# Quasinormal modes and shadow of noncommutative black hole

J. A. V. Campos<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa,
Paraíba, Brazil  M. A. Anacleto<EMAIL_ADDRESS>Departamento de
Física, Universidade Federal de Campina Grande Caixa Postal 10071, 58429-900
Campina Grande, Paraíba, Brazil  F. A. Brito<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de Campina Grande Caixa Postal
10071, 58429-900 Campina Grande, Paraíba, Brazil  Departamento de Física,
Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa,
Paraíba, Brazil  E. Passos<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal de Campina Grande Caixa Postal 10071, 58429-900 Campina
Grande, Paraíba, Brazil

###### Abstract

In this paper we investigate quasinormal modes (QNM) for a scalar field around
a noncommutative Schwarzschild black hole. We verify the effect of
noncommutativity on quasinormal frequencies by applying two procedures widely
used in the literature. The first is the Wentzel-Kramers-Brillouin (WKB)
approximation up to sixth order. In the second case we use the continuous
fraction method developed by Leaver. Besides, we also show that due to
noncommutativity, the shadow radius is reduced when we increase the
noncommutative parameter. In addition, we find that the shadow radius is
nonzero even at the zero mass limit for finite noncommutative parameter.

##  I Introduction

Initial studies for black hole perturbations were done by Regge and Wheeler
[1] and Zerilli [2] for Schwarzschild geometry, as well as for the Kerr black
hole [3]. In 1970 Vishveshwara [4] identified a type of disturbance subject to
special conditions such as outgoing waves in the spatial infinity and ingoing
waves in the vicinity of the event horizon. These disturbances were called
quasinormal modes valid only for a group of complex frequencies [5]. These
quasinormal frequencies present a real part that provides the oscillation
frequency while the imaginary part determines the damping rate of the modes.
The dominant quasinormal modes can be seen in gravitational wave signals and
in this case, the emitted waves are related to many physical processes such as
astrophysical phenomena involving the evolution of binary systems and stellar
oscillations or other highly dense objects in the early universe. In this way
these quasinormal modes have been observed experimentally by LIGO/VIRGO [6,
7].

The analysis of quasinormal modes has been widely explored in the literature
[8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] by using different mechanisms, such
as the WKB approximation and numerical methods. The first works using the WKB
approximation to find quasinormal modes were done by Schutz and Will [19].
Improvements in the method were made by Iyer and Will by adding corrections up
to third order [20], and so the results for Schwarzschild black hole are very
close to those obtained by the numerical method of Leaver [21] in $l\geq 4$
regime. Aiming at a new improvement in the WKB approach, studies made by
Konoplya [22] extended the method up to sixth order leading to more accurate
results. Currently, we can find extensions of the WKB approximation up to
thirteenth order [23]. As aforementioned, the numerical method is another way
to obtain quasinormal frequency modes. Therefore, the first numerical approach
to calculate quasinormal frequencies was described by Leaver, and the applied
mathematical procedure is called a continuous fraction [24]. In [21], Leaver
has obtained quasinormal modes for Schwarzschild and Kerr black holes and also
for Reissner-Nordström black hole in [25]. Several works [26, 27, 28, 29] have
applied this numerical method that has presented a good precision.

In addition to quasinormal modes, in recent years several authors have devoted
themselves to study the shadow of the black hole [30, 31, 32, 33, 34]. This
shadow requires information about the geometry around the black hole, which
makes its study a very important way to understand the properties near the
event horizon. Moreover with the advancement and improvements of the
experimental techniques allowed us the first image of a supermassive black
hole in the center of the M87 galaxy by the Event Horizon Telescope [35, 36]
by using the properties of the shadow. These experimental results have been
studied by several authors [37, 38, 39] stimulated by the possibility of
understanding phenomena in the regime close to the event horizon.

In this work we will use the shadow ray to better understand the proximity of
the noncommutative Schwarzschild black hole horizon. The noncommutative
gravity has been extensively investigated, particularly in black hole physics,
mainly due to the possibility of better understanding the final stage of the
black hole — see [40, 41, 42, 43] for further details. We know that there is a
relationship between quasinormal modes and of the black hole shadow and that
several investigations contributed to this understanding. One of the first
studies that certainly served as a basis for structuring this relationship was
made by Mashhoon [44], which describes an alternative method to calculate
quasinormal modes at the eikonal limit. The most detailed geodesics study is
shown in Cardoso et al [45], which shows that the real part of the quasinormal
modes is related to the angular velocity of the null circular orbit and the
imaginary part is associated with the Lyapunov exponent. Stefanov et al [46]
in the eikonal regime stablished a connection between black hole quasinormal
modes and lensing in the strong deflection limit. Currently, important results
have been obtained at the eikonal limit, such as the relation between the real
part of quasinormal frequencies and the black hole shadow radius [47, 48, 49].

Studies related to quasinormal modes of noncommutative black holes have been
extensively carried out by several authors [50, 51, 52, 53, 54, 55]. In this
paper, we aim to determine the quasinormal modes of noncommutative
Schwarzschild black hole via Lorentzian mass distribution in order to verify
the changes caused by the noncommutative parameter. Moreover, we show that
contrary to the case of the Schwarzschild black hole, the shadow radius
presents a non-zero result at the zero mass limit. Therefore, at this limit,
the shadow radius is proportional to a minimum mass. This result has not been
obtained analytically by using the Gaussian distribution. Thus, considering
the Lorentzian distribution, some results in an analytical way are more easily
explored than in the Gaussian case where this is done numerically.
Furthermore, in [56, 57], we have also found a similar result when
investigating the zero mass limit in the black hole absorption process. In
that case, we have obtained a non-zero absorption at the zero mass limit. By
considering a Lorentzian mass distribution to introduce the noncommutativity,
in [58], we have explored the process of scattering and absorption of scalar
waves through a noncommutative Schwarzschild black hole. Moreover, in [59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74], the thermodynamics of
the BTZ and Schwarzschild black holes in the noncommutative background has
been investigated by using the WKB approach in tunneling formalism [75, 76,
77]. An advantage of using the Lorentzian distribution in analytical calculus
has been investigated in [59] and [60] where logarithmic corrections for
entropy and the condition for black hole remnant formation were obtained.

