# Quasinormal spectrum of $(2+1)$-dimensional asymptotically flat, dS and AdS
black holes

Milena Skvortsova<EMAIL_ADDRESS>Peoples’ Friendship University of Russia
(RUDN University), 6 Miklukho-Maklaya Street, Moscow, 117198, Russia

###### Abstract

While $(2+1)$-dimensional black holes in the Einstein theory allow for only
the anti-de Sitter asymptotic, when the higher curvature correction is tuned
on, the asymptotically flat, de Sitter and anti-de Sitter cases are included.
Here we propose first comprehensive study of the stability and quasinormal
spectra of the scalar field perturbations around such black holes with all
three asymptotics. Calculations of the frequencies are fulfilled with the help
of the 6th order WKB method with Pade approximants, Bernstein polynomial
method and time-domain integration. Results obtained by all three methods are
in a very good agreement in their common range of applicability. When the
multipole moment $k$ is equal to zero, the purely imaginary, i.e. non-
oscillatory, modes dominate in the spectrum for all types of the asymptotic
behavior, while the spectrum at higher $k$ resembles that in four-dimensional
spacetime with the corresponding asymptotic.

black holes, quasinormal modes

##  I Introduction

Recent and forthcoming observations of black holes in the gravitational and
electromagnetic spectra Abbott  _et al._ (2016); Akiyama  _et al._ (2019);
Goddi  _et al._ (2016); Auclair  _et al._ (2023) make a great impetus to
theoretical study of characteristic frequencies of black holes, called
quasinormal modes Berti  _et al._ (2009); Nollert (1999); Kokkotas and Schmidt
(1999); Konoplya and Zhidenko (2011).

While the main object of theoreticians is spectra of $(3+1)$\- dimensional
black holes, there were always attempts to formulate a simpler, lower
dimensional models which would carry the properties of the four-dimensional
objects. The most striking example is the $(2+1)$-dimensional asymptotically
anti-de Sitter (AdS) black holes suggested in Banados  _et al._ (1992) and
called BTZ black holes after the names of the authors of the metric.
Quasinormal modes of of these BTZ spacetimes were extensively studied (see,
for instance, Govindarajan and Suneeta (2001); Cardoso and Lemos (2001);
Konoplya (2004); Fontana (2023), and references therein) and the spectra
proved out to be qualitatively different from the asymptotically flat and de
Sitter one. In Birmingham  _et al._ (2002) it was shown that the quasinormal
modes of the BTZ black holes Govindarajan and Suneeta (2001); Cardoso and
Lemos (2001) exactly coincide with the poles of the retarded Green function in
the two-dimensional dual Conformal Field Theory, owing to the AdS/CFT
correspondence Maldacena (1998). Although it was much easier to find the
spectrum for the BTZ spacetime (in comparison with the four and higher
dimensional solutions), it did not bare the features of the asymptotically
flat space and was of limited interest mainly within the realm of holography
and quantum gravity of lower-dimensional systems.

The $(2+1)$-dimensional black-hole metric we are interested in this work
allows for all three asymptotics: flat, de Sitter and anti-de Sitter. It was
obtained in Konoplya and Zhidenko (2020) in a similar fashion with the
regularization procedure suggested in Glavan and Lin (2020) of the Einstein-
Gauss-Bonnet equations of motion. Then it was generalized in Hennigar  _et
al._ (2020, 2021). Although it turned out that the naive regularization of
Glavan and Lin (2020) does not produce a consistent theory, the black hole
solution obtained by this way, satisfied also the well-behaved gravitational
theory Aoki  _et al._ (2020). The Gauss-Bonnet corrections in the
asymptotically AdS black hole spacetimes play an important role when searching
for dual hydrodynamic description at intermediate couplings Grozdanov  _et
al._ (2021); Konoplya and Zhidenko (2017).

Here we will study quasinormal modes of the $(2+1)$-dimensional black-holes
Konoplya and Zhidenko (2020) for all three types of asymptotics. One of the
important aspects on which we will concentrate, is dynamical stability of
perturbations under consideration. Quasinormal modes were always an effective
tool to study (in)stability of black holes (see, for example, Takahashi and
Soda (2010); Ishihara  _et al._ (2008); Dyatlov (2011)). In Skvortsova (2023)
we have shown that the asymptotically flat perturbative (in the coupling
constant $\alpha$) branch is stable, despite the negative gap of the
corresponding effective potential. Here we will extend the analysis to the
other asymptotics and the non-perturbative branch. We will also study the
spectrum of a massive scalar field. The letter is known to possess peculiar
features in four dimensions, such as arbitrarily long lived quasinormal modes,
called quasiresonances which are known to exist for asymptotically flat black
holes Ohashi and Sakagami (2004), but not for asymptotically de Sitter ones
Konoplya and Molina (2005). Although quasinormal normal modes of massive (or
effectively massive) fields have been extensively studied in the background of
the four dimensional black holes Bolokhov (2023a, b); Konoplya and Fontana
(2008); Zinhailo (2018); Kokkotas  _et al._ (2011), including even the case
with the Gauss-Bonnet correction Konoplya and Zinhailo (2020); Konoplya  _et
al._ (2019a), no such analysis was performed for the $2+1$ dimensional
asymptotically flat or de Sitter black holes. At the same time, numerious
works on quasinormal modes of BTZ-like black holes Chen  _et al._ (2023);
Panotopoulos (2018); Huang  _et al._ (2018); Gupta  _et al._ (2017); Prasia
and Kuriakose (2017); Becar  _et al._ (2014); Kim  _et al._ (2012); Myung  _et
al._ (2012) did not include higher curvature corrections and were mainly
devoted to asymptotically AdS spacetimes.

