir operators of the
$SL(2,R)_{L}\times SL(2,R)_{R}$ conformal symmetry. The conformal weights of
the field $\Psi$ are

$(h_{L},h_{R})=(l,l).$ (9)

The $2\pi$ periodic identification of the angular coordinate $\phi$ requires
the ground state of the dual CFT to be at temperatures

$T_{L}=\frac{r_{+}+r_{-}}{4\pi a},\quad T_{R}=\frac{r_{+}-r_{-}}{4\pi a}.$
(10)

The conformal symmetry above has been supported by the black hole entropy and
scattering amplitudes Castro:2010fd  and confirmed using the covariant phase
space formalism Haco:2018ske .

##  3 Scalar radiation from a star orbiting in the near region

In this section, we view a star in a circular orbit in the near region as a
test particle and couple it to a massless scalar field. The resulting scalar
radiation at the horizon is computed using the gravity and the CFT analyses,
and two computations are shown to agree. Finally, we reattach the far region
and compute the radiation power at the future null infinity.

###  3.1 Gravity analysis

Consider a star in a circular orbit in the near region. The radius of the
orbit is $r_{0}$, which is required not to be much larger than $M$. The
corresponding geodesic $x^{\mu}_{*}(\tau)$ satisfies

$\displaystyle x_{*}^{r}(\tau)$ $\displaystyle=r_{0},$ (11) $\displaystyle
x_{*}^{\theta}(\tau)$ $\displaystyle=\pi/2,$ $\displaystyle x_{*}^{t}(\tau)$
$\displaystyle=\frac{r_{0}^{\frac{3}{2}}+aM^{\frac{1}{2}}}{r_{0}^{\frac{3}{4}}(r_{0}^{\frac{3}{2}}-3Mr_{0}^{\frac{1}{2}}+2aM^{\frac{1}{2}})^{\frac{1}{2}}}\tau,$
$\displaystyle x_{*}^{\phi}(\tau)$
$\displaystyle=\frac{M^{\frac{1}{2}}}{r_{0}^{\frac{3}{4}}(r_{0}^{\frac{3}{2}}-3Mr_{0}^{\frac{1}{2}}+2aM^{\frac{1}{2}})^{\frac{1}{2}}}\tau+\phi_{0},$

where $\tau$ is the proper time.

Coupling the star to a massless scalar field $\Psi$, the action of this system
can be written as

$S=\frac{1}{2}\int
d^{4}x\sqrt{-g}\left[\left(\partial\Psi(x)\right)^{2}+8\pi\lambda\Psi(x)\mathcal{S}(x)\right],$
(12)

where $\lambda$ is a coupling constant, and

$\mathcal{S}(x)=\int d\tau(-g)^{-1/2}\delta^{(4)}(x-x_{*}(\tau)).$

Then, the scalar wave equation becomes

$\square\Psi=4\pi\mathcal{T},$ (13)

where the source term is (setting $\phi_{0}=0$)

$\mathcal{T}=-\frac{\lambda(r_{0}^{\frac{3}{2}}-3Mr_{0}^{\frac{1}{2}}+2aM^{\frac{1}{2}})^{\frac{1}{2}}}{r_{0}^{\frac{5}{4}}(r_{0}^{\frac{3}{2}}+aM^{\frac{1}{2}})}\delta(r-r_{0})\,\delta(\theta-\pi/2)\,\delta\left(\phi-\frac{M^{\frac{1}{2}}}{r_{0}^{\frac{3}{2}}+aM^{\frac{1}{2}}}t\right).$
(14)

Using the mode expansions

$\displaystyle\Psi=$ $\displaystyle\sum_{l,m}e^{im\phi-i\omega
t}S_{l}(\theta)R_{lm}(r),$ (15) $\displaystyle\mathcal{T}=$
$\displaystyle\frac{1}{4\pi\rho^{2}}\sum_{l,m}e^{im\phi-i\omega
t}S_{l}(\theta)T_{lm}(r),$

with the following relation due to the symmetry of the source term

$\omega=\frac{mM^{\frac{1}{2}}}{r_{0}^{\frac{3}{2}}+aM^{\frac{1}{2}}},$ (16)

Eq. (13) in the low-energy and the near-region limits can be separated into

$\left[\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta})-\frac{m^{2}}{\sin^{2}\theta}\right]S_{l}=-K_{l}\,S_{l},$
(17)

and

$\left[\partial_{r}(\Delta\partial_{r})+\frac{(2Mr_{+}\omega-
am)^{2}}{(r-r_{+})(r_{+}-r_{-})}-\frac{(2Mr_{-}\omega-
am)^{2}}{(r-r_{-})(r_{+}-r_{-})}\right]R_{lm}=K_{l}\,R_{lm}+T_{lm}.$ (18)

Since $e^{im\phi}S_{l}(\theta)$ are spherical harmonics, we obtain

$\displaystyle T_{lm}$ $\displaystyle=\frac{1}{2\pi}\int\sin\theta d\theta
d\left(\phi-\frac{M^{\frac{1}{2}}}{r_{0}^{\frac{3}{2}}+aM^{\frac{1}{2}}}t\right)4\pi\rho^{2}\mathcal{T}e^{-im\phi+i\omega
t}S_{l}(\theta)$ (19) $\displaystyle=-\frac{2\lambda
r_{0}^{\frac{3}{4}}(r_{0}^{\frac{3}{2}}-3Mr_{0}^{\frac{1}{2}}+2aM^{\frac{1}{2}})^{\frac{1}{2}}}{r_{0}^{\frac{3}{2}}+aM^{\frac{1}{2}}}\delta(r-r_{0})S_{l}(\pi/2).$

Eq. (18) is a Sturm-Liouville problem, and we will show that it can be solved
via the standard Green function method for given boundary conditions. Firstly,
two linearly independent solutions to the homogeneous equation (with
$T_{lm}=0$) can be written as

$\displaystyle R_{lm}^{(1)}=$
$\displaystyle\left(\frac{r-r_{+}}{r-r_{-}}\right)^{-i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}}\left(1-\frac{r-r_{+}}{r-r_{-}}\right)^{l+1}$ (20)
$\displaystyle\times\
_{2}F_{1}\left(1+l-i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}},1+l-i2M\omega;1-2i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}};\frac{r-r_{+}}{r-r_{-}}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{r-r_{+}}{r-r_{-}}\right)^{-i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}}\left(1-\frac{r-r_{+}}{r-r_{-}}\right)^{l+1}$
$\displaystyle\times\bigg{[}A\
_{2}F_{1}\left(1+l-i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}},1+l-i2M\omega;2l+2;1-\frac{r-r_{+}}{r-r_{-}}\right)$
$\displaystyle+B\left(1-\frac{r-r_{+}}{r-r_{-}}\right)^{-2l-1}\
_{2}F_{1}\left(-l-i2M\omega,-l-i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}};-2l;1-\frac{r-r_{+}}{r-r_{-}}\right)\bigg{]},$
$\displaystyle R_{lm}^{(2)}=$
$\displaystyle\left(\frac{r-r_{+}}{r-r_{-}}\right)^{i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}}\left(1-\frac{r-r_{+}}{r-r_{-}}\right)^{l+1}$ (21)
$\displaystyle\times\
_{2}F_{1}\left(1+l+i2M\omega,1+l+i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}};1+2i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}};\frac{r-r_{+}}{r-r_{-}}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{r-r_{+}}{r-r_{-}}\right)^{i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}}\left(1-\frac{r-r_{+}}{r-r_{-}}\right)^{l+1}$
$\displaystyle\times\bigg{[}C\
_{2}F_{1}\left(1+l+i2M\omega,1+l+i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}};2l+2;1-\frac{r-r_{+}}{r-r_{-}}\right)$
$\displaystyle+D\left(1-\frac{r-r_{+}}{r-r_{-}}\right)^{-2l-1}\
_{2}F_{1}\left(-l+i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}},-l+i2M\omega;-2l;1-\frac{r-r_{+}}{r-r_{-}}\right)\bigg{]},$

where $\ {}_{2}F_{1}(\alpha,\beta;\gamma;z)$ is the hypergeometric function,
and

$\displaystyle A$ $\displaystyle=\frac{\Gamma\left(1-2i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}\right)\,\Gamma(-2l-1)}{\Gamma(-l-i2M\omega)\,\Gamma\left(-l-i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}}\right)},$
(22) $\displaystyle B$
$\displaystyle=\frac{\Gamma\left(1-2i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}\right)\,\Gamma(2l+1)}{\Gamma\left(1+l-i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}}\right)\,\Gamma(1+l-i2M\omega)},$
(23) $\displaystyle C$
$\displaystyle=\frac{\Gamma\left(1+2i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}\right)\,\Gamma(-2l-1)}{\Gamma\left(-l+i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}}\right)\,\Gamma(-l+i2M\omega)},$
(24) $\displaystyle D$
$\displaystyle=\frac{\Gamma\left(1+2i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}\right)\,\Gamma(2l+1)}{\Gamma(1+l+i2M\omega)\,\Gamma\left(1+l+i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}}\right)}.$
(25)

Here, we have used the following identity for the convenience of obtaining the
large-$r$ behaviors of the solutions

$\displaystyle\ {}_{2}F_{1}(\alpha,\beta;\gamma;z)=$
$\displaystyle\frac{\Gamma(\gamma)\,\Gamma(\gamma-\alpha-\beta)}{\Gamma(\gamma-\alpha)\,\Gamma(\gamma-\beta)}\
_{2}F_{1}(\alpha,\beta;\alpha+\beta-\gamma+1;1-z)$ (26)
$\displaystyle+(1-z)^{\gamma-\alpha-\beta}\frac{\Gamma(\gamma)\,\Gamma(\alpha+\beta-\gamma)}{\Gamma(\alpha)\,\Gamma(\beta)}\
_{2}F_{1}(\gamma-\alpha,\gamma-\beta;\gamma-\alpha-\beta+1;1-z).$