We organize the paper as follows: In Sec. II we implemented the effect of
noncommutativity in the Schwarzschild black hole metric by a Lorentzian
smeared mass distribution, and we analyze the results for quasinormal
frequencies. In Sec. III we apply the null geodetic method to determine the
shadow of the noncommutative black hole. In Sec. IV we make our final
considerations.

##  II Noncommutative black hole with Lorentzian smeared mass distribution

On this section we begin by considering a Lorentzian distribution [43, 78]
given by

$\displaystyle\rho_{\theta}(r)=\frac{M\sqrt{\theta}}{\pi^{3/2}(r^{2}+\pi\theta)^{2}},$
(1)

where $\theta$ is the noncommutative parameter of dimension ${length}^{2}$ and
$M$ is the total mass diffused throughout the region of linear size
$\sqrt{\theta}$. Thus, the smeared mass distribution function becomes [58]

$\displaystyle{\cal M}_{\theta}$ $\displaystyle=$
$\displaystyle\int_{0}^{r}\rho_{\theta}(r)4\pi r^{2}dr,$ (2) $\displaystyle=$
$\displaystyle\frac{2M}{\pi}\left[\tan^{-1}\left(\frac{r}{\sqrt{\pi\theta}}\right)-\frac{r\sqrt{\pi\theta}}{\pi\theta+r^{2}}\right],$
(3) $\displaystyle=$ $\displaystyle
M-\frac{4M\sqrt{\theta}}{\sqrt{\pi}r}+{\cal O}(\theta^{3/2}).$ (4)

Hence, the line element of the Schwarzschild black hole in the noncommutative
background is now given by

$\displaystyle ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega^{2},$ (5)

with

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1-\frac{2M}{r}+\frac{8M\sqrt{\theta}}{\sqrt{\pi}r^{2}},$ (6) $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}\left(r-r_{+}\right)\left(r-r_{-}\right),$ (7)

where

$\displaystyle r_{+}=M+\sqrt{M^{2}-8M\sqrt{\theta/\pi}}\approx
2M-4\sqrt{\frac{\theta}{\pi}},$ (8) $\displaystyle
r_{-}=M-\sqrt{M^{2}-8M\sqrt{\theta/\pi}}\approx 4\sqrt{\frac{\theta}{\pi}},$
(9)

which represent the radius of the event horizon and the Cauchy horizon,
respectively.

The next step we consider the case of the massless scalar field described by
the Klein-Gordon equation in the background (7)

$\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\right)\Psi=0.$
(10)

Thus, we apply the separation of variables method in the above equation by
using the following Ansatz

$\displaystyle\Psi(\textbf{r},t)=\dfrac{R_{\omega
l}(r)}{r}Y_{lm}(\vartheta,\phi)e^{-i\omega t},$ (11)

where $\omega$ is the frequency and $Y_{lm}(\vartheta,\phi)$ are the spherical
harmonics.

Now, we can obtain a radial equation for $R_{\omega l}(r)$:

$\left(r-r_{-}\right)\left(r-r_{+}\right)\dfrac{d^{2}R_{\omega
l}(r)}{dr^{2}}+r^{2}\dfrac{df(r)}{dr}\dfrac{dR_{\omega
l}(r)}{dr}+\left[\dfrac{\omega^{2}r^{4}}{\left(r-r_{-}\right)\left(r-r_{+}\right)}-r\dfrac{df(r)}{dr}-l(l+1)\right]R_{\omega
l}(r)=0.$ (12)

We can reduce the radial equation (12) into a Schrödinger like equation by
introducing a new coordinate (called tortoise coordinate) given by
$dr_{*}=f(r)^{-1}dr$ and

$r_{*}=r+\dfrac{r_{-}^{2}}{r_{-}-r_{+}}\log(r-r_{-})-\dfrac{r_{+}^{2}}{r_{-}-r_{+}}\log(r-r_{+}),$
(13)

so that the radial equation becomes

$\displaystyle\dfrac{d^{2}R_{\omega
l}(r_{*})}{dr_{*}^{2}}+\left[\omega^{2}-V_{eff}\right]R_{\omega l}(r_{*})=0,$
(14)

where

$V_{eff}=\dfrac{f(r)}{r}\dfrac{df(r)}{dr}+\dfrac{f(r)l(l+1)}{r^{2}}.$ (15)

In the following sections, we will obtain the quasinormal frequencies by two
methods that are widely used in the literature. The first uses a sixth order
WKB approximation, and the second method introduced by Leaver and improved by
Nollert [82], consists of using the continuous fraction method to find
numerically quasinormal modes.

###  II.1  WKB approximation

Quasinormal modes correspond to solutions of the wave equation (14) that
satisfy the conditions of the purely outgoing waves at infinity and purely
incoming waves at the event horizon, i.e.,

$R_{\omega l}(r_{*})\sim e^{\pm i\omega r_{*}},\qquad
r_{*}\rightarrow\pm\infty.$ (16)

In this section we will use the WKB approximation to find the quasinormal
modes. The first works using the WKB approximation to evaluate the quasinormal
modes were done by Schutz and Will [19]. Improvements in the method were made
using corrections up to third order [20, 79] and up to sixth order by Konoplya
[22]. The quasinormal modes are obtained by using the sixth order corrections
for the WKB approximation as follows

$\dfrac{i\left(\omega_{n}^{2}-\bar{V}_{eff}\right)}{\sqrt{-2\bar{V}_{eff}^{\prime\prime}}}-\sum_{j=2}^{6}\Omega_{j}=n+\dfrac{1}{2},$
(17)

where $\Omega_{j}$ are the correction terms of the model. We have that
$\bar{V}_{eff}$ is the maximum effective potential at point $\bar{r}_{*}$ and
$(^{\prime\prime})$ refers to the second derivative with respect to the
tortoise coordinate. We can obtain the values of $\bar{r}_{*}$ by making
$\bar{V}_{eff}^{\prime}=0$. In figure 1 we show the curves of the effective
potential for $l=1,2$ and $\Theta=0.0,0.05,0.10,0.12$, where we define
$\Theta=\sqrt{\theta}/(M\sqrt{\pi})$ for $M=1$. In the tables (1 \- 3) we
present the tabulated quasinormal frequencies using sixth order WKB method.