Here we will compute quasinormal frequencies with the help of three methods:
6th order WKB method with Pade approximants, time-domain integration and the
Bernstein polynomial methods. While the WKB method in the form suggested in
Schutz and Will (1985); Iyer and Will (1987); Konoplya (2003) cannot be
applied to asymptotically AdS spacetimes, the other two methods can be
effectively used for all three types of asymptotic behavior. We will show that
the results obtained by all the three methods are in a very good agreement in
the common range of their applicability.

The paper is organized in the following way. In sec. II we briefly discuss the
properties of the black-hole metric under consideration and the wave like
equation for a, generally, massive scalar field. Sec. III is devoted the
review of methods for calculation of quasinormal frequencies, while the Sec.
IV discusses the obtained results for all three asymptotics. Finally, in the
Conclusions we summarize the obtained results.

##  II Black hole metrics and wavelike equations

The metric of the (2+1)-dimensional black hole has the following general line
element

$\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+f(r)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}x^{2},$
(1)

where the metric function which includes the Gauss-Bonnet corrections was
found in Konoplya and Zhidenko (2020) (see also Hennigar  _et al._ (2020))

$f(r)=1-\frac{r^{2}}{2\alpha}\left(-1\pm\sqrt{\frac{4\alpha\left(\Lambda\left(r^{2}-r_{H}^{2}\right)+\frac{\alpha}{r_{H}^{2}}+1\right)}{r^{2}}+1}\right).$
(2)

Here, $r_{H}$ is the event horizon radius, $\Lambda$ is the cosmological
constant, $\alpha$ is the Gauss-Bonnet coupling constant.

The above metric includes two branches: the perturabtive in $\alpha$, once
$1+2\alpha/r_{H}^{2}>0$ and a ”plus” sign before the square root, and the non-
perturbative one, if one chooses ”minus” and $1+2\alpha/r_{H}^{2}<0$.

(a) In the perturbative branch $\Lambda$ is constrained as follows:

$\Lambda<-\frac{\alpha}{r_{H}^{4}},$ (3)

which means that if $\alpha\geq 0$ the cosmological constant must be negative
and consequently the black hole spacetime must be asymptotically anti-de
Sitter. The asymptotic form of the metric produces the constrain

$1+4\alpha\Lambda\geq 0.$ (4)

At $\alpha<0$ and $\Lambda=0$, the metric is asymptotically flat, while if
$\alpha\rightarrow 0$, the function $f(r)$ vanishes and the solution does not
exist which reflects the fact that at zero coupling constant there are only
asymptotically AdS (BTZ Banados  _et al._ (1992) ) black holes. If $\alpha<0$
and $\Lambda>0$, the metric is asymptotically de Sitter and the cosmological
constant is constrained according to the above inequality (3,4).

(b) In the non-perturbative branch one always has $\alpha<0$, which means the
upper limit for the black hole radius,

$r_{H}^{2}<-2\alpha.$ (5)

However, the black hole solution appears only if the charge is non-zero and
larger than some minimal value determined in Konoplya and Zhidenko (2020).
Here we will consider the neutral case only, so that from here and on we will
be limited by the perturbative branch.

The general-covariant Klein-Gordon equation can be written as follows,

$\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi\right)=0.$
(6)

After the separation of variables it can be reduced to the wave-like equation,

$\dfrac{d^{2}\Psi}{dr_{*}^{2}}+(\omega^{2}-V(r))\Psi=0,$ (7)

where the “tortoise coordinate” $r_{*}$ is defined as:

$dr_{*}\equiv\frac{dr}{f(r)}.$ (8)

The effective potential has the form

$V(r)=f(r)\left(\frac{k^{2}}{r^{2}}+\frac{f^{\prime}(r)}{2r}-\frac{f(r)}{4r^{2}}\right),$
(9)

where $k=0,1,2,...$ is the multipole number. Examples of the effective
potentials for asymptotically flat and AdS black holes at $k=0$ are shown in
figs. 1,5. The presence of the negative gap for asymptotically flat and de
Sitter black holes means that the further study of the stability is necessary
for those cases. Higher values of $k$ as well as the asymptotically AdS black
hole at all $k$ provide positive definite effective potentials and are
therefore guaranteed to be stable.

---
Figure 1: Left: Effective potential for asymptotically flat black holes with
$k=0$, $\alpha=-0.1$ (bottom, black), $\alpha=-0.2$, and $\alpha=-0.3$ (top,
red) $r_{H}=1$. Right: Effective potential for asymptotically AdS black holes
with $k=0$, $\alpha=0.3$ (top, black), $\alpha=0.2$, and $\alpha=0.3$ (bottom,
red), $r_{H}=1$, $\Lambda=-0.5$.

##  III The three methods for calculation of quasinormal modes

Quasinormal modes are proper oscillation frequencies with the boundary
conditions which are the same at the event horizon, but different, depending
on the asymptotic of the spacetime. At the event horizon we always require the
purely incoming wave, because classical event horizon (when neglecting the
Hawking radiation) has complete absorption of the incoming wave. In far zone
the boundary conditions are:

  * •

purely outgoing wave at infinity or de Sitter horizon for asymptotically flat
or de Sitter case respectively, and,

  * •

vanishing wave function (Dirichlet condition) at the AdS horizon.