The asymptotic behaviors of the solutions are

$\displaystyle R_{lm}^{(1)}$
$\displaystyle\rightarrow\left(\frac{r-r_{+}}{r_{+}-r_{-}}\right)^{-i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}}\,,$ $\displaystyle r\rightarrow r_{+},$ (27)
$\displaystyle\rightarrow
A(r_{+}-r_{-})^{l+1}\,r^{-l-1}+B(r_{+}-r_{-})^{-l}\,r^{l}\,,$ $\displaystyle
r\rightarrow\infty,$ $\displaystyle R_{lm}^{(2)}$
$\displaystyle\rightarrow\left(\frac{r-r_{+}}{r_{+}-r_{-}}\right)^{i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}}\,,$ $\displaystyle r\rightarrow r_{+},$ (28)
$\displaystyle\rightarrow
C(r_{+}-r_{-})^{l+1}\,r^{-l-1}+D(r_{+}-r_{-})^{-l}\,r^{l}\,,$ $\displaystyle
r\rightarrow\infty.$

Notice that at the horizon $R_{lm}^{(1)}$ is purely ingoing, while
$R_{lm}^{(2)}$ is purely outgoing. Both solutions have one rise and one
falloff at infinity. Let us define

$R_{lm}^{(3)}\equiv DR_{lm}^{(1)}-BR_{lm}^{(2)},$ (29)

whose asymptotic behaviors are

$\displaystyle R_{lm}^{(3)}\rightarrow$
$\displaystyle\,\,D\left(\frac{r-r_{+}}{r_{+}-r_{-}}\right)^{-i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}}-B\left(\frac{r-r_{+}}{r_{+}-r_{-}}\right)^{i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}}\,,$ $\displaystyle r\rightarrow r_{+},$ (30)
$\displaystyle\rightarrow$ $\displaystyle\,\,(DA-
BC)(r_{+}-r_{-})^{l+1}\,r^{-l-1}\,,$ $\displaystyle r\rightarrow\infty.$

We see that $R_{lm}^{(3)}$ has only one falloff at infinity, which we will
refer to as the Neumann boundary condition as in Porfyriadis:2014fja . In
contrast, the Dirichlet boundary condition has only one rise at infinity.
Then, the solution to Eq. (18), which is purely ingoing at the horizon and
obeys the Neumann boundary condition at infinity, is given by

$R_{lm}(r)=\frac{1}{W}\left(X\Theta(r_{0}-r)R_{lm}^{(1)}(r)+Z\Theta(r-r_{0})R_{lm}^{(3)}(r)\right),$
(31)

where $W$ is the $r$-independent Wronskian

$W=\Delta\Bigl{(}R_{lm}^{(1)}(r)\partial_{r}R_{lm}^{(3)}(r)-R_{lm}^{(3)}(r)\partial_{r}R_{lm}^{(1)}(r)\Bigr{)},$
(32)

and

$\displaystyle X$ $\displaystyle=-\frac{2\lambda
r_{0}^{\frac{3}{4}}(r_{0}^{\frac{3}{2}}-3Mr_{0}^{\frac{1}{2}}+2aM^{\frac{1}{2}})^{\frac{1}{2}}}{r_{0}^{\frac{3}{2}}+aM^{\frac{1}{2}}}S_{l}(\pi/2)R_{lm}^{(3)}(r_{0}),$
(33) $\displaystyle Z$ $\displaystyle=-\frac{2\lambda
r_{0}^{\frac{3}{4}}(r_{0}^{\frac{3}{2}}-3Mr_{0}^{\frac{1}{2}}+2aM^{\frac{1}{2}})^{\frac{1}{2}}}{r_{0}^{\frac{3}{2}}+aM^{\frac{1}{2}}}S_{l}(\pi/2)R_{lm}^{(1)}(r_{0}).$

Thus, we have

$\displaystyle\Psi(r\rightarrow r_{+})$
$\displaystyle=\sum_{l,m}e^{im\phi-i\omega
t}S_{l}(\theta)\frac{X}{W}\left(\frac{r-r_{+}}{r_{+}-r_{-}}\right)^{-i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}},$ (34) $\displaystyle\Psi(r\rightarrow\infty)$
$\displaystyle=\sum_{l,m}e^{im\phi-i\omega t}S_{l}(\theta)\frac{Z}{W}(DA-
BC)(r_{+}-r_{-})^{l+1}r^{-l-1}.$

The Klein-Gordon particle number flux, which is defined as

$\mathcal{F}\equiv\int\sqrt{-g}J^{r}d\theta d\phi\,,\quad\text{with
}J^{\mu}\equiv\frac{i}{8\pi}(\Psi^{*}\nabla^{\mu}\Psi-\Psi\nabla^{\mu}\Psi^{*}),$
(35)

vanishes at infinity as required by the Neumann boundary condition. The flux
at the horizon is

$\mathcal{F}_{lm}=\frac{1}{2}(2Mr_{+}\omega-am)\frac{|X|^{2}}{|W|^{2}},$ (36)

for $m>0$.

###  3.2 CFT analysis

As analyzed in Porfyriadis:2014fja , the near region with an orbiting star is
dual to a deformed CFT. To compute the deformation explicitly, we need to
extend the solution in the region $r<r_{0}$ to the boundary at infinity, where
the extended solution will contain both the Neumann and the Dirichlet modes
satisfying the corresponding boundary conditions respectively. Since we chose
the Neumann boundary condition at infinity in the previous section, the
Neumann modes will play the role of the CFT operators, while the Dirichlet
modes can be viewed as the deformation.

The action of the deformed CFT is

$S=S_{CFT}+\sum_{l}\int
dt^{+}dt^{-}J_{l}(t^{+},t^{-})\mathcal{O}_{l}(t^{+},t^{-}),$ (37)

where $S_{CFT}$ is the original CFT action, and $\mathcal{O}_{l}$ are the CFT
operators with the left and the right weights $(l,\,l)$. A comparison between
conformal coordinates of the Kerr metric Castro:2010fd  and those of
$\mathrm{AdS}_{3}$ Maldacena:1998bw  implies

$t^{+}=\phi-\frac{a}{2M^{2}}t,\quad t^{-}=\phi.$ (38)

Considering the symmetry of the source term $\mathcal{T}$, we expand the
deformation

$J_{l}(t^{+},t^{-})=\sum_{m}J_{lm}e^{i\omega_{L}t^{+}-i\omega_{R}t^{-}},$ (39)

where

$\omega_{L}=\frac{2M^{2}\omega}{a},\ \omega_{R}=\frac{2M^{2}\omega-ma}{a}.$
(40)

As discussed above, the coefficients $J_{lm}$ are determined by the solution
extended from $r<r_{0}$ to $r\rightarrow\infty$:

$R^{ext}_{lm}(r)=\frac{X}{W}R_{lm}^{(1)}(r)\rightarrow\frac{X}{W}\left(A(r_{+}-r_{-})^{l+1}r^{-l-1}+B(r_{+}-r_{-})^{-l}r^{l}\right)\,,\quad
r\rightarrow\infty.$ (41)

We read off $J_{lm}$ from the Dirichlet modes

$J_{lm}=\frac{X}{W}B.$ (42)

Thus, the deformed action can be written as

$S=S_{CFT}+\sum_{l,m}\int
dt^{+}dt^{-}J_{lm}e^{i\omega_{L}t^{+}-i\omega_{R}t^{-}}\mathcal{O}_{l}(t^{+},t^{-}).$
(43)

The time-dependent perturbation pumps both the left and the right energies
into the CFT. Fermi’s golden rule gives the transition rate out of the thermal
state Maldacena:1997ih ; Gubser:1997cm

$\mathcal{R}=2\pi\sum_{l,m}|J_{lm}|^{2}\int
dt^{+}dt^{-}e^{-i\frac{2M^{2}\omega}{a}t^{+}+i\frac{2M^{2}\omega-
ma}{a}t^{-}}G(t^{+},t^{-}),$ (44)

where
$G(t^{+},t^{-})=\langle\mathcal{O}^{\dagger}(t^{+},t^{-})\mathcal{O}(0,0)\rangle_{T_{L},T_{R}}$
is the finite-temperature two-point function of the CFT. Using
$T_{L}=(r_{+}+r_{-})/4\pi a$ and $T_{R}=(r_{+}-r_{-})/4\pi a$, with the
appropriate $i\epsilon$ prescription, we have Bredberg:2009pv ;
Porfyriadis:2014fja ; Hadar:2014dpa ; Hadar:2015xpa

$\displaystyle\mathcal{R}_{lm}=$
$\displaystyle\,\,4\pi^{2}C_{\mathcal{O}}^{2}\left(\frac{M}{a}\right)^{2l-1}\left(\frac{\sqrt{M^{2}-a^{2}}}{a}\right)^{2l-1}(2Mr_{+}\omega-
am)(r_{+}-r_{-})^{-1}\frac{|X|^{2}}{|W|^{2}}$ (45)
$\displaystyle\,\,\times\frac{e^{\pi\frac{4M\omega
r_{+}-2am}{r_{+}-r_{-}}}}{\sinh\pi\frac{4Mr_{+}\omega-2am}{r_{+}-r_{-}}}\frac{4l^{2}}{\left(l^{2}+\left(\frac{4M^{2}\omega-2am}{r_{+}-r_{-}}\right)^{2}\right)\,\left(l^{2}+(2M\omega)^{2}\right)}$
$\displaystyle\approx$
$\displaystyle\,\,\frac{16\pi^{2}}{al^{2}}C_{\mathcal{O}}^{2}\left(\frac{M}{a}\right)^{2l-1}\left(\frac{\sqrt{M^{2}-a^{2}}}{a}\right)^{2l-2}(2Mr_{+}\omega-
am)\frac{|X|^{2}}{|W|^{2}}$
$\displaystyle\,\,\times\frac{1}{1-e^{-2\pi\frac{4Mr_{+}\omega-2am}{r_{+}-r_{-}}}},$

where $C_{\mathcal{O}}^{2}$ is an undetermined operator normalization, and in
the last step, we have dropped some subleading terms in the limit $\omega M\ll
1$.