Figure 1: The effective potential $V_{eff}$ as function of the tortoise
coordinate $r_{*}$ (a) $l=1$ and (b) $l=2$.

Another possibility is to study scattering by using the WKB method done in
[20]. In order to develop this investigation, we use the boundary conditions
for equation (14) in the form:

$R_{\omega l}(r_{*})=\begin{cases}A_{int}e^{-i\omega r_{*}}+A_{out}e^{i\omega
r_{*}},\qquad r_{*}\rightarrow\infty,\\\ A_{tr}e^{-i\omega r_{*}},\hskip
76.82234ptr_{*}\rightarrow-\infty.\end{cases}$ (18)

Notice that as we want to obtain the reflection and transmission coefficients,
as done for tunneling in quantum mechanics, we need the condition $A_{int}\neq
0$. The quantity $\left(\omega^{2}-V_{eff}\right)$ in $\eqref{ERtor}$ is
assumed to be purely real, and with these imposed conditions we can find

$\mathcal{K}=\dfrac{i\left(\omega^{2}-\bar{V}_{eff}\right)}{\sqrt{-2\bar{V}_{eff}^{\prime\prime}}}-\sum_{j=2}^{6}\Omega_{j}(\mathcal{K}),$
(19)

where $\omega$ is purely real and $\Omega_{j}(\mathcal{K})$ are coefficients
that depend on the effective potential and $\mathcal{K}$ is a purely imaginary
quantity. This way of studying scattering by using the WKB approximation can
be also found in [23, 80]. Thus, using the relationship between $\mathcal{K}$
and the reflection and transmission coefficients obtained in [20] we get the
following:

$\displaystyle|R|^{2}=\dfrac{|A_{out}|^{2}}{|A_{int}|^{2}}$ $\displaystyle=$
$\displaystyle\dfrac{1}{1+e^{-2i\pi\mathcal{K}}},\qquad 0<|R|^{2}<1,$ (20)
$\displaystyle|T|^{2}=\dfrac{|A_{tr}|^{2}}{|A_{int}|^{2}}$ $\displaystyle=$
$\displaystyle\dfrac{1}{1+e^{2i\pi\mathcal{K}}}=1-|R|^{2}.$ (21)

Now to find the coefficients, we just calculate the value $\mathcal{K}$ that
can be obtained by solving the equation (19). This method has a good
approximation for $l>0$ as we can see in Figure 2, where we have a comparison
between the numerical results and the WKB approximation for the transmission
coefficient for $l=1,2,3$ and $\Theta=0.05,0.12$. Notice that the curves
obtained by the WKB approximation are very close to those obtained by the
numerical method used in the paper [58], showing that the method presents
excellent results. The sixth-order WKB approximation does not show good
results for $l=0$, improving only when we take large $\omega$. This problem is
also mentioned in [23].

Figure 2: Transmission coefficients for three multipole $l=1,2,3$ (from left
to right) and $\Theta=0.05,0.12$. The approximation between the two methods is
very good.

###  II.2 Leaver’s continued fraction

The numerical method is another way to obtain the quasinormal frequency, and
this procedure has been described by Leaver [21, 25], and which is also found
in other works [26, 27] showing that it is a method with good precision.

We start analyzing the radial equation (12) which is subject to boundary
conditions at infinity $r\rightarrow\infty$ and near the event horizon
$r\rightarrow r_{+}$, such that one obtains the asymptotic solutions

$R_{\omega l}(r)\approx\begin{cases}e^{i\omega
r}r^{i\omega(r_{-}+r_{+})},\hskip 183.52019ptr\rightarrow\infty,\\\
e^{-i\omega r_{+}}\left(r_{+}-r_{-}\right)^{-i\omega
r_{-}^{2}/(r_{-}-r_{+})}\left(r-r_{+}\right)^{i\omega
r_{+}^{2}/(r_{-}-r_{+})},\qquad r\rightarrow r_{+}.\end{cases}$ (22)

We can obtain a solution that has the desired behavior on the horizon
($r=r_{+}$), and that can be written in the form

$R=\frac{r}{r-r_{-}}\left(r-r_{-}\right)^{i\omega(r_{+}+r_{-})}\left(\frac{r-r_{+}}{r-r_{-}}\right)^{\frac{-i\omega
r_{+}^{2}}{r_{+}-r_{-}}}e^{i\omega
r}\sum_{k=0}^{\infty}a_{k}\left(\frac{r-r_{+}}{r-r_{-}}\right)^{k}.$ (23)

By replacing the solution (23) in the equation (12), we obtain the recurrence
relation

$\alpha_{0}a_{1}+\beta_{0}a_{0}=0,$ (24)

and

$\alpha_{k}a_{k+1}+\beta_{k}a_{k}+\gamma_{k}a_{k-1}=0,\qquad k\geq 1.$ (25)

The recurrence relation coefficients $\alpha_{k}$, $\beta_{k}$ and
$\gamma_{k}$ are simple functions of $k$ and the parameters $\omega$, $l$ and
the radius $r_{-}$ and $r_{+}$:

$\displaystyle\alpha_{k}$ $\displaystyle=$
$\displaystyle\left(1+k\right)\left[r_{-}\left(k+1\right)-r_{+}\left(1+k-2i\omega
r_{+}\right)\right],$ (26) $\displaystyle\beta_{k}$ $\displaystyle=$
$\displaystyle r_{+}\left[l(l+1)+1+2\left(k-2i\omega
r_{+}\right)\left(1+k-2i\omega
r_{+}\right)\right]-r_{-}\left[l(l+1)+1-2i\omega r_{+}+2k\left(1+k-2i\omega
r_{+}\right)\right],$ (27) $\displaystyle\gamma_{k}$ $\displaystyle=$
$\displaystyle\left[k-2i\omega\left(r_{-}+r_{+}\right)\right]\left[k\left(r_{-}-r_{+}\right)+2i\omega
r_{+}^{2}\right].$ (28)

See that the solution (23) applied directly to the radial equation returns a
three-term recurrence relation which makes it easier to use continued
fractions. This is because our metric for a non-commutative black hole results
in a generalized spheroidal wave equation whose solutions are connected by
three-term recurrence relations — see Appendix V for further details. However,
the Leaver’s method is not necessarily limited to recurrence relations of this
type as shown in [25], where it was considered an equation that describes odd-
parity perturbations of a charged black hole, which equation has series
solutions whose coefficients are connected by four-term recurrence relations.