###  III.1 Time-domain integration

The evolution of perturbations in time can be analyzed via integration of the
wave equation at a fixed value of the radial coordinate in time domain. This
way contribution of all modes at a given multipole moment is included and the
instability, if any, can be detected via observation of growing profiles. Here
we will use the time-domain integration method suggested in Gundlach  _et al._
(1994). This method was further used in a number of works (see, for example,
Konoplya and Fontana (2008); Churilova and Stuchlik (2020); Bronnikov and
Konoplya (2020); Ishihara  _et al._ (2008); Konoplya and Molina (2005)) and
showed a very good agreement with the precise methods for the fundamental
mode. The method for asymptotically flat spacetime is based on the integration
of the master wave-like equation in the null-cone coordinates $u=t-r_{*}$,
$v=t+r_{*}$ and usage of the discretization scheme Gundlach  _et al._ (1994),

$\Psi\left(N\right)=\Psi\left(W\right)+\Psi\left(E\right)-\Psi\left(S\right)-\Delta^{2}V\left(S\right)\frac{\Psi\left(W\right)+\Psi\left(E\right)}{4}+{\cal
O}\left(\Delta^{4}\right).$ (10)

In the above equation we used the following definition of the points:
$N\equiv\left(u+\Delta,v+\Delta\right)$, $W\equiv\left(u+\Delta,v\right)$,
$E\equiv\left(u,v+\Delta\right)$, $S\equiv\left(u,v\right)$. Then, the
Gaussian wave is supposed to propagate along the null surfaces $u=u_{0}$ and
$v=v_{0}$. For asymptotically AdS black holes the discretization scheme by
Molina et. al. Wang  _et al._ (2004) was used.

###  III.2 WKB method

For checking the results obtained by the time-domain integration at $k>0$ we
used the 6th order WKB method with Pade approximants Konoplya  _et al._
(2019b); Matyjasek and Opala (2017); Konoplya (2003). This method was used in
great number of publications (see, for instance, Bolokhov (2023c, b, a);
Konoplya  _et al._ (2019a) ) showing good concordance with accurate results
whenever $k\geq n$ and the effective potential has a single maximum. It is
based on the expansion of the wave function in the Taylor series near the peak
of the effective potential and further matching it with the asymptotic WKB
expansions near the event horizon and at infinity (or de Sitter horizon). This
method implies that there are two turning points, so strictly speaking it
should not be applied to the massive scalar field, for which the third turning
point appears. Nevertheless, comparison with precise calculations shows that
usually it provides reasonably good accuracy for relatively small values of
the scalar field mass Konoplya  _et al._ (2019b); Konoplya and Fontana (2008).
The WKB formula of the $n$-th order requires taking of the derivatives of the
effective potential up to the $2n$-th order. The explicit form of the WKB
formula as well as further details on this method can be found in a review
Konoplya  _et al._ (2019b).

###  III.3 Bernstein polynomial method

For asymptotically flat or AdS black holes, we introduce the compact
coordinate $u$, as prescribed in Fortuna and Vega (2020),

$u\equiv\frac{1}{r}.$

Then, we extract the regular part of the wave function $\Psi(u)$, defined as
$y(u)$ multiplied by some function which keeps the singularities, in a similar
fashion with Konoplya  _et al._ (2023), and write $y(u)$ as a sum

$y(u)=\sum_{k=0}^{N}C_{k}B_{k}^{N}(u),$ (11)

where

$B_{k}^{N}(u)\equiv\frac{N!}{k!(N-k)!}u^{k}(1-u)^{N-k}$

are the Bernstein polynomials. For asymtotically de Sitter case the choice of
the compact coordinate is different Konoplya and Zhidenko (2023). Using the
Chebyschev collocation grid with $N+1$ points, one finds a set of linear
equations with respect to $C_{k}$, which has nontrivial solutions once the
coefficient matrix is singular. Then one has to solve numerically the
eigenvalue problem of a matrix pencil with respect to $\omega$ and find
coefficients $C_{k}$. Once the polynomial (11) is found, it can be used to
approximate solutions to the corresponding wave equation Fortuna and Vega
(2020). This method is known to be especially effective when searching for the
purely imaginary, i.e. non-oscillatory modes. Here we used the package for the
Bernstein spectral method which can be applied to all three types of the
asymptotic behavior and is publicly available from Konoplya and Zhidenko
(2023).

##  IV Quasinormal modes and stability

Here we will discuss the quasinormal frequencies obtained by all three
methods, stability and evolution of perturbations in time domain, including
the intermediate and asymptotic tails, for the three types of asymptotic
behaviour.

