# Quasi-normal modes of a **natural** AdS wormhole in Einstein–Born–Infeld Gravity

**natural**

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## Abstract

We study the matter perturbations of a new AdS wormhole in \((3+1)\)-dimensional Einstein–Born–Infeld gravity, called “natural wormhole”, which does not require exotic matters. We discuss the stability of the perturbations by numerically computing the quasi-normal modes (QNMs) of a massive scalar field in the wormhole background. We investigate the dependence of quasi-normal frequencies on the mass of scalar field as well as other parameters of the wormhole. It is found that the perturbations are always stable for the wormhole geometry which has the general relativity (GR) limit when the scalar field mass *m* satisfies a certain, tachyonic mass bound \(m^2>m_*^2\) with \(m_*^2<0\), analogous to the Breitenlohner-Freedman (BF) bound in the global-AdS space, \(m_{\mathrm{BF}}^2=3 {\Lambda }/4\). It is also found that the BF-like bound \(m_*^2\) shifts by the changes of the cosmological constant \({\Lambda }\) or angular-momentum number *l*, with a *level crossing* between the lowest complex and pure-imaginary modes for zero angular momentum \(l=0\). Furthermore, it is found that the unstable modes can also have oscillatory parts as well as non-oscillatory parts *depending on whether the real and imaginary parts of frequencies are dependent on each other or not*, contrary to arguments in the literature. For wormhole geometries which do not have the GR limit, the BF-like bound does not occur and the perturbations are stable for arbitrary tachyonic and non-tachyonic masses, up to a critical mass \(m_{c}^2>0\) where the perturbations are completely frozen.

## 1 Introduction

Oscillations in closed systems with conserved energies are described by normal modes with real frequencies. Free fields propagating in a confining box correspond to those cases. On the other hand, in open systems with energy dissipations, oscillations are described by quasi-normal modes (QNMs) with complex frequencies. Perturbations of fields propagating in the background of a black hole correspond to these cases. In the presence of a black hole, the (gravitational or matter) fields can fall into the black hole so that their energies can dissipate into the black holes and the fields decay

The complex frequencies of QNMs carry the characteristic properties of a black hole like mass, charge, and angular momentum, *independent* of the initial perturbations. Because the computational details of QNMs depend much on the based gravity theories, QNMs can reflect the information about the gravity theory itself as well as that of the background spacetime. In the recent detection of gravitational waves by LIGO and VIRGO, which are thought to be radiated from mergers of binary black holes or neutron stars, there is a regime, called *ring-down phase*, which can be described by QNMs [1, 2]. As more precise data be available in the near future, QNMs will be one of the key tools for the test of general relativity (GR) as well as the black hole spacetime [3].

QNMs of black hole systems have been studied for a long time and much have been known in various cases (for some recent reviews, see [4, 5]). There is another important system, *wormhole* spacetime, which corresponds to an open system so that QNMs appear too. Fields entering into a wormhole without return would be also observed as the modes with dissipating energy and decay.

Because the characteristics of the metric near the throat of a wormhole is different from those near the horizon of a black hole, one can distinguish them by comparing their QNMs of the gravitational waves with the same boundary condition at asymptotic infinity. It has been found that wormhole geometries can also show the similar gravitational wave forms as in the black hole systems up to some early ring-down phase but some different wave forms at later times [6, 7] (see also [8, 9, 10, 11, 12, 13] for relevant discussions). In the near future, with increased precisions of gravitational wave detections, it may be possible to distinguish those two systems by investigating their late-time behaviors.

Compared to QNMs in the black hole systems, there are several issues about the wormholes themselves, which are still thought to be some hypothetical objects without any conclusive observational evidence, even though they can be exact solutions of Einstein’s equation. In the conventional approaches to construct wormholes, there are the “naturalness problems” due to (i) the hypothetical exotic matters which support the throat of a wormhole but violate energy conditions [14] and (ii) the artificial construction of the wormhole throat by cuts and pastes [15, 16]. Actually, in the recent analysis of gravitational waves from wormholes [6, 7], the considered wormholes are known as the “thin–shell” wormholes which are quite artificial [15, 16].