The boundary condition at infinity will be satisfied for quasinormal frequency
values $\omega=\omega_{n}$, so that the series in (23) is absolutely
convergent. Hence, we have a recurrence relation with three terms to determine
the coefficient $a_{k}$, and we can write in terms of a continuous fraction
[81]

$\dfrac{a_{k+1}}{a_{k}}=\dfrac{-\gamma_{k+1}}{\beta_{k+1}-\dfrac{\alpha_{k+1}\gamma_{k+2}}{\beta_{k+2}-\dfrac{\alpha_{k+2}\gamma_{k+3}}{\beta_{k+2}-\cdots}}},$
(29)

which can also be found as follows

$\dfrac{a_{k+1}}{a_{k}}=\dfrac{-\gamma_{k+1}}{\beta_{k+1}-}\dfrac{\alpha_{k+1}\gamma_{k+2}}{\beta_{k+2}-}\dfrac{\alpha_{k+2}\gamma_{k+3}}{\beta_{k+3}-\cdots}.$
(30)

We can obtain the characteristic equation for quasinormal frequencies by
assigning $k=0$ in (30) and comparing with reason
$a_{1}/a_{0}=-\beta_{0}/\alpha_{0}$ obtained from (24),

$0=\dfrac{\beta_{0}}{\alpha_{0}}-\dfrac{\gamma_{1}}{\beta_{1}-}\dfrac{\alpha_{1}\gamma_{2}}{\beta_{2}-}\dfrac{\alpha_{2}\gamma_{3}}{\beta_{3}-\cdots}.$
(31)

With the equation above, we can obtain the quasinormal frequencies
$\omega_{n}$, by just calculating its roots numerically. However, the equation
(31) is more used to find the fundamental frequency in the case of the more
stable root, and another way of finding the modes is to invert this equation
to a large number of $k$ as follows

$\beta_{k}-\dfrac{\alpha_{k-1}\gamma_{k}}{\beta_{k-1}-}\dfrac{\alpha_{k-2}\gamma_{k-1}}{\beta_{k-2}-}\cdots\dfrac{\alpha_{0}\gamma_{1}}{\beta_{0}}=\dfrac{\alpha_{k}\gamma_{k+1}}{\beta_{k+1}-}\dfrac{\alpha_{k+1}\gamma_{k+2}}{\beta_{k+2}-}\cdots.$
(32)

To complement the analysis, we will check the behavior for very large $k$, as
done in [82]. Now, we have to reorganize the equation (25) dividing $\alpha$,
$\beta$ and $\gamma$ by $a_{k}$ to obtain

$\alpha_{k}\dfrac{a_{k+1}}{a_{k}}+\beta_{k}+\gamma_{k}\dfrac{a_{k-1}}{a_{k}}=0.$
(33)

We can see that, $\lim\limits_{k\to\infty}(a_{k+1}/a_{k})\simeq 1$, and we can
get a more complete expression by expanding $a_{k+1}/a_{k}$ in power series in
terms of $\sqrt{k}$,

$\lim\limits_{k\to\infty}\dfrac{a_{k+1}}{a_{k}}=\sum_{i=0}^{\infty}C_{i}k^{-i/2}=C_{0}+\dfrac{C_{1}}{\sqrt{k}}+\dfrac{C_{2}}{k}+\dfrac{C_{3}}{k^{3/2}}+\cdots.$
(34)

By considering the series up to the third term and admitting $C_{0}=1$ we have

$\displaystyle\lim\limits_{k\to\infty}\dfrac{a_{k+1}}{a_{k}}$
$\displaystyle\approx$ $\displaystyle
1+\dfrac{C_{1}}{\sqrt{k}}+\dfrac{C_{2}}{k}+\cdots,$ (35)
$\displaystyle\lim\limits_{k\to\infty}\dfrac{a_{k-1}}{a_{k}}$
$\displaystyle\approx$ $\displaystyle
1-\dfrac{C_{1}}{\sqrt{k}}+\dfrac{(C_{1})^{2}-C_{2}}{k}+\dfrac{C_{1}\left(2C_{2}-1/2-(C_{1})^{2}\right)}{k^{3/2}}+\cdots.$
(36)

Now, to obtain the values of $C_{1}$ and $C_{2}$, we replace these two
expressions above into equation (33) in the very large $k$ regime by making
the multiplications and restricting up to terms of the order $k^{-3/2}$ to get

$\dfrac{(C_{1})^{2}-2i\omega\left(r_{-}-r_{+}\right)}{k}-\dfrac{C_{1}}{2}\dfrac{\left[-3+2(C_{1})^{2}-4C_{2}-4i\omega\left(r_{-}+r_{+}\right)\right]}{k^{3/2}}\approx
0,$ (37)

such that we find the following:

$\displaystyle\left(C_{1}\right)^{2}$ $\displaystyle=$ $\displaystyle
2i\omega\left(r_{-}-r_{+}\right),$ (38) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle-2i\omega r_{+}-\dfrac{3}{4}.$ (39)

Now we have,

$\lim\limits_{k\to\infty}\dfrac{a_{k+1}}{a_{k}}\approx
1\pm\dfrac{\sqrt{2i\omega\left(r_{-}-r_{+}\right)}}{\sqrt{k}}-\dfrac{2i\omega
r_{+}+3/4}{k}.$ (40)

Here, we see that at the limit $r_{-}\rightarrow 0$ and $r_{+}\rightarrow 1$,
we get the result for the Schwarzschild case initially found by Leaver.

###  II.3 Results

In the Tables (1 \- 3), we present some results for the quasinormal modes
computed using the sixth-order WKB approximation, and the Leaver’s continues
fraction method described in the previous section by admitting $M=1$ for
various values of $\Theta=\sqrt{\theta/\pi}$, $l$ and $n$. We see that the
results between the methods approach when $l>1$, this is, due to the
instability of the WKB method for small multipole numbers mainly close to zero
this instability can also be seen in the graphs of Fig. 3. An important detail
is in the sign of the imaginary part that is always negative when the
frequency is associated with the scalar field. A justification for this is due
to the exponential drop of the quasinormal modes over time by losing energy in
the form of scalar waves.