###  IV.1 Asymptotically flat case

---
Figure 2: Left: Semi-logarithmic plot of absolute value of wave function as a
function of time. Right: logarithmic plot. Here we have $k=1$, $\Lambda=0$,
$\alpha=-0.49$, $r_{H}=1$, $\mu=0$.
---
Figure 3: Logarithmic plot of absolute value of wave function as a function of
time together with the line $\sim t^{-(1+2k)}$. Here we have $k=0$ (left) and
$k=1$ (right), $\Lambda=0$, $\alpha=-0.25$, $r_{H}=1$, $\mu=0.1$.
---
Figure 4: Logarithmic plot of absolute value of wave function as a function of time together with the line $\sim t^{-1}$. Here we have $k=0$ (left) and $k=1$ (right), $\Lambda=0$, $\alpha=-0.25$, $r_{H}=1$, $\mu=5$. At this choice of the parameters, the effective potentials are dominated by the mass term $\mu^{2}$ and almost indistinguishable. Consequently, the time-domain profiles practically do not differ. $\alpha$ | WKB-6th | WKB 6th $(\tilde{m}=4)$ | WKB 6th $(\tilde{m}=5)$ | Time-domain
---|---|---|---|---
-0.05 | 0.113474-0.018831 i | 0.113474-0.018832 i | 0.113474-0.018832 i | 0.113475 - 0.0188308 i
-0.1 | 0.162438-0.040272 i | 0.162481-0.040287 i | 0.162482-0.040284 i | 0.162494 - 0.0402705 i
-0.15 | 0.199705-0.065110 i | 0.200795-0.064978 i | 0.200583-0.064855 i | 0.200706 - 0.0647788 i
-0.2 | 0.222710-0.097917 i | 0.232438-0.092848 i | 0.231375-0.092840 i | 0.232197 - 0.09272 i
-0.25 | 0.198068-0.179425 i | 0.258296-0.124127 i | 0.253298-0.122829 i | 0.257531 - 0.124026 i
-0.3 | 0.147112-0.510272 i | 0.279402-0.159111 i | 0.256398-0.143391 i | 0.276146 - 0.157778 I
-0.35 | 0.238216-1.258200 i | 0.294961-0.202321 i | 0.182390-0.117272 i | 0.287704 - 0.191991 i
-0.4 | 0.94005-2.17251 i | 0.292964-0.261626 i | 0.040493-0.142892 i | 0.293085 - 0.224448 i
-0.45 | 17.3639-0.9183 i | 0.195434-0.333651 i | 0.0118962-0.1108563 i | 0.294249 - 0.253671 i
-0.49 | – | – | – | 0.293476 - 0.274684 i
Table 1: Asymptotically flat case: Fundamental quasinormal modes ($n=0$) for $k=1$ massless perturbations at $\Lambda=0$ obtained by the WKB method and time-domain integration. $\mu$ | 6WKB | WKB 6th ($\tilde{m}=4$) | WKB 6th ($\tilde{m}=5$)
---|---|---|---
0. | 0.199705-0.065110 i | 0.200795-0.064978 i | 0.200583-0.064855 i
0.05 | 0.200163-0.064893 i | 0.201245-0.064626 i | 0.201038-0.064638 i
0.1 | 0.201539-0.064246 i | 0.202452-0.063934 i | 0.202403-0.063987 i
0.15 | 0.203835-0.063174 i | 0.204691-0.062886 i | 0.204680-0.062908 i
0.2 | 0.207053-0.061684 i | 0.207875-0.061398 i | 0.207871-0.061409 i
0.25 | 0.211199-0.059790 i | 0.211981-0.059496 i | 0.211979-0.059506 i
0.3 | 0.216275-0.057508 i | 0.217008-0.057205 i | 0.217007-0.057220 i
0.35 | 0.222288-0.054860 i | 0.222956-0.054548 i | 0.222958-0.054575 i
0.4 | 0.229239-0.051872 i | 0.229830-0.051556 i | 0.229839-0.051604 i
0.45 | 0.237130-0.048574 i | 0.237630-0.048260 i | 0.237656-0.048337 i
0.5 | 0.245964-0.045002 i | 0.246360-0.044698 i | 0.246415-0.044809 i
0.55 | 0.255738-0.041196 i | 0.256023-0.040916 i | 0.256118-0.041057 i
0.6 | 0.266452-0.037201 i | 0.266627-0.036963 i | 0.266765-0.037122 i
0.65 | 0.278104-0.033065 i | 0.278188-0.032889 i | 0.278348-0.033052 i
0.7 | 0.290690-0.028841 i | 0.290720-0.028734 i | 0.290850-0.028888 i
Table 2: Asymptotically flat case: Fundamental quasinormal modes ($n=0$) for
$k=1$ massive perturbations obtained by the WKB method; $\alpha=-0.15$.

Asymptotically flat black holes in the perturbative branch are characterized
by two qualitatively different types of modes. The modes corresponding to
$k\geq 1$ have both real and imaginary parts representing decaying
oscillations, while the $k=0$ modes of the massless field are purely
imaginary, i.e. exponentially decaying in time, as can be seen from figs. 2.
The $k=1$ modes computed by the WKB and time-domain integration methods are
shown in table I. The discrepancy between the methods grows as $\alpha$
approaches its extreme value and as the WKB method converges only
asymptotically, but not in each order, the results must be interpreted in
favour of the time-domain integration method. In the near extreme regime
$\alpha=0.49$ the WKB formula has very large error producing senseless
results. Nevertheless, for small and moderate values of the coupling constant
the agreement between the two methods is quite good.

The instability of a static background is governed by the non-oscillatory,
i.e. purely imaginary, growing mode Konoplya  _et al._ (2008). Positive
definite effective potential guarantee stability of the perturbation.
Therefore, $k=1,2,..$ multipoles are stable. Time-domain profiles for $k=0$
shown in Skvortsova (2023), as well as here in figs. 2 signify the stability
of perturbations even for the near extreme values of the coupling constant.

When the massive term is turned on, the damping rate is suppressed (see table
II) indicating the existence of quasi-resonances, similar to the ones observed
in Ohashi and Sakagami (2004) and subsequent papers in four dimensions.
Nevertheless, the clear evidence of the arbitrarily long lived modes must be
done with the help of a quickly convergent method of calculationZhidenko
(2006), such as the Leaver method Leaver (1985). Here it cannot be applied to
the wave-like equation in its present form, because the Leaver method requires
polynomial form of the master differential equation.