Recently a new type of wormhole solutions was proposed to avoid the problem caused by the exotic matters [17, 18]. In the new type of solutions, named as “ natural wormholes” [18], the throat is defined as the place where the solutions are *smoothly* joined. The metric and its derivatives are continuous so that the exotic matters are not introduced at the throat. From the new definition, throat can not be constructed arbitrarily contrary to the conventional cuts and pastes approach. The purpose of this paper is to study QNMs for these natural wormholes.

In this paper, we consider the recently constructed Anti-de Sitter (AdS) wormholes in Einstein–Born–Infeld gravity [19] and compute QNMs of a *massive* scalar field perturbation in the wormhole background. We consider the asymptotically AdS case since it is simpler than that of the asymptotically flat or de Sitter. Moreover there are several interesting aspects which are absent or unclear in other cases.

For example, it is well known that there exists a tachyonic mass bound, called Breitenlohner–Freedman (BF) bound, \(m^2 > m_{\mathrm{BF}}^2\), for the “conserved and positive” energy of perturbations of a massive scalar field with mass *m* in the global AdS background [20, 21, 22, 23] (see [24, 25] for generalization to higher spins). The solutions exist with discrete real frequencies of ordinary normal modes above the BF bound and the perturbation becomes unstable below the BF bound. What we are going to study in this paper is about what happens in the BF bound for wormholes in the asymptotically AdS spacetime. It would be physically clear that the *local* deformation of a spacetime by the presence of wormholes would not change the stability properties of the whole spacetime much from the stability property at asymptotic infinity which is governed by the BF bound: It is hard to imagine a *smooth* spacetime where the (matter) perturbations with \(m^2<m_{\mathrm{BF}}^2\) are *partly* unstable at infinity but also *partly* stable near wormholes, by some local effects. This implies that the perturbations of massive fields in the background of AdS wormholes would show both QNMs and BF bound. We will numerically compute these for a minimally-coupled real scalar field based on the approach of Horowitz and Hubeny for AdS space [26]. Moreover, the asymptotically AdS case would be interesting in the string theory contexts of the AdS/CFT correspondence. QNMs for AdS black hole spacetimes have been much studied in this context [27, 28, 29, 30] but little is known for AdS wormhole spacetimes.

The existence of large charged black holes or wormholes would be questionable since our universe seems to be charge-neutral in the large scales. However in the small scales, the charged black holes or wormholes may exist, as the charged elementary particles do. The well-known charged black hole is Reissner–Nordstrom (RN) black hole with the usual Maxwell’s electromagnetic field in GR. But at short distances, we need some modifications of GR for a consistent quantum theory, i.e., (renormalizable) quantum gravity [18]. We may also need modification of Maxwell’s electromagnetism as an effective description of quantum effects or genuine classical modifications at short distances. The non-linear generalization of Maxwell’s theory by Born and Infeld (BI) corresponds to the latter case [31, 32] and in this set up we may consider the generalized charged black holes and wormholes which include the RN case as the GR limit [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. On the other hand, with the advent of D-branes, the BI-type action has also attracted renewed interests as an effective description of low energy superstring theory [45, 46].

The organization of the paper is as follow. In Sect. 2, we consider the new AdS wormhole in Einstein–Born–Infeld (EBI) gravity. In Sect. 3, we set up a formalism to calculate QNMs of a massive scalar field. In Sect. 4, we set up the formula for numerical computation of QNMs. In Sect. 5, we summarize our numerical results of QNMs. In Sect. 6, we conclude with some discussions. Throughout this paper, we use the conventional units for the speed of light *c* and the Boltzman’s constant \(k_B\), \(c=k_B=1\), but keep the Newton’s constant *G* and the Planck’s constant \(\hbar \) unless stated otherwise.