We can see the influence in the quasinormal modes for the noncommutative case
by admitting values for $\Theta$ where $\Theta=0$ returns to the Schwarzschild
case. With the increase of the non-commutative parameter $\Theta$, we have an
increase in the real part of the quasinormal frequency, while the imaginary
part begins to grow and then decreases. Another way of visualizing the effects
of the noncommutative parameter is through the graphs of Fig. 3, these plots
were obtained using the WKB method. We have quasinormal modes where in the
plot we depicted the real part (top) and the imaginary part (bottom). The
modes are based on $n$ for the following multipoles numbers $l=1,2,3$ and $4$,
for which we see that the results become more linear with the results varying
$\Theta$ and $l=3$ and $4$. Thus, in Fig. 3 we observe that, by varying $l$
and the parameter $\theta$, the imaginary part of the frequency does not cross
the horizontal axis or change sign, thus indicating that the black hole
remains stable due to scalar perturbation. In addition, it is interesting to
make a plot for the complex plane as in Fig. 4 where we consider three
families of multipoles $l=1,2,3$ and varying $\Theta$ as follows $0$ (black),
$0.05$ (red), $0.10$ (blue), $0.12$ (green). The left panel was obtained by
the WKB approximation and the right panel by the continuous fraction method
(numerical). We can see that the frequency curves incline more closely when we
use the WKB approximation. We also see that for the extreme case $\Theta=0.12$
the curve tilts more to the left in both methods.

Table 1: QN frequencies for $l=1$ |  $\omega_{0}$ |  $\omega_{1}$ |  $\omega_{2}$
---|---|---|---
$\Theta$ | 6th order WKB | numerical | 6th order WKB | numerical | 6th order WKB | numerical
0.00 | 0.292910 - 0.097762i | 0.292936 - 0.0976600i | 0.264471 - 0.306518i | 0.264449 - 0.306257i | 0.231014 - 0.542166i | 0.229539 - 0.540133i
0.05 | 0.316239 - 0.099311i | 0.316243 - 0.0992441i | 0.290671 - 0.309586i | 0.290591 - 0.309431i | 0.260297 - 0.543374i | 0.258427 - 0.542162i
0.10 | 0.351481 - 0.097397i | 0.351435 - 0.0973776i | 0.330103 - 0.300074i | 0.329892 - 0.300209i | 0.300773 - 0.519039i | 0.299190 - 0.519345i
0.12 | 0.371967 - 0.091934i | 0.371932 - 0.0919646i | 0.346286 - 0.282086i | 0.346358 - 0.282153i | 0.300083 - 0.491675i | 0.301082 - 0.491385i
Table 2: QN frequencies for $l=2$ |  $\omega_{0}$ |  $\omega_{1}$ |  $\omega_{2}$
---|---|---|---
$\Theta$ | 6th order WKB | numerical | 6th order WKB | numerical | 6th order WKB | numerical
0.00 | 0.483642 - 0.096766i | 0.483644 - 0.0967588i | 0.463847 - 0.295627i | 0.463851 - 0.295604i | 0.430386 - 0.508700i | 0.430544 - 0.508558i
0.05 | 0.521842 - 0.098436i | 0.521844 - 0.0984288i | 0.504088 - 0.300022i | 0.504087 - 0.299997i | 0.474038 - 0.514099i | 0.474067 - 0.513965i
0.10 | 0.580028 - 0.096808i | 0.580028 - 0.0968027i | 0.565520 - 0.293676i | 0.565511 - 0.293660i | 0.540100 - 0.499152i | 0.540039 - 0.499083i
0.12 | 0.615658 - 0.091534i | 0.615659 - 0.0915307i | 0.599809 - 0.276911i | 0.599817 - 0.276896i | 0.569259 - 0.469194i | 0.569307 - 0.469133i
Table 3: QN frequencies for $l=3$ |  $\omega_{0}$ |  $\omega_{1}$ |  $\omega_{2}$
---|---|---|---
$\Theta$ | 6th order WKB | numerical | 6th order WKB | numerical | 6th order WKB | numerical
0.00 | 0.675366 - 0.096501i | 0.675366 - 0.0964996i | 0.660671 - 0.292288i | 0.660671 - 0.292285i | 0.633591 - 0.496011i | 0.633626 - 0.496008i
0.05 | 0.728594 - 0.098197i | 0.728594 - 0.0981957i | 0.715414 - 0.297060i | 0.715414 - 0.297056i | 0.691124 - 0.502924i | 0.691137 - 0.502911i
0.10 | 0.809869 - 0.096642i | 0.809869 - 0.0966413i | 0.799161 - 0.291636i | 0.799159 - 0.291633i | 0.779124 - 0.491455i | 0.779117 - 0.491442i
0.12 | 0.860294 - 0.091410i | 0.860294 - 0.0914089i | 0.848871 - 0.275415i | 0.848871 - 0.275412i | 0.826463 - 0.462957i | 0.826463 - 0.462950i

Figure 3: Real (top) and imaginary (bottom) parts of the quasinormal
frequencias as function of the $n$. We see that for $l=1$ the frequency curves
are very dispersed with the increase of $n$, while for $l=3,4$ the curves of
the real part of the frequency are more constant.

Figure 4: Complex plane of the QNMs. In plot (a) we have the results obtained
by the WKB approximation, while in (b) we use continuous fraction. The markers
denote the multipole number as: $l=1$(circle), $l=2$(square) and
$l=3$(diamond), while the colors denote the value of the noncomutative
parameter $\Theta=0$(black), $\Theta=0.05$(red), $\Theta=0.10$(blue) and
$\Theta=0.12$(green).

##  III Null geodesic and Shadow of a noncommutative black hole

We know that in the vicinity of a black hole all the photons are absorbed so
that a distant observer looking at the black hole, in absence of any other
source, will see a spot created by this absorption, and this spot is usually
called the shadow of the black hole. These shadows have been studied long ago
by Synge [83] and Luminet [84] who started studies for Schwarzschild black
hole while Kerr black holes were studied by Bardeen [85].