In four dimensional spacetimes at asymptotically late times, a massless scalar
decay according to the power-law Price (1972), while the massive scalar field
Koyama and Tomimatsu (2001, 2002) is oscillatory and the envelope does not
depend on the multipole number:

$|\Psi|\sim t^{-5/6}\sin(\mu t),\quad t\rightarrow\infty,\quad D=4.$ (12)

The same law was observed for a number of other backgrounds and spin of the
field Moderski and Rogatko (2001); Konoplya  _et al._ (2007); Jing (2005);
Seahra  _et al._ (2005) and others. At intermediately late times in four
dimensions (or, correspondingly, small $\mu M$) the decay law is

$|\Psi|\sim t^{-(\frac{8}{6}+\ell)}\sin(A(\mu)t),\quad D=4.$ (13)

In $2+1$-dimensional asymptotically flat case we observe that at intermediate
late times, as can be seen in figs. 3, the decay law is

$|\Psi|\sim t^{-(1+2k)}\sin(A(\mu)t),\quad D=3.$ (14)

The decay law at asymptotic times does not depend on $k$, as can be seen in
figs. 4:

$|\Psi|\sim t^{-1}\sin(A(\mu)t),\quad t\rightarrow\infty,\quad D=3.$ (15)

The integration until the decayed asymptotic tails demonstrate the stability
of the perturbation. Notice, that the same asymptotic decay law has been
recently observed for the Bardeen spacetime Bolokhov (2023b).

###  IV.2 Asymptotically de Sitter case

---
Figure 5: Left: Effective potential. Right: Logarithmic plot of absolute value
of wave function as a function of time. Here we have $k=0$ (left),
$\Lambda=0.24$, $\alpha=-0.25$, $r_{H}=1$.

In tables III and IV we can see that the positive cosmological constant
suppresses both real and imaginary parts of quasinormal modes in the
perturbative branch. A similar effect was observed also for asymptotically de
Sitter black holes in four and higher dimensions Zhidenko (2004); Konoplya and
Zhidenko (2004); Giammatteo and Moss (2005). The time domain profiles shown in
figs. 5 signifies that the at the near extreme asymptotically de Sitter black
holes purely imaginary modes with very small damping rate dominate in the
signal. This agrees with observations in $D=4$ and higher dimensional cases,
for which asymptotic tails for Schwarzschild-de Sitter black hole decay
exponentially Molina  _et al._ (2004); Brady  _et al._ (1999), which also
could be interpreted as dominance of the purely imaginary modes Konoplya and
Zhidenko (2022). However, as in our case the damping rate is tiny in the near
extreme regime, there is still a risk of instability in the extreme case.

The massive scalar field does not allow for arbitrarily long-lived quasinormal
modes for asymptotically de Sitter black holes. This could be shown
analytically exactly in the same way as in Konoplya and Zhidenko (2005).

$\Lambda$ | WKB-6th ($\tilde{m}=5$) | Time-domain
---|---|---
0.01 | 0.247756 - 0.125308 i | 0.247676 - 0.123123 i
0.05 | 0.203529 - 0.116081 i | 0.206496 - 0.115736 i
0.1 | 0.153240 - 0.098081 i | 0.155070 - 0.098524 i
0.15 | 0.102442 - 0.073853 i | 0.103707 - 0.074847 i
0.2 | 0.051553 - 0.042960 i | 0.051540 - 0.043093 i
0.24 | 0.010047 - 0.009668 i | 0.010091 - 0.009704 i
Table 3: Asymptotically de Sitter case: Fundamental quasinormal modes ($n=0$) for $k=1$ massless perturbations obtained by the WKB method and time-domain integration; $\alpha=-0.25$. $\Lambda$ | WKB-6th ($\tilde{m}=5$) | Time-domain
---|---|---
0.01 | 0.543925-0.126157 i | 0.543858 - 0.126648 i
0.05 | 0.463678-0.117548 i | 0.463611 - 0.117138 i
0.1 | 0.358309-0.099902 i | 0.357864 - 0.101339 i
0.15 | 0.246702-0.075731 i | 0.246749 - 0.075648 i
0.2 | 0.127664-0.043281 i | 0.127674 - 0.043309 i
0.24 | 0.026269-0.009707 i | 0.026269 - 0.009670 i
Table 4: Asymptotically de Sitter case: Fundamental quasinormal modes ($n=0$)
for $k=2$ massless perturbations obtained by the WKB method and time-domain
integration; $\alpha=-0.25$.

###  IV.3 Asymptotically anti-de Sitter case

---
Figure 6: Time=domain profile for the asymptotically AdS black hole at $k=1$,
$\Lambda=0.1$, $\alpha=-0.5$, $r_{H}=1$. The dominant mode extracted by the
Prony method is $\omega=-0.719279i$.

This case takes place in the perturbative branch once $\alpha\geq 0$. We apply
the time-domain integration and the Bernstein polynomial methods to analyze
the evolution of perturbations in this case.

When $\alpha\rightarrow\infty$, the metric function is reduced to the BTZ one
Banados  _et al._ (1992),

$f(r)\rightarrow(r_{H}^{2}-r^{2})\Lambda$ (16)

and the spectrum known from Cardoso and Lemos (2001) is reproduced with great
accuracy by the Bernstein polynomial method, as can be seen, for example, from
table V. The general exact analytical formula for the quasinormal modes of the
BTZ black holes in the units we are using here has the following form:

$\omega_{n}=\pm\sqrt{\Lambda}k-2\Lambda(n+1)i,$ (17)

which coincides with Cardoso and Lemos (2001), if one goes over to the units
of the AdS radius $l=1/\sqrt{\Lambda}$ and uses $M=r_{H}^{2}/l^{2}$.