###  3.3 Comparison

Unlike the extremal case, the dual CFT we deal with here is at nonzero
temperature. When we compute the transition rate, the effect of stimulated
emission should be subtracted. The net result is Gubser:1997cm

$\mathcal{R}_{lm}^{net}=\mathcal{R}_{lm}(1-e^{-\beta_{L}\cdot\omega_{L}-\beta_{R}\cdot\omega_{R}}),$
(46)

where

$\beta_{L}=1/T_{L},\quad\beta_{R}=1/T_{R}.$ (47)

Comparing Eq. (36) with Eq. (46), we see that by choosing the normalization

$C_{\mathcal{O}}=\frac{\sqrt{a}l}{4\sqrt{2}\pi}\Bigl{(}\frac{a}{M}\Bigr{)}^{l-1/2}\Bigl{(}\frac{a}{\sqrt{M^{2}-a^{2}}}\Bigr{)}^{l-1},$
(48)

which is independent of $m$ and $\omega$, we have exactly

$\mathcal{R}_{lm}^{net}=\mathcal{F}_{lm},$ (49)

as predicted by the holographic dictionary.

After finding the agreement between the gravity and the CFT computations, let
us discuss the extremal limit $a=M$. In this limit, it is tempting to suspect
that the hidden conformal symmetry reduces to the conformal symmetry of the
NHEK geometry. However, it should be emphasized that the near region for the
non-extremal Kerr black hole is defined by $r-r_{+}\ll 1/\omega$. In contrast,
the near-horizon region of the extremal case describes the part $r-r_{+}\ll
M$, which is stricter in the low-energy limit. Thus, the hidden conformal
symmetry is not equal to the NHEK conformal symmetry even if $a=M$.

In fact, the results obtained in this paper do not work well when taking
$a\rightarrow M$ and $(r-r_{+})/M\rightarrow 0$. The modes that can propagate
in the NHEK geometry are those with frequencies very close to the superradiant
bound $\omega=m/2M$. However, the low-energy limit requires $\omega M\ll 1$,
which means $m\ll 1$ near the superradiant bound and contradicts the fact that
$m$ is a positive integer. Thus, we do not expect that non-extremal results in
this paper can be extended to the extremal limit.

###  3.4 Reattaching the far region

In the previous subsections, we impose the Neumann boundary condition at the
asymptotic boundary of the near region. Here, we reattach the far region and
compute the scalar radiation flux at the future null infinity.

Using the mode expansions (15), the radial part of Eq. (13) can be written as

$\partial_{r}(\Delta\partial_{r}R_{lm})+\left(\frac{H^{2}}{\Delta}+2am\omega-a^{2}\omega^{2}\right)R_{lm}=K_{l}R_{lm}+T_{lm},$
(50)

where $H=(r^{2}+a^{2})\omega-am$. Considering the low-energy limit $\omega
M\ll 1$ and the far-region limit $r\gg M$, we have

$\partial_{r}(r^{2}\partial_{r}R_{lm})+(r^{2}\omega^{2}-l(l+1))R_{lm}=0.$ (51)

Two linearly independent solutions are

$\frac{1}{\sqrt{\omega r}}J_{l+1/2}(\omega r),\quad\frac{1}{\sqrt{\omega
r}}J_{-l-1/2}(\omega r),$ (52)

with the asymptotic behaviors

$\displaystyle\frac{1}{\sqrt{\omega r}}J_{l+1/2}(\omega r)$
$\displaystyle\rightarrow\omega^{l}r^{l}\frac{2^{-l-1/2}}{\Gamma(l+3/2)}\,,$
$\displaystyle r\rightarrow 0,$ (53)
$\displaystyle\rightarrow\sqrt{\frac{2}{\pi}}\frac{1}{\omega r}\sin(\omega
r-\pi l/2)\,,$ $\displaystyle r\rightarrow\infty,$
$\displaystyle\frac{1}{\sqrt{\omega r}}J_{-l-1/2}(\omega r)$
$\displaystyle\rightarrow\omega^{-l-1}r^{-l-1}\frac{2^{l+1/2}}{\Gamma(-l+1/2)}\,,$
$\displaystyle r\rightarrow 0,$ (54)
$\displaystyle\rightarrow\sqrt{\frac{2}{\pi}}\frac{1}{\omega r}\cos(\omega
r+\pi l/2)\,,$ $\displaystyle r\rightarrow\infty.$

The solution of Eq. (51) that is purely outgoing at the null infinity can be
written as

$R^{far}_{lm}=\frac{P}{\sqrt{\omega r}}J_{l+1/2}(\omega
r)+\frac{Q}{\sqrt{\omega r}}J_{-l-1/2}(\omega r),$ (55)

where

$\frac{P}{Q}=ie^{-i\pi l}.$ (56)

Since the far and the near regions overlap in the region $M\ll r\ll 1/\omega$,
we can match the inner part of the far-region solution with the outer part of
the near-region solution. The near-region solution that is purely ingoing at
the horizon satisfies the form of Eq. (31) plus $c_{0}R_{lm}^{(1)}$, where
$c_{0}$ is a constant. Then, we can fix

$\displaystyle Q=$ $\displaystyle\frac{Z}{W}(DA-BC)(r_{+}-r_{-})^{l+1}$ (57)
$\displaystyle\times\left[\omega^{-l-1}\frac{2^{l+1/2}}{\Gamma(-l+1/2)}-ie^{-i\pi
l}\omega^{l}(r_{+}-r_{-})^{2l+1}\frac{2^{-l-1/2}A}{\Gamma(l+3/2)B}\right]^{-1}.$

Thus,

$\Psi(r\rightarrow\infty)=\sum_{l,m}e^{im\phi-i\omega
t}S_{l}(\theta)Q\sqrt{\frac{2}{\pi}}\frac{1}{\omega r}e^{i\omega r}e^{i\pi
l/2}.$ (58)

The outgoing radiation flux at the future null infinity is given by

$\frac{dE}{dt}=\frac{1}{\pi}Q^{2}.$ (59)

##  4 Gravitational radiation from a star orbiting in the near region

In this section, we generalize the analysis of the scalar radiation in the
previous section to the gravitational case.

###  4.1 Gravity analysis

It is convenient to analyze gravitational waves in the Newman-Penrose
formalism. In Boyer-Lindquist coordinates, we choose the Kinnersley tetrad
Teukolsky:1973ha , which has the components $(t,r,\theta,\phi)$

$\displaystyle l^{\mu}$
$\displaystyle=\left(\frac{r^{2}+a^{2}}{\Delta},1,0,\frac{a}{\Delta}\right),$
(60) $\displaystyle n^{\mu}$
$\displaystyle=\frac{1}{2\rho^{2}}(r^{2}+a^{2},-\Delta,0,a),$ $\displaystyle
m^{\mu}$
$\displaystyle=\frac{1}{\sqrt{2}(r+ia\cos\theta)}\left(ia\sin\theta,0,1,\frac{i}{\sin\theta}\right).$

Note that $l\cdot n=-m\cdot\bar{m}=-1$. In the standard notation, the nonzero
spin coefficients are

$\displaystyle\rho=-\frac{1}{r-ia\cos\theta},\
\tau=-\frac{ia\rho\bar{\rho}\sin\theta}{\sqrt{2}},\
\mu=\frac{(r^{2}-2Mr+a^{2})\rho^{2}\bar{\rho}}{2},$ (61)
$\displaystyle\pi=\frac{ia\rho^{2}\sin\theta}{\sqrt{2}},\
\beta=-\frac{\bar{\rho}\cot\theta}{2\sqrt{2}},\
\gamma=\mu+\frac{(r-M)\rho\bar{\rho}}{2},\ \alpha=\pi-\bar{\beta},$

while the Weyl scalars are

$\psi_{0}^{(0)}=\psi_{1}^{(0)}=\psi_{3}^{(0)}=\psi_{4}^{(0)}=0,\
\psi_{2}^{(0)}=M\rho^{3}.$ (62)

To avoid confusion, in this section, $\rho$ only refers to one spin
coefficient rather than the coordinate in the Kerr metric.

Teukolsky showed Teukolsky:1973ha  that the perturbation $\psi_{4}^{(1)}$ of
the Weyl scalar

$\psi_{4}^{(0)}=C_{\alpha\beta\gamma\delta}n^{\alpha}\bar{m}^{\beta}n^{\gamma}\bar{m}^{\delta}$
(63)

in any Type D background (including the Schwarzschild and Kerr black holes)
obeys

$\left[(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})(D+4\epsilon-\rho)-(\bar{\delta}-\bar{\tau}+\bar{\beta}+3\alpha+4\pi)(\delta-\tau+4\beta)-3\psi_{2}^{(0)}\right]\psi_{4}^{(1)}=4\pi
T_{4},$ (64)

where

$\displaystyle T_{4}=$
$\displaystyle(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})\left[(\bar{\delta}-2\bar{\tau}+2\alpha)T_{n\bar{m}}-(\Delta+2\gamma-2\bar{\gamma}+\bar{\mu})T_{\bar{m}\bar{m}}\right]$
(65)
$\displaystyle+(\bar{\delta}-\bar{\tau}+\bar{\beta}+3\alpha+4\pi)\left[(\Delta+2\gamma+2\bar{\mu})T_{n\bar{m}}-(\bar{\delta}-\bar{\tau}+2\bar{\beta}+2\alpha)T_{nn}\right].$

Here, $D\equiv l^{\mu}\partial_{\mu}$, $\Delta\equiv n^{\mu}\partial_{\mu}$,
$\delta\equiv m^{\mu}\partial_{\mu}$, and
$\bar{\delta}\equiv\bar{m}^{\mu}\partial_{\mu}$. The source terms are given by

$T_{n\bar{m}}=T_{\mu\nu}n^{\mu}\bar{m}^{\nu},\
T_{\bar{m}\bar{m}}=T_{\mu\nu}\bar{m}^{\mu}\bar{m}^{\nu},\
T_{nn}=T_{\mu\nu}n^{\mu}n^{\nu}.$ (66)