As these shadows correspond to the apparent shape of the photon capture
orbits, the space-time metric itself is enough to determine them and thereby
better understand the geometry of the near horizon. One method of determining
the apparent shape of the black hole is through the shadow boundary that can
be studied by the equations of null geodesics. Similar studies have been done
in different contexts [87, 88, 89, 90, 91, 92]

###  III.1 Null geodesic

We can find the geodesics from equation (7) by taking a Lagrangian in the form

$\mathcal{L}\equiv\dfrac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}.$ (41)

Thus, we have

$2\mathcal{L}=f(r)\dot{t}^{2}-\dfrac{\dot{r}^{2}}{f(r)}-r^{2}\left(\dot{\vartheta}^{2}+\sin^{2}\vartheta\dot{\phi}^{2}\right),$
(42)

where the “$\cdot$” is the derivative with respect to an affine parameter.

We are interested in the path of a ray of light in the described metric, which
is spherically symmetrical, so if we analyze in a plane, any ray of light that
begins with a certain angle $\vartheta$ must remain with the same angle. We
will then consider an equatorial plane by setting the angle $\vartheta$ to
$\pi/2$.

Thus, two equations are enough to describe the movement of a beam of light. We
can put together a system with these equations that give rise to two geodesic
motion constants $E$ and $L$, which correspond to energy and angular momentum
respectively:

$E=f(r)\dot{t},\qquad\quad L=r^{2}\dot{\phi}.$ (43)

Now, as our goal is to study the null geodesics we have to
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0$, and using the equations (43) we can
write

$\dot{r}^{2}+f(r)\dfrac{L^{2}}{r^{2}}=E^{2}.$ (44)

Introducing a new variable $u=1/r$ we can write the orbit equation as follows

$\dfrac{du}{d\phi}=\sqrt{\dfrac{1}{b^{2}}-u^{2}+2Mu^{3}-\dfrac{8M\sqrt{\theta}}{\sqrt{\pi}}u^{4}},$
(45)

where $b=L/E$ is the impact parameter defined as the perpendicular distance
(measured at infinity) between the geodesic and a parallel line that passes
through the origin. So differentiating (45) we have,

$\dfrac{d^{2}u}{d\phi^{2}}=-u+3Mu^{2}-\dfrac{16M\sqrt{\theta}}{\sqrt{\pi}}u^{3}.$
(46)

By solving the equations (45) and (46) numerically, we can obtain the behavior
of the geodesic lines for different values of the impact parameter $b$. In the
figure 5, we verify the change of the geodesic lines for different impact
parameters $b$ and also by varying the values of the noncommutative parameter.
In the figures, we have a black disk that represents the limit of the event
horizon, the internal dotted circle is the radius for the photon sphere
(critical radius), and the external dashed circle is the critical impact
parameter (shadow). Hence, we see that the noncommutative parameter decreases
the effect of the black hole on the light beams. For a similar effect see also
[93].

Figure 5: Geodesics surrounding a noncommutative black hole. The impact
parameters defined as $b=3.6,4.1,4.7,5.2,5.9$ and $6.5$ are the same for all
graphs, assuming $M=1$. We can clearly see the influence of the noncommutative
parameter on the geodesic curves from (a) $\Theta=0$ (Schwarzschild case) to
(d) $\Theta=0.12$.

###  III.2 Critical orbit and shadows

It is known that the shadow of the black hole is directly related to the
impact parameter for the photon orbit, as we will see below. So to determine
the shadow limit, we will start by studying the effective potential that
satisfies the equation of the null geodesic as follows

$\dot{r}^{2}+{\cal V}_{eff}(r)=0,$ (47)

where using (44), we have

$\displaystyle{\cal V}_{eff}(r)=f(r)\dfrac{L^{2}}{r^{2}}-E^{2}.$ (48)

In this case, we can obtain a critical radius or critical circular orbit for a
photon $r_{c}$ and critical impact parameter $b_{c}$, by using the following
conditions: ${\cal V}_{eff}(r_{c})=0$ and $\dfrac{d{\cal
V}_{eff}(r_{c})}{dr}=0$. So we find

$\displaystyle r_{c}$ $\displaystyle=$
$\displaystyle\frac{3}{4}\left(r_{+}+r_{-}\right)+\frac{3}{4}\sqrt{(r_{+}+r_{-})^{2}-\frac{32r_{+}r_{-}}{9}},$
(49) $\displaystyle=$
$\displaystyle\frac{3M}{2}+\frac{3M}{2}\sqrt{1-\frac{64\sqrt{\theta}}{9M\sqrt{\pi}}},$
(50) $\displaystyle b_{c}$ $\displaystyle=$
$\displaystyle\dfrac{r_{c}}{\sqrt{f(r_{c})}}=\frac{r^{2}_{c}}{\sqrt{(r_{c}-r_{+})(r_{c}-r_{-})}}.$
(51)

Let us now compute the size of the black hole shadow that can be expressed via
celestial coordinates as follows [85]

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\lim\limits_{r_{o}\to\infty}\left[-r_{o}^{2}\sin\vartheta_{o}\dfrac{d\phi}{dr}\Bigr{\rvert}_{\vartheta=\vartheta_{o}}\right],$
(52) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\lim\limits_{r_{o}\to\infty}\left[r_{o}^{2}\dfrac{d\vartheta}{dr}\Bigr{\rvert}_{\vartheta=\vartheta_{o}}\right],$
(53)

where $\left(r_{o},\vartheta_{o}\right)$ is the observer position at infinity.

As our study is restricted to the equatorial plane, the radius that delimits
the size of the shadow is equivalent to the critical impact parameter, and so
we have

$R_{s}\equiv\sqrt{\alpha^{2}+\beta^{2}}=b_{c},$ (54)

where

$R_{s}=\left[\dfrac{\left(3M+\sqrt{9M^{2}-{64M\sqrt{\theta}}/{\sqrt{\pi}}+{128{\theta}}/{{\pi}}}\right)^{4}}{8\left(3M^{2}-{16M\sqrt{\theta}}/{\sqrt{\pi}}+{32\theta}/{\pi}+M\sqrt{9M^{2}-{64M\sqrt{\theta}}/{\sqrt{\pi}}+{128{\theta}}/{{\pi}}}\right)}\right]^{1/2}.$
(55)

In a semiclassical description of the scattering [86], the impact parameter is
associated with each partial wave $b=(l+1/2)/\omega$ in the large $l$ regime.
As shown in [45], the real part of the quasinormal frequencies at the eikonal
limit corresponds to the angular velocity for the last null circular orbit
$\Omega_{c}$ and the imaginary part is associated with the Lyapunov exponent
$\lambda$ which determines the unstable timescale of the orbit