$\Lambda$ |  $n=0$ (B) |  $n=1$ (B) |  $n=2$ (B) |  $n=0$ (T-D)
---|---|---|---|---
-0.5 | 0.707107 - 0.999999 i | 0.707107 - 2.000000 i | 0.707107 - 3.000000 i | 0.707130 - 0.999985 i
-1 | 1.000000 - 2.000000 i | 0.999999 - 4.000000 i | 1.000000 - 5.999999 i | 1.000225 - 1.999804 i
-1.5 | 1.224745 - 3.000000 i | 1.224745 -6.000000 i | 1.224746 -9.000000 i | 1.225700 - 2.994060 i
-2 | 1.414214 - 4.000000 i | 1.414214 -8.000000 i | 1.414220 - 12.00000 i | 1.409750 - 3.996330 i
Table 5: Asymptotically anti-de Sitter (BTZ) black hole: Quasinormal modes for $k=1$ massless perturbations obtained by the Bernstein polynomial method (B) and time-domain integration (T-D); $\alpha=0$, $r_{H}=1$. $\Lambda$ | $\alpha=0.1$ | $\alpha=0.2$ | $\alpha=0.3$
---|---|---|---
-0.5 | \- 0.719279 i | \- 0.387343 i | \- 0.212103 i
-1 | \- 1.351860 i | \- 0.902496 I | –
-1.5 | \- 1.87552 i | – | –
-2 | \- 2.36785 i | – | –
Table 6: Asymptotically anti-de Sitter case: Quasinormal modes for $k=1$
massless perturbations obtained by the time-domain integration. Empty space
corresponds to the regime $1+4\alpha\Lambda<0$ which is outside the constrain
given by eq. (4).

From table VI one can see that the fundamental modes are purely imaginary,
which agrees with the exponentially decaying time-domain profile in fig. 6,
and roughly proportional to $\Lambda$.

##  V Conclusions

Here we have analyzed quasinormal modes of a scalar field in the background of
the asymptotically flat, de Sitter and anti-de Sitter black holes inspired by
the Gauss-Bonnet correction to the Einstein action. Usually $2+1$-dimensional
black holes allow for only anti-de Sitter asymptotic behavior. We have shown
that the quasinormal spectrum of the $2+1$ asymptotically flat and de Sitter
black holes keeps a number of features of the correspondingly asymptotically
flat or de Sitter four dimensional spacetimes. In particular,

  * •

Perturbations of asymptotically flat black holes is not equidistant in $n$,
like it happens for the asymptotically AdS case, but resembles the spectrum of
the four and higher dimensional asymptotically flat black holes. The same is
true for asymptotically de Sitter case.

  * •

At late times, when the quasinormal modes for asymptotically flat case are
suppressed by the late-time tails, the field decays at $k>0$ according to the
power law, though via a different from the four-dimensional Price’s law Price
(1972). Nevertheless, $k=0$ perturbations are exceptional, as they are
governed by the purely imaginary, i.e. non-oscillatory, exponentially decaying
modes at all times.

  * •

When the massive term is tuned on, an arbitrarily long-lived modes exist for
asymptotically flat case, which are called quasi-resonances and extensively
studied in four dimensions.

  * •

The positive cosmological constant suppresses the real oscillation frequency
and the damping rate of the perturbation.

Despite the deep negative gap of the $k=0$ effective potential, the time-
domain integration did not show any instability, but instead showed stable
modes with tiny decay rate, which means that further study (possibly with
alternative more accurate and stable methods) of the stability in the near
extreme regime might be necessary.

###### Acknowledgements.

The author acknowledges R. A. Konoplya for fruitful discussions, and for
careful reading of the manuscript. This work was supported by RUDN University
research project FSSF-2023-0003.