## 2 New AdS wormholes in EBI gravity

*L*(

*F*) is the BI Lagrangian density, given by

*Q*represents the electric charge located at the origin and

*M*is the ADM mass which is composed of the intrinsic mass

*C*and (finite) self energy of a point charge \(M_0\), defined by

*M*(Fig. 1).

*natural*wormholes, the throat which connects two universes (or equivalently, two remote parts of the same universe) is defined as the place where the solutions are smoothly joined. For the reflection \((Z_2)\) symmetric universe, the new spherically symmetric wormhole metric is described by

*not*to be vanished in contrast to Morris–Thorne’s approach [14], while the quantities \({dN_{\pm }(r_0)}/{dr},~ {df_{\pm }(r_0)}/{dr}\) in (12) need

*not*to be vanished in both Morris–Thorne’s approach [14] and Visser’s cuts and pastes approach [15, 16].

*M*for given values of \(\beta , Q,\) and \({\Lambda }\). Now, from the property of the metric function [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]

*r*of \(df/dr =0\),

*M*can be expressed in terms of \(r_0\) as,

*maximum*value \(M_0\) of (10) at \(r_0=0\) (thick curves in Fig. 2). It is interesting to note that the mass of our new AdS wormhole without exotic matters can be

*negative*for large \(r_0\), similar to the conventional wormholes which require the exotic matters and violate energy conditions. This may be considered as another evidence that

*exotic matters cemented at the throat are not mandatory for constructing large scale wormholes.*

*M*in terms of the black hole horizon \(r_+\), corresponding to the largest

*r*of \(f(r)=0\),

*positive*definite. This is a monotonically increasing function of \(r_+\) with the

*minimum*\(M_0\) at \(r_+=0\) when \(\beta Q \le 1/2\) so that there exists only the Schwarzschild-like (type I) black hole with one horizon (three thin curves from below in Fig. 2). On the other hand, when \(\beta Q > 1/2\) (top thin curve in Fig. 2), there exists the RN-like (type II) black hole with two horizons as well as the Schwarzschild-like black hole with one horizon. In this latter case, the black hole mass function (17) is concave with the minimum

*M*increases as the throat radius \(r_0\) reduces, by accretion of ordinary (positive energy) matters until it reaches to the extremal black hole horizon \(r_+^*\) where the wormhole mass is equal to the black hole mass \(M(r_0)=M(r_+)=M^*\). Then, no further causal contact with the wormhole is possible “classically” afterwards since the throat is located inside the horizon \(r_+\).

^{1}Hence, for the case \(\beta Q>1/2\), the ranges of \(r_0\) and the wormhole mass \(M(r_0)\) are bounded by \(r_0 > r_+^*,~M(r_0) < M^*\) for the “observable” wormhole throat outside the black hole horizon \(r_+\), in contrast to \(r_0 \ge 0, ~M(r_0) \le M_0\) for the former case \(\beta Q \le 1/2\).

*r*, on the other hand, the metric function is expanded as

## 3 Massive scalar perturbations in the new AdS-EBI wormhole

*m*is given by

*l*.

*r*, using the asymptotic expansions (21) and (23), respectively. Here, \(\int ^\infty _{r_0}f^{-1} dr\) is the (finite) value of the tortoise coordinate \(r_*(r)\) evaluated at \(r=\infty \) and can be expanded as \(-{3}{{\Lambda }^{-1}} (r_0^{-1}+{{\Lambda }^{-1}} r_0^{-3}+ \cdots )\).

*not light-like*“generally” due to

*non-vanishing*\(f(r_0)\) and the effective potential \(\widetilde{V}(r_0)\), in contrast to the

*always-light-like*solutions at the black hole horizon \(r_+\), where \(f(r_+)\) and the effective potential (28) vanish: The effective potential \(\widetilde{V}(r_0)\)

*may*vanish when the mass of scalar field

*m*satisfies \(m^2=-l (l+1)/r_0^2\) (for example, \(m=0\) for \(l=0\), or \(m^2=-6/r_0^2\) for \(l=2\)), but generally it does not

^{2}(Fig. 4).