Suppose the rest mass of the star is $m_{0}$. The energy-momentum tensor

$T^{\mu\nu}=m_{0}\int
d\tau(-g)^{-1/2}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\delta^{(4)}\left(x-x_{*}(\tau)\right)$
(67)

has the nonzero components

$\displaystyle T_{\phi\phi}=$
$\displaystyle\,\,\frac{m_{0}M(r_{0}^{2}-2a\sqrt{Mr_{0}}+a^{2})^{2}}{r_{0}^{\frac{11}{4}}(r_{0}^{\frac{3}{2}}-3Mr_{0}^{\frac{1}{2}}+2aM^{\frac{1}{2}})^{\frac{1}{2}}(a\sqrt{M}+r_{0}^{\frac{3}{2}})}\cdot\delta(r-r_{0})\,\delta(\theta-\pi/2)\,\delta\left(\phi-\frac{M^{\frac{1}{2}}}{r_{0}^{\frac{3}{2}}+aM^{\frac{1}{2}}}t\right),$
(68) $\displaystyle T_{\phi t}=$
$\displaystyle\,\,-\frac{m_{0}\left(a^{3}M+aM(4M-r_{0})r_{0}+a^{2}\sqrt{Mr_{0}}(-4M+r_{0})+\sqrt{M}r_{0}^{\frac{5}{2}}(-2M+r_{0})\right)}{r_{0}^{\frac{11}{4}}(r_{0}^{\frac{3}{2}}-3Mr_{0}^{\frac{1}{2}}+2aM^{\frac{1}{2}})^{\frac{1}{2}}(a\sqrt{M}+r_{0}^{\frac{3}{2}})}$
$\displaystyle\times\delta(r-r_{0})\,\delta(\theta-\pi/2)\,\delta\left(\phi-\frac{M^{\frac{1}{2}}}{r_{0}^{\frac{3}{2}}+aM^{\frac{1}{2}}}t\right),$
$\displaystyle T_{tt}=$
$\displaystyle\,\,\frac{m_{0}\left(a\sqrt{M}+\sqrt{r_{0}}(-2M+r_{0})\right)^{2}}{r_{0}^{\frac{11}{4}}(r_{0}^{\frac{3}{2}}-3Mr_{0}^{\frac{1}{2}}+2aM^{\frac{1}{2}})^{\frac{1}{2}}(a\sqrt{M}+r_{0}^{\frac{3}{2}})}\cdot\delta(r-r_{0})\,\delta(\theta-\pi/2)\,\delta\left(\phi-\frac{M^{\frac{1}{2}}}{r_{0}^{\frac{3}{2}}+aM^{\frac{1}{2}}}t\right).$

To simplify Eq. (64), we define

$\psi^{(-2)}\equiv\rho^{-4}\psi_{4}^{(1)},\quad\mathcal{T}\equiv
2\rho^{-4}T_{4},$ (69)

and use the expansions

$\displaystyle\psi^{(-2)}=$ $\displaystyle\sum_{l,m}e^{im\phi-i\omega
t}S_{l}(\theta)R_{lm}(r),$ (70) $\displaystyle\mathcal{T}=$
$\displaystyle-\frac{1}{4\pi(r^{2}+a^{2}\cos^{2}\theta)}\sum_{l,m}e^{im\phi-i\omega
t}S_{l}(\theta)T_{lm}(r),$

where the relation (16) holds. Then, Eq. (64) can be separated into

$\displaystyle\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta}S_{l})-\frac{m^{2}}{\sin^{2}\theta}S_{l}-\frac{2ms\cos\theta}{\sin^{2}\theta}S_{l}-s^{2}\cot^{2}\theta
S_{l}+sS_{l}$ (71) $\displaystyle-2sa\omega\cos\theta
S_{l}+a^{2}\omega^{2}\cos^{2}\theta S_{l}=-K_{l}S_{l},$

and

$\displaystyle\Delta^{-s}\partial_{r}(\Delta^{s+1}\partial_{r}R_{lm})+\frac{(2Mr_{+}\omega-
am)^{2}}{(r-r_{+})(r_{+}-r_{-})}R_{lm}-\frac{(2Mr_{-}\omega-
am)^{2}}{(r-r_{-})(r_{+}-r_{-})}R_{lm}$ (72) $\displaystyle-is\frac{2M\omega
r_{+}-am}{r-r_{+}}R_{lm}-is\frac{2M\omega r_{-}-am}{r-r_{-}}R_{lm}$
$\displaystyle+2is\omega(r-M)R_{lm}+\left(r^{2}+2M(r+2M)\right)\omega^{2}R_{lm}=K_{l}R_{lm}+T_{lm},$

where $K_{l}$ is the separation constant and $s=-2$, compatible with the
choice of the Weyl scalar $\psi_{4}$ as the outgoing wave Teukolsky:1972my .
In the low-energy limit, Eq. (71) becomes

$\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta}S_{l})-\frac{m^{2}}{\sin^{2}\theta}S_{l}-\frac{2ms\cos\theta}{\sin^{2}\theta}S_{l}-s^{2}\cot^{2}\theta
S_{l}+sS_{l}=-K_{l}S_{l}.$ (73)

Thus, $e^{im\phi}S_{l}$ are the spin-weighted spherical harmonics with
$K_{l}=(l-s)(l+s+1)$ Goldberg:1966uu , and we can obtain

$\displaystyle T_{lm}$ $\displaystyle=-\frac{1}{2\pi}\int\sin\theta d\theta
d(\phi-\frac{M^{\frac{1}{2}}}{r_{0}^{\frac{3}{2}}+aM^{\frac{1}{2}}}t)4\pi(r^{2}+a^{2}\cos^{2}\theta)\mathcal{T}e^{-im\phi+i\omega
t}S_{l}(\theta)$ (74)
$\displaystyle=a_{0}\delta(r-r_{0})+a_{1}\delta^{\prime}(r-r_{0})+a_{2}\delta^{\prime\prime}(r-r_{0}),$

where the expressions of $a_{0}$, $a_{1}$, and $a_{2}$ are omitted, whose
explicit forms are irrelevant in the rest of this paper. In the low-energy and
the near-region limits, Eq. (72) reduces to

$\displaystyle\Delta^{-s}\partial_{r}(\Delta^{s+1}\partial_{r}R_{lm})+\frac{(2Mr_{+}\omega-
am)^{2}}{(r-r_{+})(r_{+}-r_{-})}R_{lm}-\frac{(2Mr_{-}\omega-
am)^{2}}{(r-r_{-})(r_{+}-r_{-})}R_{lm}$ (75) $\displaystyle-is\frac{2M\omega
r_{+}-am}{r-r_{+}}R_{lm}-is\frac{2M\omega
r_{-}-am}{r-r_{-}}R_{lm}=K_{l}R_{lm}+T_{lm}.$

Eq. (75) is also a Sturm-Liouville problem, which can be solved via the
standard Green function method. Two linearly independent solutions to the
homogeneous equation ($T_{lm}=0$) are given by hypergeometric functions

$\displaystyle R_{lm}^{(1)}$
$\displaystyle=\left(\frac{r-r_{+}}{r-r_{-}}\right)^{-i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}-s}\left(1-\frac{r-r_{+}}{r-r_{-}}\right)^{l+s+1}$ (76)
$\displaystyle\quad\times\
_{2}F_{1}\left(1+l-i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}},1+l-s-i2M\omega;1-s-2i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}};\frac{r-r_{+}}{r-r_{-}}\right)$
$\displaystyle=\left(\frac{r-r_{+}}{r-r_{-}}\right)^{-i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}-s}\left(1-\frac{r-r_{+}}{r-r_{-}}\right)^{l+s+1}$
$\displaystyle\quad\times\bigg{[}A\
_{2}F_{1}\left(1+l-i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}},1+l-s-i2M\omega;2l+2;1-\frac{r-r_{+}}{r-r_{-}}\right)$
$\displaystyle\quad+B\,\left(1-\frac{r-r_{+}}{r-r_{-}}\right)^{-2l-1}\
_{2}F_{1}\left(-l-s-i2M\omega,-l-i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}};-2l;1-\frac{r-r_{+}}{r-r_{-}}\right)\bigg{]},$
$\displaystyle R_{lm}^{(2)}$
$\displaystyle=\left(\frac{r-r_{+}}{r-r_{-}}\right)^{i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}}\left(1-\frac{r-r_{+}}{r-r_{-}}\right)^{l+s+1}$ (77)
$\displaystyle\quad\times\
_{2}F_{1}\left(1+l+s+i2M\omega,1+l+i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}};1+s+2i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}};\frac{r-r_{+}}{r-r_{-}}\right)$
$\displaystyle=\left(\frac{r-r_{+}}{r-r_{-}}\right)^{i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}}\left(1-\frac{r-r_{+}}{r-r_{-}}\right)^{l+s+1}$
$\displaystyle\quad\times\bigg{[}C\
_{2}F_{1}\left(1+l+s+i2M\omega,1+l+i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}};2l+2;1-\frac{r-r_{+}}{r-r_{-}}\right)$
$\displaystyle\quad+D\,\left(1-\frac{r-r_{+}}{r-r_{-}}\right)^{-2l-1}\
_{2}F_{1}\left(-l+i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}},-l+s+i2M\omega;-2l;1-\frac{r-r_{+}}{r-r_{-}}\right)\bigg{]},$

where

$\displaystyle A$ $\displaystyle=\frac{\Gamma\left(1-s-2i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}\right)\,\Gamma(-2l-1)}{\Gamma(-l-s-i2M\omega)\,\Gamma\left(-l-i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}}\right)},$
(78) $\displaystyle B$
$\displaystyle=\frac{\Gamma\left(1-s-2i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}\right)\,\Gamma(2l+1)}{\Gamma\left(1+l-i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}}\right)\,\Gamma(1+l-s-i2M\omega)},$
(79) $\displaystyle C$
$\displaystyle=\frac{\Gamma\left(1+s+2i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}\right)\,\Gamma(-2l-1)}{\Gamma\left(-l+i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}}\right)\,\Gamma(-l+s+i2M\omega)},$
(80) $\displaystyle D$
$\displaystyle=\frac{\Gamma\left(1+s+2i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}\right)\,\Gamma(2l+1)}{\Gamma(1+l+s+i2M\omega)\,\Gamma\left(1+l+i\frac{4M^{2}\omega-2am}{r_{+}-r_{-}}\right)}.$
(81)