## References

  * Abbott  _et al._ (2016) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc] .
  * Akiyama  _et al._ (2019) K. Akiyama _et al._ (Event Horizon Telescope), Astrophys. J. Lett. 875, L1 (2019), arXiv:1906.11238 [astro-ph.GA] .
  * Goddi  _et al._ (2016) C. Goddi _et al._ , Int. J. Mod. Phys. D 26, 1730001 (2016), arXiv:1606.08879 [astro-ph.HE] .
  * Auclair  _et al._ (2023) P. Auclair _et al._ (LISA Cosmology Working Group), Living Rev. Rel. 26, 5 (2023), arXiv:2204.05434 [astro-ph.CO] .
  * Berti  _et al._ (2009) E. Berti, V. Cardoso, and A. O. Starinets, Class. Quant. Grav. 26, 163001 (2009), arXiv:0905.2975 [gr-qc] .
  * Nollert (1999) H.-P. Nollert, Class. Quant. Grav. 16, R159 (1999).
  * Kokkotas and Schmidt (1999) K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel. 2, 2 (1999), arXiv:gr-qc/9909058 .
  * Konoplya and Zhidenko (2011) R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys. 83, 793 (2011), arXiv:1102.4014 [gr-qc] .
  * Banados  _et al._ (1992) M. Banados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992), arXiv:hep-th/9204099 .
  * Govindarajan and Suneeta (2001) T. R. Govindarajan and V. Suneeta, Class. Quant. Grav. 18, 265 (2001), arXiv:gr-qc/0007084 .
  * Cardoso and Lemos (2001) V. Cardoso and J. P. S. Lemos, Phys. Rev. D 63, 124015 (2001), arXiv:gr-qc/0101052 .
  * Konoplya (2004) R. A. Konoplya, Phys. Rev. D 70, 047503 (2004), arXiv:hep-th/0406100 .
  * Fontana (2023) R. D. B. Fontana, (2023), arXiv:2306.02504 [gr-qc] .
  * Birmingham  _et al._ (2002) D. Birmingham, I. Sachs, and S. N. Solodukhin, Phys. Rev. Lett. 88, 151301 (2002), arXiv:hep-th/0112055 .
  * Maldacena (1998) J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998), arXiv:hep-th/9711200 .
  * Konoplya and Zhidenko (2020) R. A. Konoplya and A. Zhidenko, Phys. Rev. D 102, 064004 (2020), arXiv:2003.12171 [gr-qc] .
  * Glavan and Lin (2020) D. Glavan and C. Lin, Phys. Rev. Lett. 124, 081301 (2020), arXiv:1905.03601 [gr-qc] .
  * Hennigar  _et al._ (2020) R. A. Hennigar, D. Kubiznak, R. B. Mann, and C. Pollack, Phys. Lett. B 808, 135657 (2020), arXiv:2004.12995 [gr-qc] .
  * Hennigar  _et al._ (2021) R. A. Hennigar, D. Kubiznak, and R. B. Mann, Class. Quant. Grav. 38, 03LT01 (2021), arXiv:2005.13732 [gr-qc] .
  * Aoki  _et al._ (2020) K. Aoki, M. A. Gorji, and S. Mukohyama, Phys. Lett. B 810, 135843 (2020), arXiv:2005.03859 [gr-qc] .
  * Grozdanov  _et al._ (2021) S. Grozdanov, A. O. Starinets, and P. Tadić, JHEP 06, 180 (2021), arXiv:2104.11035 [hep-th] .
  * Konoplya and Zhidenko (2017) R. A. Konoplya and A. Zhidenko, JHEP 09, 139 (2017), arXiv:1705.07732 [hep-th] .
  * Takahashi and Soda (2010) T. Takahashi and J. Soda, Prog. Theor. Phys. 124, 711 (2010), arXiv:1008.1618 [gr-qc] .
  * Ishihara  _et al._ (2008) H. Ishihara, M. Kimura, R. A. Konoplya, K. Murata, J. Soda, and A. Zhidenko, Phys. Rev. D 77, 084019 (2008), arXiv:0802.0655 [hep-th] .
  * Dyatlov (2011) S. Dyatlov, Commun. Math. Phys. 306, 119 (2011), arXiv:1003.6128 [math.AP] .
  * Skvortsova (2023) M. Skvortsova, (2023), arXiv:2311.02729 [gr-qc] .
  * Ohashi and Sakagami (2004) A. Ohashi and M.-a. Sakagami, Class. Quant. Grav. 21, 3973 (2004), arXiv:gr-qc/0407009 .
  * Konoplya and Molina (2005) R. A. Konoplya and C. Molina, Phys. Rev. D 71, 124009 (2005), arXiv:gr-qc/0504139 .
  * Bolokhov (2023a) S. V. Bolokhov, (2023a), arXiv:2311.05503 [gr-qc] .
  * Bolokhov (2023b) S. V. Bolokhov,  (2023b), 10.20944/preprints202310.0517.v1.
  * Konoplya and Fontana (2008) R. A. Konoplya and R. D. B. Fontana, Phys. Lett. B 659, 375 (2008), arXiv:0707.1156 [hep-th] .
  * Zinhailo (2018) A. F. Zinhailo, Eur. Phys. J. C 78, 992 (2018), arXiv:1809.03913 [gr-qc] .
  * Kokkotas  _et al._ (2011) K. D. Kokkotas, R. A. Konoplya, and A. Zhidenko, Phys. Rev. D 83, 024031 (2011), arXiv:1011.1843 [gr-qc] .
  * Konoplya and Zinhailo (2020) R. A. Konoplya and A. F. Zinhailo, Eur. Phys. J. C 80, 1049 (2020), arXiv:2003.01188 [gr-qc] .
  * Konoplya  _et al._ (2019a) R. A. Konoplya, A. F. Zinhailo, and Z. Stuchlík, Phys. Rev. D 99, 124042 (2019a), arXiv:1903.03483 [gr-qc] .
  * Chen  _et al._ (2023) C. Chen, Q. Pan, and J. Jing, Phys. Lett. B 846, 138186 (2023), arXiv:2302.05861 [gr-qc] .