*shifted*tortoise coordinate \({\widetilde{r}}_{*} \equiv r_*- \int ^{\infty }_{r_0} f^{-1}dr\), which approaches zero as \(r \rightarrow \infty \).

^{3}

*global*AdS space,

^{4}\(m^2>m^2_{\mathrm{BF}}\) with the BF bound \(m^2_{\mathrm{BF}}=3 {\Lambda }/4\) [20, 21, 22, 23, 49]. Moreover, note that the solution (37) with the bound (40), satisfies the vanishing Dirichlet boundary condition at \(r=\infty \) (\({\widetilde{r}}_*=0\)) even though the effective potential \(\widetilde{V}\) is not positive infinite (Fig. 4): at \(r=\infty \), \(\widetilde{V}\) is positive infinite for \(m^2 >2 {\Lambda }/3\) but zero or negative infinite for \(2 {\Lambda }/3 \ge m^2 > 3 {\Lambda }/4\). The usual stability criterion based on the positivity of the effective potential is not quite correct when considering massive perturbations in AdS background for the latter mass range [50].

^{5}

## 4 **M**assive Quasi-Normal Modes

**M**

*reduced*effective potential

*V*(

*r*),

*V*(

*r*) is

*not*positive definite and its positivity depends on \(m^2\) (Fig. 5):

*V*(

*r*) is positive definite for the whole region of \(r \ge r_0\) only for \(m^2> 2 {\Lambda }/3\).

^{6}

*x*. Since \(f(x_0)>0\) in our wormhole system, one can remove the overall \((x-x_0)^{-1}\) factor in (46) so that \(x=x_0\) is not a singular point, whereas there is one regular singular point at the spatial infinity \(x=0\). Then one can expand Eq. (46) around the throat \(x_0\) up to the pole at \(x=0\) and solve the equation at each order of the expansion.

*s*,

*t*,

*u*around \(x=x_0\) as

^{7}

*vanishing*at the throat as \(\Phi \sim e^{-i{\omega }v}(x-x_0)\). Since in this paper we want to consider non-vanishing ingoing modes to study QNMs, we take only the \({\alpha }=0\) case.

^{8}Then the desired solution can be expanded as

*Mathematica*, the coefficients \(a_n, s_n, t_n,u_n\) can be computed up to an arbitrary order

*N*. In the next section, we present the numerical computation of QNM frequencies \({\omega }\equiv {\omega }_R-i {\omega }_I\) based on this method.

## 5 Numerical results and their interpretations

*M*and

*Q*. Here, we focus on the case of \(\beta Q>1/2\), where the well-defined GR limit of \(\beta \rightarrow \infty \) exists [19]. The result shows that the perturbations are stable (\({\omega }_I>0\)) if \(m^2\) is above certain threshold values \(m_*^2\) (\(-0.16769, -0.16720, -0.16704\) for \(M=10^{-5}, 0.1, 0.15\) with \(Q=1\) (Fig. 6); \(-0.16769, -0.16728, -0.16704\) for \(Q=1, 0.9, 0.85\) with \(M=10^{-5}\) (Fig. 7)). For a given value of

*M*or

*Q*, QNM frequencies \({\omega }_I\) and \({\omega }_R\) increase as \(m^2\) increases above \(m_*^2\). Here, we note that the critical mass \(m_*^2\) is close to the BF bound of (40), \(m^2_{\mathrm{BF}}=-3 {\Lambda }/4=-0.15\) [20, 21, 22, 23]. On the other hand, for a given value of \(m^2\), \({\omega }_I\) and \({\omega }_R\) increase as

*M*decreases or

*Q*increases, corresponding to increasing throat radius \(r_0\) from (16) (Fig. 2). Neglecting the small differences of \(m_*^2\) for different

*M*and

*Q*, we may approximately fit the numerical result of the \(m^2\) dependence in Figs. 6 and 7 to the

*analytic*functions

*M*and

*Q*. The result shows that the critical mass squared shifts as \(m_*^2 \approx -0.16769, -0.11156,-0.08294\) for \({\Lambda }=-0.2,-0.13,-0.1\), respectively. This is consistent with the shifts of the corresponding

*BF*bounds, \(m_{\mathrm{BF}}^2=3 {\Lambda }/4=-0.15,-0.1,-0.075\).