Then the asymptotic behaviors of the solutions are

$\displaystyle R_{lm}^{(1)}$
$\displaystyle\rightarrow\left(\frac{r-r_{+}}{r_{+}-r_{-}}\right)^{-i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}-s}\,,$ $\displaystyle r\rightarrow r_{+},$ (82)
$\displaystyle\rightarrow
A\,(r_{+}-r_{-})^{l+s+1}\,r^{-l-s-1}+B\,(r_{+}-r_{-})^{-l+s}\,r^{l-s}\,,$
$\displaystyle r\rightarrow\infty,$ $\displaystyle R_{lm}^{(2)}$
$\displaystyle\rightarrow\left(\frac{r-r_{+}}{r_{+}-r_{-}}\right)^{i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}}\,,$ $\displaystyle r\rightarrow r_{+},$ (83)
$\displaystyle\rightarrow
C\,(r_{+}-r_{-})^{l+s+1}\,r^{-l-s-1}+D\,(r_{+}-r_{-})^{-l+s}\,r^{l-s}\,,$
$\displaystyle r\rightarrow\infty.$

Like the scalar field case, we define

$R_{lm}^{(3)}\equiv DR_{lm}^{(1)}-BR_{lm}^{(2)},$ (84)

whose asymptotic behaviors are

$\displaystyle R_{lm}^{(3)}\rightarrow$
$\displaystyle\,\,D\,\left(\frac{r-r_{+}}{r_{+}-r_{-}}\right)^{-i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}-s}-B\,\left(\frac{r-r_{+}}{r_{+}-r_{-}}\right)^{i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}}\,,$ $\displaystyle r\rightarrow r_{+},$ (85)
$\displaystyle\rightarrow$ $\displaystyle\,\,(DA-
BC)\,(r_{+}-r_{-})^{l+s+1}\,r^{-l-s-1}\,,$ $\displaystyle r\rightarrow\infty.$

The solution to Eq. (75), which is purely ingoing at the horizon and obeys
Neumann boundary conditions at infinity, is given by

$R_{lm}(r)=\frac{1}{\Delta_{0}^{-s}W}\left(X\Theta(r_{0}-r)R_{lm}^{(1)}(r)+Z\Theta(r-r_{0})R_{lm}^{(3)}(r)\right)+\frac{a_{2}}{\Delta_{0}}\delta(r-r_{0}),$
(86)

where $W$ is the $r$-independent Wronskian

$W=\Delta^{s+1}\left(R_{lm}^{(1)}(r)\partial_{r}R_{lm}^{(3)}(r)-R_{lm}^{(3)}(r)\partial_{r}R_{lm}^{(1)}(r)\right)\,,$
(87)

and

$\displaystyle X=$
$\displaystyle\,\,R_{lm}^{(3)}(r_{0})\left(a_{0}-s\frac{(2r_{0}-2M)a_{1}}{\Delta_{0}}+2s\frac{a_{2}}{\Delta_{0}}+s(s-1)\frac{(2r_{0}-2M)^{2}}{\Delta_{0}}\frac{a_{2}}{\Delta_{0}}-\frac{a_{2}}{\Delta_{0}}V(r_{0})\right)$
(88)
$\displaystyle+\partial_{r}R_{lm}^{(3)}(r_{0})\left(-a_{1}+(s-1)\frac{(2r_{0}-2M)a_{2}}{\Delta_{0}}\right),$
$\displaystyle Z=$
$\displaystyle\,\,R_{lm}^{(1)}(r_{0})\left(a_{0}-s\frac{(2r_{0}-2M)a_{1}}{\Delta_{0}}+2s\frac{a_{2}}{\Delta_{0}}+s(s-1)\frac{(2r_{0}-2M)^{2}}{\Delta_{0}}\frac{a_{2}}{\Delta_{0}}-\frac{a_{2}}{\Delta_{0}}V(r_{0})\right)$
$\displaystyle+\partial_{r}R_{lm}^{(1)}(r_{0})\left(-a_{1}+(s-1)\frac{(2r_{0}-2M)a_{2}}{\Delta_{0}}\right).$

Here,

$\displaystyle\Delta_{0}=$ $\displaystyle\,\,r_{0}^{2}-2Mr_{0}+a^{2}\,,$
$\displaystyle V(r)=$ $\displaystyle\,\,\frac{(2Mr_{+}\omega-
am)^{2}}{(r-r_{+})(r_{+}-r_{-})}-\frac{(2Mr_{-}\omega-
am)^{2}}{(r-r_{-})(r_{+}-r_{-})}$ $\displaystyle-is\frac{2M\omega
r_{+}-am}{r-r_{+}}-is\frac{2M\omega r_{-}-am}{r-r_{-}}-(l-s)(l+s+1)\,.$

Thus, we have

$\displaystyle\psi^{(-2)}(r\rightarrow r_{+})$
$\displaystyle=\sum_{l,m}e^{im\phi-i\omega
t}S_{l}(\theta)\frac{X}{\Delta_{0}^{-s}W}\left(\frac{r-r_{+}}{r_{+}-r_{-}}\right)^{-i\frac{2Mr_{+}\omega-
am}{r_{+}-r_{-}}-s},$ (89) $\displaystyle\psi^{(-2)}(r\rightarrow\infty)$
$\displaystyle=\sum_{l,m}e^{im\phi-i\omega
t}S_{l}(\theta)\frac{Z}{\Delta_{0}^{-s}W}(DA-
BC)(r_{+}-r_{-})^{l+s+1}r^{-l-s-1}.$

The equivalent particle number flux at the horizon, studied in
Teukolsky:1974yv , is

$\mathcal{F}_{lm}=\frac{128(2Mr_{+}\omega-
am)}{|\mathcal{C}|^{2}}\frac{|X|^{2}}{\Delta_{0}^{4}|W|^{2}}\cdot\left(\frac{(2Mr_{+}\omega-
am)^{2}}{(r_{+}-r_{-})^{2}}+\frac{1}{4}\right)\left(\frac{(2Mr_{+}\omega-
am)^{2}}{(r_{+}-r_{-})^{2}}+1\right),$ (90)

with

$\displaystyle|\mathcal{C}|^{2}=$ $\displaystyle\,\,(Q^{2}+4a\omega
m-4a^{2}\omega^{2})\left[(Q-2)^{2}+36a\omega m-36a^{2}\omega^{2}\right]$ (91)
$\displaystyle+(2Q-1)(96a^{2}\omega^{2}-48a\omega
m)+144\omega^{2}(M^{2}-a^{2}),$ $\displaystyle Q=$
$\displaystyle\,\,l^{2}+l+a^{2}\omega^{2}-2a\omega m,$

where $m>0$.

###  4.2 CFT analysis and comparison

The CFT analysis has two minor differences between the scalar and the
gravitational cases. The first difference is in the determination of the CFT
deformation. The CFT deformation is determined by the metric perturbation
itself, but the perturbation $\psi_{4}^{(1)}$ studied in the previous
subsection involves the second derivative of the metric. To obtain the metric
perturbation $h_{\mu\nu}$, we need the following relation Chrzanowski:1975wv ;
Wald:1978vm

$h_{\mu\nu}=\Theta_{\mu\nu}\Psi_{H},$ (92)

where

$\displaystyle\Theta_{\mu\nu}=$ $\displaystyle-
l_{\mu}l_{\nu}(\bar{\delta}+\alpha+3\bar{\beta}-\bar{\tau})(\bar{\delta}+4\bar{\beta}+3\bar{\tau})-{\bar{m}}_{\mu}{\bar{m}}_{\nu}(D-\bar{\rho})(D+3\bar{\rho})$
(93)
$\displaystyle+l_{(\mu}{\bar{m}}_{\nu)}\left[(D+\rho-\bar{\rho})(\bar{\delta}+4\bar{\beta}+3\bar{\tau})+(\bar{\delta}+3\bar{\beta}-\alpha-\pi-\bar{\tau})(D+3\bar{\rho})\right],$

and the Hertz potential $\Psi_{H}$ is given by

$\Psi_{H}=\frac{4}{\mathcal{C}}R^{(-2)}(r)S^{(+2)}(\theta)e^{-i\omega
t+im\phi}.$ (94)

Here,

$\displaystyle R^{(-2)}(r)=$
$\displaystyle\,\,\frac{X}{\Delta_{0}^{-s}W}R_{lm}^{(1)}(r)$ (95)
$\displaystyle\rightarrow$
$\displaystyle\,\,\frac{X}{\Delta_{0}^{-s}W}\left[A(r_{+}-r_{-})^{l+s+1}r^{-l-s-1}+B(r_{+}-r_{-})^{-l+s}r^{l-s}\right],\
r\rightarrow\infty.$

Thus, like the scalar case, the coefficients $J_{lm}$ are read off from the
Dirichlet modes

$J_{lm}=\frac{4}{\mathcal{C}}\frac{X}{\Delta_{0}^{-s}W}B.$ (96)

The second difference is that the conformal weights become

$h_{L}=l-s,\quad h_{R}=l.$ (97)

Following the steps similar to the scalar case, we obtain

$\displaystyle\mathcal{R}_{lm}=$ $\displaystyle\,\,2\pi
C_{\mathcal{O}}^{2}|J_{lm}|^{2}\left(\frac{M}{a}\right)^{2l-2s-1}\left(\frac{\sqrt{M^{2}-a^{2}}}{a}\right)^{2l-1}e^{\pi\frac{4M\omega
r_{+}-2am}{r_{+}-r_{-}}}$ (98)
$\displaystyle\times|\Gamma(l-s+i2M\omega)|^{2}\cdot\left|\Gamma\left(l+i\frac{4M^{2}\omega-2ma}{r_{+}-r_{-}}\right)\right|^{2}/\left(\Gamma(2l-2s)\,\Gamma(2l)\right)$
$\displaystyle\approx$
$\displaystyle\,\,\frac{2048\pi^{2}C_{\mathcal{O}}^{2}}{al(2l+1)(2l+2)(2l+3)(l+2)^{2}}\frac{|X|^{2}}{\mathrm{\Delta}_{0}^{4}\,|W|^{2}}\left(\frac{M}{a}\right)^{2l+3}\left(\frac{\sqrt{M^{2}-a^{2}}}{a}\right)^{2l-2}\frac{2M\omega
r_{+}-am}{|\mathcal{C}|^{2}}$
$\displaystyle\times\frac{1}{1-e^{-2\pi\frac{4M\omega
r_{+}-2am}{r_{+}-r_{-}}}}\left(\frac{(2Mr_{+}\omega-
am)^{2}}{(r_{+}-r_{-})^{2}}+\frac{1}{4}\right)\left(\frac{(2Mr_{+}\omega-
am)^{2}}{(r_{+}-r_{-})^{2}}+1\right).$

In the last step, we dropped some subleading terms in the limit $\omega M\ll
1$.