  * Panotopoulos (2018) G. Panotopoulos, Gen. Rel. Grav. 50, 59 (2018), arXiv:1805.04743 [hep-th] .
  * Huang  _et al._ (2018) L. Huang, J. Chen, and Y. Wang, Eur. Phys. J. C 78, 299 (2018).
  * Gupta  _et al._ (2017) K. S. Gupta, T. Jurić, and A. Samsarov, JHEP 06, 107 (2017), arXiv:1703.00514 [hep-th] .
  * Prasia and Kuriakose (2017) P. Prasia and V. C. Kuriakose, Eur. Phys. J. C 77, 27 (2017), arXiv:1608.05299 [gr-qc] .
  * Becar  _et al._ (2014) R. Becar, P. A. Gonzalez, and Y. Vasquez, Phys. Rev. D 89, 023001 (2014), arXiv:1306.5974 [gr-qc] .
  * Kim  _et al._ (2012) Y.-W. Kim, Y. S. Myung, and Y.-J. Park, Phys. Rev. D 85, 124018 (2012), arXiv:1204.3706 [hep-th] .
  * Myung  _et al._ (2012) Y. S. Myung, Y.-W. Kim, and Y.-J. Park, Phys. Rev. D 85, 084007 (2012), arXiv:1201.3964 [hep-th] .
  * Schutz and Will (1985) B. F. Schutz and C. M. Will, Astrophys. J. 291, L33 (1985).
  * Iyer and Will (1987) S. Iyer and C. M. Will, Phys. Rev. D35, 3621 (1987).
  * Konoplya (2003) R. A. Konoplya, Phys. Rev. D 68, 024018 (2003), arXiv:gr-qc/0303052 .
  * Gundlach  _et al._ (1994) C. Gundlach, R. H. Price, and J. Pullin, Phys. Rev. D 49, 883 (1994), arXiv:gr-qc/9307009 .
  * Churilova and Stuchlik (2020) M. S. Churilova and Z. Stuchlik, Class. Quant. Grav. 37, 075014 (2020), arXiv:1911.11823 [gr-qc] .
  * Bronnikov and Konoplya (2020) K. A. Bronnikov and R. A. Konoplya, Phys. Rev. D 101, 064004 (2020), arXiv:1912.05315 [gr-qc] .
  * Wang  _et al._ (2004) B. Wang, C.-Y. Lin, and C. Molina, Phys. Rev. D 70, 064025 (2004), arXiv:hep-th/0407024 .
  * Konoplya  _et al._ (2019b) R. A. Konoplya, A. Zhidenko, and A. F. Zinhailo, Class. Quant. Grav. 36, 155002 (2019b), arXiv:1904.10333 [gr-qc] .
  * Matyjasek and Opala (2017) J. Matyjasek and M. Opala, Phys. Rev. D 96, 024011 (2017), arXiv:1704.00361 [gr-qc] .
  * Bolokhov (2023c) S. V. Bolokhov, (2023c), arXiv:2310.12326 [gr-qc] .
  * Fortuna and Vega (2020) S. Fortuna and I. Vega, (2020), arXiv:2003.06232 [gr-qc] .
  * Konoplya  _et al._ (2023) R. A. Konoplya, Z. Stuchlik, A. Zhidenko, and A. F. Zinhailo, Phys. Rev. D 107, 104050 (2023), arXiv:2303.01987 [gr-qc] .
  * Konoplya and Zhidenko (2023) R. A. Konoplya and A. Zhidenko, Phys. Rev. D 107, 044009 (2023), arXiv:2211.02997 [gr-qc] .
  * Konoplya  _et al._ (2008) R. A. Konoplya, K. Murata, J. Soda, and A. Zhidenko, Phys. Rev. D 78, 084012 (2008), arXiv:0807.1897 [hep-th] .
  * Zhidenko (2006) A. Zhidenko, Phys. Rev. D 74, 064017 (2006), arXiv:gr-qc/0607133 .
  * Leaver (1985) E. W. Leaver, Proc. Roy. Soc. Lond. A 402, 285 (1985).
  * Price (1972) R. H. Price, Phys. Rev. D 5, 2439 (1972).
  * Koyama and Tomimatsu (2001) H. Koyama and A. Tomimatsu, Phys. Rev. D 63, 064032 (2001), arXiv:gr-qc/0012022 .
  * Koyama and Tomimatsu (2002) H. Koyama and A. Tomimatsu, Phys. Rev. D 65, 084031 (2002), arXiv:gr-qc/0112075 .
  * Moderski and Rogatko (2001) R. Moderski and M. Rogatko, Phys. Rev. D 64, 044024 (2001), arXiv:gr-qc/0105056 .
  * Konoplya  _et al._ (2007) R. A. Konoplya, A. Zhidenko, and C. Molina, Phys. Rev. D 75, 084004 (2007), arXiv:gr-qc/0602047 .
  * Jing (2005) J. Jing, Phys. Rev. D 72, 027501 (2005), arXiv:gr-qc/0408090 .
  * Seahra  _et al._ (2005) S. S. Seahra, C. Clarkson, and R. Maartens, Phys. Rev. Lett. 94, 121302 (2005), arXiv:gr-qc/0408032 .
  * Zhidenko (2004) A. Zhidenko, Class. Quant. Grav. 21, 273 (2004), arXiv:gr-qc/0307012 .
  * Konoplya and Zhidenko (2004) R. A. Konoplya and A. Zhidenko, JHEP 06, 037 (2004), arXiv:hep-th/0402080 .
  * Giammatteo and Moss (2005) M. Giammatteo and I. G. Moss, Class. Quant. Grav. 22, 1803 (2005), arXiv:gr-qc/0502046 .
  * Molina  _et al._ (2004) C. Molina, D. Giugno, E. Abdalla, and A. Saa, Phys. Rev. D 69, 104013 (2004), arXiv:gr-qc/0309079 .
  * Brady  _et al._ (1999) P. R. Brady, C. M. Chambers, W. G. Laarakkers, and E. Poisson, Phys. Rev. D 60, 064003 (1999), arXiv:gr-qc/9902010 .
  * Konoplya and Zhidenko (2022) R. A. Konoplya and A. Zhidenko, Phys. Rev. D 106, 124004 (2022), arXiv:2209.12058 [gr-qc] .
  * Konoplya and Zhidenko (2005) R. A. Konoplya and A. V. Zhidenko, Phys. Lett. B 609, 377 (2005), arXiv:gr-qc/0411059 .