Figure 9 shows the lowest QNM frequency as a function of \(m^2\) for varying angular-momentum number *l*. The result shows that the critical mass shifts as \(m_*^2 \approx -0.16769, -0.16679,-0.15524\) for \(l=2,1,0\), respectively. Especially for \(l=0\), it shows a *level crossing* between the lowest *pure-imaginary* mode representing an over-damping (purple color) and the lowest *complex* mode (pink color) at \(m^2\approx -0.1089\) before the critical mass \(m^2_*\) is being reached, so that the instability is governed by the used-to-be higher (pure) imaginary frequency modes.^{9} On the other hand, there are discontinuities in \({\omega }_I\) and \({\omega }_R\) at \(m^2 \approx -0.0784\) for the lowest *complex* frequency \({\omega }={\omega }_R-i {\omega }_I\) (green color) even before the level-crossing point.^{10}

*l*for varying \(m^2\). For small

*l*(\(l \le 5\)), \({\omega }_I\) decreases but \({\omega }_R\) increases (except for the case \(m^2<0\)) as

*l*increases, whereas for large

*l*, both \({\omega }_I\) and \({\omega }_R\) increase as

*l*increases. It is interesting to note that there is a bouncing point of \({\omega }_I\), where \({\omega }_I=0\) at \(l=5\). Here we have considered only the case \(m^2 >m_*^2\), which is stable for small

*l*.

Figure 11 shows the QNM frequency as a function of \(r_0\) for varying \(m^2\) in Fig. 6. Though the result is preliminary since there are only limited data points, i.e., for a given value of *m*, only three points of \(r_0\) which correspond to the three values of *M* in Fig. 6, the result indicates interestingly the linear dependence of QNM frequency to the throat radius.

^{11}

^{12}

*m*, should have the following form

^{13}

^{14}it seems to agree with the so-called “

*quasi-resonance modes (QRMs)*” with \({\omega }_I=0\) in massive QNMs, though the oscillatory parts are different,

*i.e.,*\({\omega }_R=0\) in our wormhole case but \({\omega }_R \ne 0\) in QRMs [51].

^{15}

^{16}and one finds the imaginary part of the integral

*light-like*at the throat, similar to the case for the black hole [52, 57]. Actually, this second case corresponds to our choice of vanishing Neumann boundary condition \(({d \varphi }/{dr_*}|_{r_*=0}=0)\) from the solution \(\varphi = {e^{i{\omega }r_*}{\widetilde{\varphi }}} \sim e^{i({\omega }-k) r_*}+i(k-{\omega }){r_*} e^{i {\omega }r_*}\). Now, for the simplicity of our discussion, let us consider only the second case,

^{17}which we have studied numerically in this paper. From (67), one finds naively that “ non-vanishing \({\omega }_R\) may imply \({\omega }_I>0\), i.e., stable modes ” [52, 57]. However, here it is important to note that this is the only case when \({\omega }_R\) (or \(k_R\)) is “independent” on \({\omega }_I\): When \({\omega }_R\) (or \(k_R\)) is

*not*independent on \({\omega }_I\), the solution \({\omega }_I>0\) may not be the