$\displaystyle\omega_{QNM}=\Omega_{c}l-i\left(n+\frac{1}{2}\right)|\lambda|,$
(56)

being the angular velocity given by

$\displaystyle\Omega_{c}=\frac{\dot{\phi}}{\dot{t}}=\frac{f(r_{c})b_{c}}{r^{2}_{c}}=\frac{1}{b_{c}}$
(57)

This suggests a relationship between quasinormal frequencies and shadow radius
as done by Jusufi [47] at the eikonal limit, showing that the real part of
quasinormal modes is inversely proportional to the radius $R_{s}$ as follows

$Re(\omega)=\lim_{l>>1}\dfrac{l}{R_{s}}.$ (58)

Note that this expression is valid only for large values of $l$ in most of the
cases although fails for Einstein-Lovelock theory as shown by Konoplya and
Stuchlik [94]. We can see in Fig. 3 that the higher the value of $l$ the
smaller the contribution of $n$. Thus by using the results obtained by the WKB
approximation we compare with the shadow radius $R_{s}$.

Table 4: The real part of quasinormal frequencies for large $l$ compared with shadow radius $R_{s}$ $M=1$ |  $Re(\omega)/\left(l+1/2\right)$ |  $\left(R_{s}\right)^{-1}$
---|---|---
$\Theta$ | $l=100$ | $l=1500$ | $----$
0.00 | 0.192450707 | 0.192450092 | 0.192450090
0.05 | 0.207583654 | 0.207582949 | 0.207582946
0.10 | 0.230750607 | 0.230749836 | 0.230749832
0.12 | 0.245314933 | 0.245314347 | 0.245314347

In the table 4 we see that by increasing the value of $l$, the results between
the real part of the quasinormal frequencies and the black hole shadow radius
approach each other. Now, we can express $R_{s}$ by considering $\theta$
small, and so we get the following approximate expression

$R_{s}\approx
3\sqrt{3}M-4\sqrt{3}\sqrt{\frac{\theta}{\pi}}+\frac{16}{3\sqrt{3}M}\frac{\theta}{\pi}+\cdots.$
(59)

Note that for $\theta=0$, we have the shadow radius for the Schwarzschild
black hole case. Therefore, we notice that the shadow radius is reduced when
we change the parameter $\theta$. In Fig. 6, we see the circles that represent
the shadow boundaries of the noncommutative black hole for different values of
$\Theta$. Note that we have a reduction in the circles when we vary $\Theta$.
Furthermore, taking $M\rightarrow 0$ in (55), we obtain a non-zero result for
the shadow radius, that is

$R_{s}\approx 8M_{min},$ (60)

where $M_{min}=\sqrt{\theta/\pi}$ is the minimal mass [60]. Therefore, at the
limit of $M\rightarrow 0$ the shadow radius is proportional to the minimum
mass and the black hole becomes a black hole remnant. We have shown this
behavior in Fig. 7. In Fig. 7, we show the behavior of the shadow radius by
keeping $\Theta$ fixed and assuming small values of $M$. In Fig. 8, We show
the behavior of the shadow radius by keeping $M$ fixed and assuming small
values of $\Theta$.

Figure 6: We see the influence of non-commutativity in the shadow admitting
$M=1$ and $\Theta=0.0,0.05,0.10$ and $0.12$.

Figure 7: We see the influence of non-commutativity in the shadow admitting
(a) $\Theta=0.01$ and $M=1.0,0.6,0.4,0.2,0.09$. (b) $\Theta=0.005$ and
$M=0.1,0.09,0.08,0.06,0.04$.

Figure 8: We see the influence of non-commutativity in the shadow admitting
(a) $M=0.03$ and $\Theta=0.000,0.001,0.002,0.003$. (b) $M=0.05$ and
$\Theta=0.000,0.001,0.002,0.003$.

##  IV Conclusions

In summary, in this work, we investigate the quasinormal frequencies for a
noncommutative Schwarzschild black hole by two different methods in order to
investigate and compare the results. Using the sixth-order WKB approximation
and Leaver’s continuous fraction, we found that there is a small difference
between the quasinormal frequencies obtained by each method mainly for small
multipoles. The effects of the noncommutative parameter $\Theta$ cause an
increase in the real part of the quasinormal frequencies, while the magnitude
of the imaginary part begins to grow and then decreases. For the black hole
shadow we use the results obtained by the WKB method to verify that in large
$l$ regimes the real part of quasinormal modes is inversely proportional to
the shadow radius. In addition, we have shown that the shadow radius is non-
zero at the zero mass limit. Therefore being proportional to a minimum mass.
However for $\theta=0$, we recover the shadow radius for the Schwarzschild
black hole case. Finally, we also notice that the shadow radius is reduced
when we increase the noncommutative parameter.

###### Acknowledgements.

We would like to thank CNPq, CAPES and CNPq/PRONEX/FAPESQ-PB (Grant nos.
165/2018 and 015/2019), for partial financial support. MAA, FAB and EP
acknowledge support from CNPq (Grant nos. 306962/2018-7 and 433980/2018-4,
312104/2018-9, 304852/2017-1).

##  V Appendix

Here we will show that the differential equation (12) can be written in the
form of a generalized spheroidal wave equation, as done by Leaver for the
Schwarzschild and Kerr cases in [95]. The generalized spheroidal wave equation
has the following form

$x(x-x_{0})\dfrac{d^{2}y}{dx^{2}}+\left(B_{1}+B_{2}x\right)\dfrac{dy}{dx}+\left[\omega^{2}x(x-x_{0})-2\eta\omega(x-x_{0})+B_{3}\right]y=0.$
(61)

The radial equation (12) can be rewritten as follows

$\displaystyle\left(r-r_{-}\right)\left(r-r_{+}\right)\dfrac{d^{2}R(r)}{dr^{2}}+\dfrac{\left(r(r_{+}+r_{-})-2r_{-}r_{+}\right)}{r}\dfrac{dR(r)}{dr}+\Big{[}\dfrac{\omega^{2}r^{4}}{\left(r-r_{-}\right)\left(r-r_{+}\right)}-\dfrac{\left(r(r_{+}+r_{-})-2r_{-}r_{+}\right)}{r^{2}}$
$\displaystyle-l(l+1)\Big{]}R(r)=0.$ (62)