With the normalization

$C_{\mathcal{O}}=\frac{\sqrt{al(2l+1)(2l+2)(2l+3)}(l+2)}{4\pi}\left(\frac{a}{M}\right)^{l+3/2}\left(\frac{a}{\sqrt{M^{2}-a^{2}}}\right)^{l-1},$
(99)

we find again

$\mathcal{R}_{lm}(1-e^{-\beta_{L}\cdot\omega_{L}-\beta_{R}\cdot\omega_{R}})=\mathcal{F}_{lm}.$
(100)

###  4.3 Reattaching the far region

In this subsection, we reattach the far region and compute the gravitational
radiation flux at the future null infinity. Eq. (72) can be written as

$\mathrm{\Delta}^{-s}\partial_{r}(\mathrm{\Delta}^{s+1}\partial_{r}R_{lm})+\left[\frac{H^{2}-2is(r-M)H}{\mathrm{\Delta}}+4is\omega
r+2am\omega-a^{2}\omega^{2}\right]R_{lm}=K_{l}R_{lm}+T_{lm},$ (101)

where $H=(r^{2}+a^{2})\omega-am$. Considering the low-energy and the far-
region limits, we have

$r^{-2s}\partial_{r}(r^{2s+2}\partial_{r}R_{lm})+\left(r^{2}\omega^{2}+2is\omega
r-(l-s)(l+s+1)\right)R_{lm}=0.$ (102)

Two linearly independent solutions are

$\displaystyle R_{lm}^{(1,far)}=$ $\displaystyle r^{l-s}e^{-i\omega r}\
_{1}F_{1}(l+1-s,2l+2;2i\omega r),$ (103) $\displaystyle R_{lm}^{(2,far)}=$
$\displaystyle r^{-l-s-1}e^{-i\omega r}\ _{1}F_{1}(-l-s,-2l;2i\omega r),$

with the asymptotic behaviors

$\displaystyle R_{lm}^{(1,far)}$ $\displaystyle\rightarrow\ r^{l-s}\,,$
$\displaystyle r\rightarrow 0,$ (104)
$\displaystyle\rightarrow\frac{(-2i\omega)^{-1-l+s}\,\Gamma(2+2l)}{\Gamma(1+l+s)}r^{-1}e^{-i\omega
r}+\frac{(2i\omega)^{-1-l-s}\,\Gamma(2+2l)}{\Gamma(1+l-s)}r^{-1-2s}e^{i\omega
r}\,,$ $\displaystyle r\rightarrow\infty,$ $\displaystyle R_{lm}^{(2,far)}$
$\displaystyle\rightarrow\ r^{-l-s-1}\,,$ $\displaystyle r\rightarrow 0,$
(105)
$\displaystyle\rightarrow\frac{(-2i\omega)^{l+s}\,\Gamma(-2l)}{\Gamma(-l+s)}r^{-1}e^{-i\omega
r}+\frac{(2i\omega)^{l-s}\,\Gamma(-2l)}{\Gamma(-l-s)}r^{-1-2s}e^{i\omega
r}\,,$ $\displaystyle r\rightarrow\infty.$

The solution of Eq. (102) that is purely outgoing at the null infinity can be
written as

$R_{lm}^{(far)}(r)=PR_{lm}^{(1,far)}+QR_{lm}^{(2,far)},$ (106)

where

$P/Q=-(-2i\omega)^{2l+1}\frac{\Gamma(1+l+s)\,\Gamma(-2l)}{\Gamma(-l+s)\,\Gamma(2l+2)}.$
(107)

Matching the far-region solution with the near-region solution, we obtain

$Q=\frac{Z}{\Delta_{0}^{-s}W}(DA-
BC)(r_{+}-r_{-})^{l+s+1}\,\left[1+(-2i\omega)^{2l+1}\frac{\Gamma(1+l+s)\,\Gamma(-2l)}{\Gamma(-l+s)\,\Gamma(2l+2)}\frac{A}{B}(r_{+}-r_{-})^{2l+1}\right]^{-1}.$
(108)

Thus,

$\Psi(r\rightarrow+\infty)=\sum_{l,m}\ e^{im\phi-i\omega
t}S_{l}(\theta)\,Q\,\frac{(2i\omega)^{l-s}\,\Gamma(-2l)}{\Gamma(-l-s)}\left[1+\frac{\Gamma(1+l+s)}{\Gamma(1+l-s)}\right]\,r^{-1-2s}e^{i\omega
r}.$ (109)

The outgoing radiation flux at the future null infinity, studied in
Teukolsky:1974yv , is

$\frac{dE}{dt}=2^{2l+3}\omega^{2l+2}|Q|^{2}\,\left|\frac{\Gamma(-2l)}{\Gamma(-l+2)}\left[1+\frac{\Gamma(1+l-2)}{\Gamma(1+l+2)}\right]\right|^{2}.$
(110)

##  5 Discussion

In this paper, we compute gravitational waves emitted by a star orbiting near
the horizon of a non-extremal Kerr black hole from the gravity and the dual
CFT perspectives, respectively. Considering the stimulated emission at finite
temperature, two results agree. This strongly supports the hidden conformal
symmetry of the Kerr black hole. The stimulated emission factor at finite
temperature is notable. In previous studies, since the extreme Kerr black hole
is dual to a CFT at zero temperature, this factor does not appear explicitly
in the results.

To show the hidden conformal symmetry, we take two limits in this paper. The
near-region limit is easy to understand. When we are far enough away from the
black hole, the effect of the black hole is negligible. Therefore, the dual
CFT mainly contains information about the black hole and the spacetime near
it. In other words, the black hole and its near region are dual to the CFT.
The low-energy limit seems unnatural. However, suppose we are close enough to
a black hole, and its effect on us is large enough. In that case, all physical
activities (such as the generation and propagation of gravitational waves) are
in low energy compared to the effect of the black hole. Thus, both limits
imply the same condition: the black hole dominates the physical phenomena we
consider.

Although we succeeded in computing gravitational waves using the hidden
conformal symmetry in this paper, there are still many difficulties in
developing a useful CFT tool to compute gravitational waves in practice.
Firstly, for the generic non-extremal Kerr black hole, the conformal symmetry
of the spacetime is destroyed, so relating different orbits like the extremal
case is not easy. The solution space where the conformal symmetry exists might
be a potential direction, and various gravitational waves might be related
through conformal transformations. In addition, the gravitational wave sources
in astronomical observations are not necessarily limited to a test object
orbiting near the horizon of a black hole. Extending the application of CFT to
more cases, such as the binary black hole merger, requires a complete
understanding of the Kerr/CFT correspondence.

###### Acknowledgements.

We would like to thank Jiang Long, Jianxin Lu, Wei Song, and Yueliang Wu for
helpful discussions. This work is supported by the National Key Research and
Development Program of China (Grant No. 2021YFC2201901) and in part by the
NSFC under grant No. 12147103.

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11institutetext:  School of Physical Science and Technology & Collaborative
Innovation Center of Suzhou Nano Science and Technology, Soochow University,
Suzhou 215006, China

Phases: geometric; dynamic or topological Phase transitions and critical
phenomena Ultrasonics, quantum acoustics, and physical effects of sound

# Dirac cones and higher-order topology in quasi-continuous media

Zhi-Kang Lin 11 Jian-Hua Jiang E-mail: 11<EMAIL_ADDRESS>11

###### Abstract

We consider the Dirac cones and higher order topological phases in quasi-
continuous media of classical waves (e.g., photonic and sonic crystals). Using
sonic crystals as prototype examples, we revisit some of the known systems in
the study of topological acoustics. We show the emergence of various Dirac
cones and higher order topological band gaps in the same motherboard by tuning
the geometry of the system. We provide a pedagogical review of the underlying
physics and methodology via the bulk-edge-corner correspondence, symmetry-
based indicators, Wannier representations, filling anomaly, and fractional
corner charges. In particular, the theory of the Dirac cones and the higher-
order topology are put in the same framework. These examples and the
underlying physics principles can be inspiring and useful in future study of
higher order topological metamaterials.

###### pacs:

03.65.Vf

###### pacs:

68.35.Rh

###### pacs:

43.35.+d

##  1 Introduction

The past few years have witnessed the rapid development in a new frontier of
topological phases of matter [1, 2] that is termed as the higher-order
topological insulators (HOTIs) [3]. HOTIs are firstly indicated by the
existence of chiral edge states in three-dimensional (3D) axion insulators in
magnetic fields [4, 5] and then intrigue tremendous interest due to the
celebrated multipole insulators [6, 7]. At the same time or later, several
concrete models and theories have being proposed [8, 9, 10, 11, 12, 13, 3, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37]. Prototypes of HOTIs include the two-dimensional (2D) second-
order topological insulators which host one-dimensional (1D) gapped edge
states and zero-dimensional (0D) in-gap topological corner states. Here, the
term “second-order” means the protected topological boundary states have a co-
dimension of two (i.e., the topological boundary states have two dimensions
lower than the bulk states). Such HOTIs generalize the conventional bulk-edge
correspondence to the bulk-edge-corner correspondence and unveil a
unprecedented regime of multidimensional topological physics with versatile
topological phenomena. Unlike the conventional topological insulators which
are protected by the time-reversal symmetry, HOTIs are protected by the
crystalline symmetry and characterized by the bulk topological invariants such
as the quantized multipole polarizations [6, 7]. HOTIs represent a large
category of topological crystalline insulator phases which widely emerge in
natural [3, 38, 39, 40, 41, 42, 43, 44, 45, 46] and artificial [47, 48, 49,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68,
69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,
88] materials. In many cases, the nonzero polarizations in HOTIs manifest the
filling anomaly [20] of the bulk states in finite systems, which, from a real-
space perspective, can be revealed via the Wannier centers of the Bloch bands
[20, 35].