*unique*possibility, generally. For example, if we consider \({\omega }_R\) as a function of \({\omega }_I\), \({\omega }_R={\omega }_R ({\omega }_I)\), (67) is generally a non-linear equation for one independent variable \({\omega }_I\) and its solution needs not to agree with the previous naive one without separating independent variables, which may lead to misleading solutions. This means that \({\omega }_R \ne 0,~ {\omega }_I<0\) (oscillating, unstable modes) can also be the possible solution depending on the details of \({\omega }_R={\omega }_R ({\omega }_I)\). Actually, Fig. 14 shows the relation \({\omega }_R \approx 1/2+ a~ {\omega }_I +b~ {\omega }_I^2 \cdots \) which allows the continuous solution of \({\omega }_R \ne 0, {\omega }_I<0\), as implied by the result of (58) (left), as well as the usual discontinuous solution (right, \(l=0,1\) case). This indicates the existence of more fundamental reason for the relation.

^{18}The usual result \({\omega }_R=0, {\omega }_I<0\) for the case \(l=0\) may be due to the independence of \({\omega }_I(<0)\) from \({\omega }_R\) after the level crossing of two initially different modes.

Finally, in order to obtain reliable numerical results we need to compute the partial sum with the typical truncation of the order of \(N=250\) (Fig. 15). In this work, we consider up to \(N=300\) for most computations, but we consider up to \(N=400\) for the level crossing of \(l=0\) QNMs in Fig. 9.

## 6 Concluding remarks

We have studied QNMs for a massive scalar field in the background of a *natural* AdS wormhole in EBI gravity, which has been recently constructed without exotic matters. For the case where the GR limit exists, i.e., \(\beta Q>1/2\), we have shown numerically the existence of a BF-like bound \(m_*^2<0\) so that the perturbation is unstable for a tachyonic mass \(m^2 <m_*^2\), like the perturbation in the global AdS. Furthermore, we have shown that the unstable modes (\({\omega }_I<0\)) can also have oscillatory parts (\({\omega }_R\ne 0\)) as well as non-oscillatory parts (\({\omega }_R= 0\)), depending on whether the real and imaginary parts of frequencies are dependent on each other or not, contrary to arguments in the literature. On the other hand, for the case where the GR limit does not exist, i.e., \(\beta Q\le 1/2\), the BF-like bound does not exist. In this case, the perturbation is completely “frozen” above a certain non-tachyonic mass bound \(m_c^2>0\) which is big compared to the wormhole mass *M* for \(\beta Q< 1/2\).

*level crossing*(between the lowest pure-imaginary and complex modes) of \({\omega }_I\) for \(l=0\) and a bouncing behavior of \({\omega }_I\) for higher

*l*. We have shown the linear dependence of QNMs on the throat radius, analogous to the black hole case. Even though the thermodynamic implication of this behavior is not quite clear, it would be interesting to study its implication to the corresponding boundary field theory as in the black hole case based on the

*AdS/CFT-correspondence*[58]. In particular, considering higher-order contributions for small \({\widetilde{r}}_*\) or large

*r*regime, our radial equation (36) can be approximated by a Calogero-like model [47],

*E*and it would be interesting to study its connection to some integrable theories at the boundary.

As the final remark, it would be straightforward to extend our formalism to more general perturbations with spins, including the (gravitational, spin 2) perturbations of the wormhole space-time itself. It would be interesting to study whether ring-down phases of natural wormholes can mimic those of black holes.

## Footnotes

- 1.
This implies that natural wormholes can be the factories of black holes by accretion of ordinary matters or vice versa [18].

- 2.
(i) \(l=0\) modes: For \(m=0\), \(\widetilde{V}\) is positive definite for \(r > r_0\) and vanishes at the throat \(r_0\), for \(m^2>0\), \(\widetilde{V}\) is positive definite for the whole region of \(r \ge r_0\), whereas for \(m^2<0\), \(\widetilde{V}\) is not positive definite but it depends on \(m^2\). (ii) \(l>0\) modes: For \(m^2=-l (l+1)/r_0^2\), \(\widetilde{V}\) vanishes at \(r_0\) but it is negative for \(r > r_0\), whereas for \(m^2 \ge 2 {\Lambda }/3\), \(\widetilde{V}\) is positive for the whole region of \(r \ge r_0\), otherwise, i.e., for \(-l (l+1)/r_0^2<m^2 <2 {\Lambda }/3\), \(\widetilde{V}\) is not positive definite. It is interesting to note that in the last case \(\widetilde{V}(r_0)<0\), the ingoing waves become “tachyonic” at the throat, i.e., \(k^2>{\omega }^2\) and we will see later that these perturbations are still stable if it is not too much tachyonic, i.e., \(2 {\Lambda }/3> m^2 > 3 {\Lambda }/4\).