In this appendix, by considering $2M=1$, for simplicity, we have
$r_{\pm}=\left(1\pm b\right)/2$, where $b=\sqrt{1-16\sqrt{\theta/\pi}}$.
Firstly, we have to find a suitable transformation, by using the following
solution

$\displaystyle R(r)=r(r-r_{-})^{\alpha}(r-r_{+})^{\beta}y(r-r_{-}),$ (63)

where $\alpha$ and $\beta$ are parameters to be determined. Furthermore, let
us assume that $x=r-r_{-}$, with $x\rightarrow 0$ as $r\rightarrow r_{-}$ and
$x\rightarrow b$ as $r\rightarrow r_{+}$. Now applying these changes we have
that the equation (62) can be written in the form

$x(x-b)\dfrac{d^{2}y}{dx^{2}}+\left[b(1+\alpha)+2(1+\alpha+\beta)x\right]\dfrac{dy}{dx}+U(x)y=0,$
(64)

where

$\displaystyle U(x)=$ $\displaystyle-$
$\displaystyle\dfrac{\left[(b-1)^{4}\omega^{2}+16b^{2}\alpha^{2}\right]}{16bx}+\dfrac{\left[(b+1)^{4}\omega^{2}+16b^{2}\beta^{2}\right]x}{16b^{2}\left(x-b\right)}+\dfrac{\left(b-1\right)^{3}\left(1+7b\right)\omega^{2}}{16b^{2}}+(2-b)x\omega^{2}+x^{2}\omega^{2}$
(65) $\displaystyle+$ $\displaystyle\alpha+2\alpha^{2}+\beta+2\alpha-l(l+1).$

In order to have an equation of the form (61) the first two terms of (65) must
be zero, that is, $(b-1)^{4}\omega^{2}+16b^{2}\alpha^{2}=0$ and
$(1+b)^{4}\omega^{2}+16b^{2}\beta^{2}=0$, so we have the following values for
$\alpha$ and $\beta$

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle-\dfrac{i(b-1)^{2}\omega}{4b},\qquad\dfrac{i(b-1)^{2}\omega}{4b}$
(66) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle-\dfrac{i(b+1)^{2}\omega}{4b},\qquad\dfrac{i(b+1)^{2}\omega}{4b}.$
(67)

Thus, for $\alpha=-i(b-1)^{2}\omega/4b=-ir_{-}^{2}\omega/b$ and
$\beta=-i(b+1)^{2}\omega/4b=-ir_{+}^{2}\omega/b$, the equation (64) is of the
form

$\displaystyle
x(x-b)\dfrac{d^{2}y}{dx^{2}}+\left[-b+2ir_{-}^{2}\omega+\dfrac{2}{b}(b-i\omega(r_{-}^{2}+r_{+}^{2}))x\right]\dfrac{dy}{dx}+\left[\omega^{2}(x-b)x+2\omega^{2}(x-b)\right.$
(68)
$\displaystyle\left.+\dfrac{\omega^{2}}{4b}\left(-1+4b+4b^{2}+b^{4}\right)-\dfrac{i\omega}{2b}\left(b^{2}+1\right)-l(l+1)\right]y=0.$

This implies that the equation (62) can be transformed into a generalized
spheroidal equation. By comparing it with (61) we have:

$\displaystyle B_{1}=-b+2ir_{-}^{2}\omega,\qquad
B_{2}=\dfrac{2}{b}(b-i\omega(r_{-}^{2}+r_{+}^{2})),$ (69) $\displaystyle
B_{3}=\dfrac{\omega^{2}}{4b^{2}}\left(-1+4b^{2}+4b^{3}+b^{4}\right)-\dfrac{i\omega}{2b}\left(b^{2}+1\right)-l(l+1)\qquad\text{and}\qquad\eta=-\omega.$
(70)

Thus, by using Jeffe’s regular solution to $y(x)$ as in [95] (p. 12) for a
generalized spheroidal wave equation we have

$\displaystyle y(x)=e^{i\omega
x}x^{-1+i\omega(r_{-}^{2}+r_{-}^{2})/b+i\omega}\sum_{k=0}^{\infty}a_{k}\left(\frac{x-b}{x}\right)^{k}.$
(71)

We can also compare the solution (63) with the values of $\alpha$ and $\beta$
to the solution (71), by rewriting it in the form

$R(r)=\frac{r}{r-r_{-}}\left(r-r_{-}\right)^{i\omega}\left(\frac{r-r_{+}}{r-r_{-}}\right)^{\frac{-i\omega
r_{+}^{2}}{b}}e^{i\omega(r-r_{-})}\sum_{k=0}^{\infty}a_{k}\left(\frac{r-r_{+}}{r-r_{-}}\right)^{k}.$
(72)

This is precisely our solution (23), for $2M=1$. Thus, just as we have three
coefficients in equation (41) in [95], we can also get the three coefficients
for the recurrence relation of the equation (68). So by using the solution
(71) we have

$\displaystyle\alpha_{0}a_{1}+\beta_{0}a_{0}=0,\qquad\alpha_{k}a_{k+1}+\beta_{k}a_{k}+\gamma_{k}a_{k-1}=0,\qquad
k\geq 1,$ (73)

where

$\displaystyle\alpha_{k}$ $\displaystyle=$
$\displaystyle(1+k)\left[b(1+k)-2ir_{+}^{2}\omega\right]/b,$ (74)
$\displaystyle\beta_{k}$ $\displaystyle=$ $\displaystyle-
r_{+}\left[1+l(l+1)+2k(k+1)-8r_{+}^{2}-4i\omega r_{+}-8i\omega
kr_{+}\right]/b$ $\displaystyle+r_{-}\left[1+l(l+1)+2k(k+1)-2i\omega
r_{+}-4i\omega nr_{+}\right]/b,$ $\displaystyle\gamma_{k}$ $\displaystyle=$
$\displaystyle(k-2i\omega)\left(bk-2i\omega r_{+}^{2}\right)/b.$ (76)

These coefficients are the same as those found in (26), (27) and (28), for
$2M=1$.

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