What makes HOTIs particularly attractive for material scientists are their
rich topological phenomena across dimensions. Topological phenomena at
different dimensions in the same material could provide multiplexing
applications. For instance, the gapped surface states in three-dimensional
HOTIs can provide surface states with high density of states to accelerate
surface chemical reaction processes. Meanwhile, the bulk, through filling
anomaly, gives rise to high-density surface charges. In addition, the gapless
hinge states offer robust transport on 1D hinge channels. In photonic HOTIs,
it has been shown that edge states can provide 1D waveguide channels [89],
while the 0D corner states can serve as robust cavities that may enable ultra-
low threshold lasing [93, 90, 91, 92].

Figure 1:  Acoustic analogue of various Dirac cones in SCs with (a) sixfold
($C_{6}$), (b) threefold ($C_{3}$), (c) fourfold ($C_{4}$) and (d) twofold
($C_{2}$) rotation symmetries. Only the unit cells of the SCs are illustrated
where the rotatable acoustic scatterers and the air regions are denoted by the
white and green regions, respectively. Except for the rotation angle $\theta$
of scatterers, the other geometric parameters $l_{1}=0.49a$ in (a),
$l_{2}=0.89a$ in (b), $l_{3}=0.375a$, $h_{1}=0.2a$ in (c) and $l_{4}=0.4a$,
$h_{2}=0.1a$ in (d) are fixed. Here, $a$ denotes the lattice constant for all
cases. (e)-(f): The corresponding acoustic band structures for SCs in (a)-(d)
with a double Dirac cone at the $\Gamma$ point (e), a Dirac cone at the $K$
point (f), a spin-1 Dirac cone at the $M$ point (g) and a generalized Dirac
cone at the $M$ point (h). The respective rotation angle $\theta$ is listed on
the top-left corner of each Figure. Insets show the Brillouin zone and the 3D
band dispersion around the degenerate points.

On the other hand, Dirac cones [94] that emerge in various photonic and
phononic artificial materials (such as photonic crystals and phononic
crystals) have been the focus of researches in the study of metamaterials,
because of their unique properties. For instance, in photonic crystals, Dirac
cones can be utilized to realize all-dielectric zero refractive index medium
that has low dissipation and enables extraordinary manipulation of light [95,
96, 97, 98].

Since there are plenty of studies on how to construct HOTIs from tight-binding
models, here we emphasize the construction of HOTIs in the Bragg scattering
regime. In this regime, which we denote as the quasi-continuous regime, waves
are nearly free, while the Bloch bands are formed by Bragg scatterings.
Examples include photonic crystals, sonic crystals (SCs) and phononic crystals
(including phonons in natural and artificial solid lattice systems), as well
as some other metamaterials. In such a regime, although there is no tight-
binding picture, HOTIs can still be realized and manipulated by tuning the
geometry of the scatterers in each unit-cell. Here, by revisiting some of the
recently studied acoustic systems, we show that Dirac cones and HOTIs can be
realized in the same system in quasi-continuous media. We further illustrate
how to obtain the topological invariants (i.e., the symmetry indicators of the
Bloch bands, the filling anomaly from the Wannier orbital picture, and the
fractional corner charges) that characterize the higher-order topology of the
Bloch bands.

##  2 Dirac cones in 2D SCs

We specifically consider four 2D airborne SCs with $C_{n}$ rotation symmetries
($n=2,3,4,6$) as illustrated in Figs. 1(a)-(d). The rotation symmetries play
an essential role in the theory of higher-order topological phases [20, 15, 9,
7, 22]. In these SCs, the band inversion at one of the high symmetry points
(HSPs) can be triggered by tuning the rotation angles of the scatterers
$\theta$. Such band inversions lead to topological transitions and the
emergence of various Dirac cones.

Here, we focus on the 2D airborne SCs where the acoustic scatterers are placed
in the air background. The wave equation for the acoustic wave is given in
details in Supplementary information. The scatterers here are realized by
photosensitive resins via the commercial 3D printing technology. As depicted
in Figs. 1(a)-(d), four designed SCs are introduced, which separately enjoy
the sixfold ($C_{6}$), fourfold ($C_{4}$), threefold ($C_{3}$) and twofold
($C_{2}$) rotation symmetries.

It is found that at certain angles, those SCs host various Dirac cones [see
Figs. 1(e)-(h)]. For instance, a double Dirac cone with fourfold degeneracy
can appear in the $C_{6}$-symmetric SC at the Brillouin zone center. At each
corner of the hexagonal unit-cell, there is a scatterer made of epoxy [see
Fig. 1(a)], which is an equilateral triangle with the side length
$l_{1}=0.49a$. While keeping the $C_{6}$ rotation symmetry and rotating the
scatterers, the acoustic bands can be tuned. At the rotation angle
$\theta=21.6^{\circ}$, a double Dirac cone emerges at the Brillouin zone
center [Fig. 1(e)]. Such a double Dirac cone is initially studied for zero-
index metamaterials [96] and then plays an pivotal role in the discovery of
the analogous spin Hall effect in various photonic, and phononic systems [100,
101].

The SC with the $C_{3}$ rotation symmetry in Fig. 1(b) is known for the
acoustic realization of the conventional [102, 103] and higher-order [87]
valley Hall insulators. In these SCs, there is a single epoxy scatterer at the
center of the unit-cell. This scatterer is an equilateral triangle with the
side length $l_{2}=0.89a$. At $\theta=0^{\circ}$, Dirac cones emerge at the
$K$ and $K^{\prime}$ points [see Fig. 1(f)].

The $C_{4}$-symmetric SC in Fig. 1(c) simulates an acoustic Lieb lattice. In a
unit-cell, there is a single epoxy scatterer at the center. This scatterer is
designed to be a cross shape characterized by the length $l_{3}=0.375a$ and
the width $h_{1}=0.2a$. By tuning the rotation angle to $\theta=24.8^{\circ}$,
a spin-1 Dirac cone with threefold degeneracy, emerges at the $M$ point [Fig.
1(g)]. The spin-1 Dirac cone is composed of two linear bands intersecting with
a nearly flat band and has been studied with great interest as a candidate for
double-zero index metamaterials [95, 97, 104, 105, 98].

The $C_{2}$-symmetric SC in Fig. 1(d) is originally proposed and
experimentally realized as a second-order topological insulator [53]. There
are four scatterers in each unit-cell. The topological transition takes place
at $\theta=0^{\circ}$ where a generalized Dirac cone with fourfold degeneracy
emerges at the $M$ point. We use the term of “generalized Dirac cone” here due
to the existence of another nodal lines crossing it. We remark that the linear
dispersion of all the Dirac cones discussed above can be illustrated by the
$k\cdot p$ method near the degenerate points[106].

Figure 2: Schematic of the topological phase transitions of bulk states and
the emergent gapped edge states. (a)-(d) The topological phase diagrams of
bulk states for the SCs in Figs. 1(a)-(d). The frequencies of the concerned
states with different symmetry representations versus the rotation angle are
plotted as the red dashed and green solid curves, respectively. Separately,
the symmetry representations are $E_{1}$ and $E_{2}$ in (a), ${}^{2}E$ and
${}^{1}E$ in (b), $E$ and $B$ in (c) and $B$ and $A$ in (d), as labelled on
the top of each Figure. The labels of the representations are based on the
character tables on the Bilbao Crystallographic Server [99]. Insets illustrate
the location of real-space Wannier centers of the lower-frequency Bloch bands
below the opened band gap. The red dots denote the Wyckoff positions. (e)-(h)
The corresponding projected band structures along the wavevectors calculated
from the ribbon-shaped structures as sketched on the top of each Figure. The
domain walls in one direction are between two topologically distinct phases
with specific rotation angles. The periodic boundary condition remains in the
other direction. The orange curves denote the gapped edge states whose wave
functions localize at the domain walls. The blue regions represent the
frequency ranges of the edge band gaps. Insets schematically illustrate the
Wannier centers around the domain-wall boundaries.

##  3 Topological transitions and edge states

We note that the periodicity of the rotation angle is $120^{\circ}$ for both
the $C_{6}$\- and $C_{3}$-symmetric SCs, while $90^{\circ}$ and $180^{\circ}$
for the $C_{4}$\- and $C_{2}$-symmetric SCs, respectively. By tuning the angle
$\theta$ while preserving the rotation symmetries, the band gaps can be closed
and reopen, leading to the topological transitions. This process is well
captured by the flip of the eigen-frequencies of the Bloch states with
different symmetry representations, as shown in Figs. 2(a)-(d).

Specifically, for the $C_{6}$-symmetric SC, the parity inversion occurs
between two doubly degenerate representations $E_{1}$ and $E_{2}$ of the
$C_{6}$ point group. The parity inversion leads to the emergence of the double
Dirac cone at the $\Gamma$ point at $\theta=21.6^{\circ}$ for the parameters
adopted here.

For the $C_{3}$-symmetric SC, tuning the rotation angle $\theta$ can trigger
the inversion between the nondegenerate representations ${}^{1}E$ and
${}^{2}E$ at the $K$ and $K^{\prime}$ points. The transition leads to the
emergence of the Dirac cones at $\theta=0^{\circ}$ where the ${}^{1}E$ and
${}^{2}E$ representations become degenerate.

In the case of the $C_{4}$-symmetric SC, by tuning the angle $\theta$,
inversion of the doubly degenerate representation $E$ and the nondegenerate
representation $B$ can be triggered at the Brillouin zone corner (i.e., the
$M$ point). At the transition point, $\theta=24.8^{\circ}$, the $E$ and $B$
representations are degenerate, leading to a triply-degenerate Dirac cone.

The $C_{2}$-symmetric SC experiences a parity inversion at the $M$ point when
the rotation angle $\theta$ is tuned. This is associated with the flip of the
doubly degenerate $2B$ and $2A$ representations. At the transition point, the
degeneracy of the $2B$ and $2A$ representations leads to the generalized Dirac
cone at the $M$ point.

The band topology of the $C_{n}$-symmetric insulators can be distinguished by
the rotation symmetry representations of the bulk Bloch states at the HSPs in
the Brillouin zone, which contribute to several topological indices [20]. The
topological indices of different phases for the four SCs are provided in
details in Supplementary Information.

We present in Figs. 2(e)-(f) the calculated acoustic edge states. The
corresponding ribbon-shaped supercells are schematically depicted on the top
of each figure, where the domain wall is between two topologically distinct
SCs with specific rotation angles. Specifically, the zigzag domain-wall
boundaries are formed in the $C_{6}$\- and $C_{3}$-symmetric cases. The
projected bands for all cases exhibit two edge states in the band gap, which
are gapped because there is no symmetry at the edge that protects them to be
gapless. Finally, we remark that the edge band gap can be closed for the
$C_{3}$\- and $C_{2}$-symmetric SCs, when the edge boundary has the emergent
glide symmetry, as shown in Refs. [53, 85]. The edge gap closing leads to an
independent topological transition at the edges, which is not related with the
bulk topological transitions.

Figure 3: Corner states as a direct demonstration of the higher-order band
topology. (a)-(d) The corresponding eigen spectra of the finite box-shaped
structures in (e)-(h). The corner states emerge in the edge band gaps. The
bulk, edge and corner states are denoted by the grey, blue and red dots,
respectively. (e)-(f) The profiles of the acoustic pressure fields of one
corner state for the $C_{6}$, $C_{3}$, $C_{4}$ and $C_{2}$-symmetric cases,
respectively. The profiles show the localization at corners. The rotation
angles of inner and outer SCs are labelled in Figures.

##  4 Wannier representations

In the Wannier-representable HOTIs studied in this work, the emergence of the
edge states can be associated with the real-space Wannier orbitals. In other
words, the two branches of the edge states can be understood as evolved from
the two Wannier orbitals exposed at the edge boundary. The band representation
theories give the $C_{n}$-symmetric Wannier orbitals located at the HSPs of
the unit cell (i.e., Wyckoff positions) according to the “band
representations” (i.e., the symmetry representations of the Bloch bands). The
band representations are listed in details in the Bilbao Crystallographic
Server [107]. The Wannier centers for different gapped phases of the four SCs
are illustrated in the insets of Figs. 2(e)-(f).

Specifically, for the $C_{6}$-symmetric SC, there are always three Bloch bands
below the gap. The three corresponding Wannier centers are at the center of
the unit cell (i.e., the Wyckoff position $a$) if
$\theta\in(-21.6^{\circ},21.6^{\circ})$, or the edges of the unit-cell (i.e.,
the Wyckoff positions $c$,$c^{\prime}$ and $c^{\prime\prime}$) if
$\theta\in(21.6^{\circ},98.4^{\circ})$. The former Wannier configuration
corresponds to the trivial atomic insulator, while the latter corresponds to
an obstructed atomic insulator. For the $C_{3}$-symmetric case, only one Bloch
band is below the concerned band gap. The single Wannier center is at the
Wyckoff position $c$ if $\theta\in(-60^{\circ},0^{\circ})$, or at the Wyckoff
position $b$ if $\theta\in(0^{\circ},60^{\circ})$. Such two gapped phases are
both topological, since for both of them the Wannier center is away from the
unit-cell center. For the $C_{4}$-symmetric SC, there is one band below the
gap if $\theta\in(-24.8^{\circ},24.8^{\circ})$. However, if
$\theta\in(24.8^{\circ},65.2^{\circ})$, there are two Bloch bands below the
gap. In the former case, the Wannier center is at the corner of the unit cell
(i.e., the Wyckoff position $b$), whereas in the latter case, the Wannier
centers are at the edge centers of the unit-cell (i.e., the Wyckoff positions
$c$ and $c^{\prime}$). Both gapped phases are topological, but of distinct
properties. For the $C_{2}$-symmetric SC, there are always two bulk bands
below the gap. If $\theta\in(-90^{\circ},0^{\circ})$, the two Wannier centers
are at the edge centers. In comparison, if $\theta\in(0^{\circ},90^{\circ})$,
one of the Wannier center is at the unit-cell center, while the other is at
the unit-cell corner.

For all the cases, there are two Wannier centers exposed to the edge boundary
within an edge supercell when two SCs with topologically distinct band gaps
are placed together. This is schematically illustrated in the insets in Figs.
2(e)-(f) where the domain-wall boundaries are marked with red lines. The two
Wannier orbitals exposed to the edge boundary are responsible for the
emergence of the edge states [20].

Figure 4: Fractional edge and corner charges. The Wannier centers of inner and
outer SCs in finite box-shaped structures are denoted by green and orange
dots, respectively. The hollow dots represent the Wannier centers occupied at
the domain wall between inner and outer SCs. In (a), for the $C_{6}$-symmetric
case, one unit cell at each corner manifests the $\frac{1}{2}$ charge.
Similarly, the fractional edge and corner charges are labelled for unit cells
in (b) for the $C_{3}$-symmetric case and in (c) for the $C_{2}$-symmetric
case. The $C_{4}$-symmetric case is similar to the $C_{2}$-symmetric case
expect for one more Wannier centers occupied at the unit-cell center of inner
SCs which denote no fractional charges.

##  5 Higher-order topology, corner states and fractional corner charges

The corner states may appear in the edge band gap as a direct manifestation of
the higher-order band topology. For all the four cases, the edge and corner
boundaries are formed in the finite-sized systems, as shown in Figs. 3(e)-(h).
The rotation angles of the inner and outer SCs are chosen as the same as those
in Fig. 2. Specifically, the topological SCs are placed in the inner region
for the $C_{6}$-symmetric case. For the other three cases, the Wannier centers
of the inner SCs, respectively, reside at the Wyckoff position $b$ for the
$C_{3}$-symmetric case and the unit-cell corner for both the $C_{4}$\- and
$C_{2}$-symmetric cases.

The calculated eigen-spectra for these finite systems are presented in Figs.
3(a)-(d) which manifest the bulk-edge-corner correspondence with the corner
states emerging in the edge band gap. The localization of the corner states
are confirmed by the acoustic pressure profiles in Figs. 3(e)-(h). We remark
that these features remain intact if the inner SC exchanges with the outer SC
for each configuration. Note that in those calculations, the hard-wall
boundary condition is set at the outer most boundary, to keep the full system
closed and Hermitian.

In addition to the above spectral signatures, which although in many cases are
not fragile but strictly speaking not completely protected by the higher-order
band topology, there are two robust bulk properties protected by the band
topology: filling anomaly and fractional corner charges. These two properties
are interrelated with each other. In simple words, filling anomaly refers to
the phenomena that in a finite system, the number of bulk eigenstates is
different from the number of Wannier orbitals in the system. The filling
anomaly is primarily due to the Wannier orbitals occupied at the edge or
corners. A direct consequence of the filling anomaly is the emergence of the
fractional charges at the edge and corner boundaries.

Following Ref. [20], the corner charges are formulated based on the
topological indices, as given in Supplementary information. However, they rely
on the specific shapes of the finite-sized systems [20]. Instead, it is
convenient to utilize the Wannier orbital distribution to detect the
fractional edge and corner charges. As depicted in Fig. 4, the Wannier centers
of the inner and outer regions are denoted by the green and orange dots,
respectively. Specially, the Wannier centers occupied at the edges and corners
are denoted by the hollow dots. We consider the charges of each unit cell,
which can be inferred by counting the number of Wannier centers dropped in the
unit cell. For instance, as shown in Fig. 4 (a) for the $C_{6}$-symmetric SC,
the single unit cell in the inner bulk region has six half of the Wannier
centers and hence carries the $\frac{1}{2}\times 6=3$ charges. At the corner
unit cell, the hollow dots denote no charges and therefore the whole charges
are $\frac{1}{2}\times 3\ mod\ 1=\frac{1}{2}$. The same derivation can apply
to the $C_{3}$-, $C_{4}$\- and $C_{2}$-symmetric cases, their distributions of
the fractional charges are presented in Figs. 4 (b) and (c). The fractional
charges around the corners are the primary cause of the corner states. Note
that there is no edge fractional charge in the $C_{6}$-symmetric case, while
the edge states still emerge in the bulk band gap. Such a feature indicates
that the filling anomaly may not manifest as the fractional charges. In
contrary, the fractional charges at the edge or corner always indicate the
filling anomaly. It’s also worth noting that the the fractional charges
calculated from the Wannier centers are theoretical values in the limit with
extremely localized Wannier orbitals. The real charges are approximate values
compared to them, which can be calculated from the first-principle numerical
method, as elaborated in Supplementary information.

##  6 Conclusions and outlook

In this paper, we revisit the higher-order topological phases in quasi-
continuous media. Using SCs as prototype examples, we show that the Dirac
cones and the higher-order topological insulator phases can emerge in the same
motherboard of SCs, upon tuning the geometry of the SCs. In particular, we
review the underlying physics and theoretical methods used in the literature
to analyze the higher-order topology and Dirac cones in the a unified
framework, i.e., the symmetry-based analysis of the Bloch bands and the
underlying Wannier representations. In addition to the known spectral bulk-
edge-corner correspondence, we emphasize that the higher-order topological
insulator phases gives rise to filling anomaly and fractional charges at the
edge and corner boundaries. The results presented in this paper is useful for
future studies on quasi-continuous topological metamaterials.

We remark that in most cases the phononic bands do not have chiral symmetry.
In some extreme cases, the edge and corner states can be destroyed by the
absence of the chiral symmetry. However, in most cases, the corner states and
the edge states remain robust in the bulk band gap. On the other hand,
although it is not easy to probe the fractional corner changes, using photonic
analogs people are now able to measure the fractional corner charges. New
ideas of probing the higher-order topology and filling anomaly were also
proposed and realized very recently (e.g., topological Wannier cycles [108,
109].)

Finally, we would like to mention that there are also fragile topological
insulators which are not Wannier representable but also supports the spectral
bulk-edge-corner correspondence, filling anomaly, fractional edge and corner
charges. From the experimental side, it is hard to distinguish the difference
between HOTIs and fragile topological insulators. Finally, it would be very
interesting as well to study the non-Hermitian regime for the higher-order and
Dirac cone phases. Due to their tunability and subwavelength nature as well as
compatibilities for various applications, quasi-continuous media are expected
to play an important role in future study and applications of higher-order
topological metamaterials.

###### Acknowledgements.

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