- 3.
- 4.
In some literature (cf. [20, 21, 22, 23]), the limiting case \(m^2=3 {\Lambda }/4\) or more generally in

*D*dimensions, \(m^2=(D-1) {\Lambda }/2(D-2)\) or \(-(D-1)^2 \ell ^{-2}/4\) with \(\ell ^{-2}=-2 {\Lambda }/(D-1)(D-2)\), has been classified as the stable one due to positivity of the energy functional but it would not be a physically viable fluctuation due to the divergence of its energy functional, which is related to the divergent norm of the solution as discussed in the above footnote No. 3. - 5.
The bound \(m^2 > 3 {\Lambda }/4\) corresponds to the absence of “genuine” tachyonic modes in the global AdS background [20, 21, 22, 23]. But this does not mean the absence of genuine tachyonic modes

*locally*. Actually, for \(2 {\Lambda }/3> m^2 > 3 {\Lambda }/4\), the ingoing modes at the throat are tachyonic as noted in the footnote No. 2, in contrast to QNMs in black hole background which are always light-like at the black hole horizon. - 6.
The metric approaches to that of \(AdS_4\) in Poincare patch \(ds^2=(dz^2+\eta ^{\mu \nu }dx_{\mu } dx_{\nu })/z^2\) with the three-dimensional flat Minkowski metric \(\eta ^{\mu \nu }\) and the radial

*AdS*coordinate \(z\sim x\) for an appropriate choice of scalings. (cf. [49]). - 7.
For the black hole cases, the coefficients are obtained as \(s_{0} =2 {\kappa }x_{+}^2,~ s_{1} = x_{+} [8 {\kappa }-\frac{1}{2}x_{+}^3 {\ddot{f}}(x_{+}) ],~ t_0=2 x_{+}^2 ( {\kappa }-i {\omega }),~ t_1=-x_{+} [-12 {\kappa }+ x_{+}^3 {\ddot{f}}(x_{+})+4i {\omega }],~ t_2=-2i {\omega }+ ~{(\mathrm{non-}{\omega }~ \mathrm{terms})}, ~ t_{n(>2)}=~{(\mathrm{non-}{\omega }~ \mathrm{terms})},~ u_1 = l(l+1) x_{+}^2 +2{\kappa }x_{+} + m^2, ~ u_2 =x_{+} [2 l(l+1)+6 {\kappa }x_{+}^{-1}-x_{+}^2 {\ddot{f}}(x_{+})]\). Compared with (49), the most important qualitative difference is the absence of \(s_{-1}\) in the black hole case so that the horizon \(x_+=1/r_+\) becomes a regular singular point. Other coefficients look similar, with the role of the surface gravity \({\kappa }\) at the black hole horizon replaced by \(x_0 f(x_0)\) in our natural wormhole case: this may be understood from the direct relation \(x_0 f(x_0)=2 {\kappa }|_{r_+ \rightarrow r_0}=4 \pi {T_H}|_{r_+ \rightarrow r_0}\) in (14).

- 8.
The \({\alpha }=1\) case corresponds to

*normal modes*without any wave flow, i.e., energy loss at the throat. It is interesting that our wormhole system allows also this solution as well as QNMs with the ingoing mode solutions of \({\alpha }=0\). This is in contrast to the black hole system, where outgoing mode solutions are allowed instead, as well as the ingoing modes at the horizon. - 9.
In this paper we have so far considered only the lowest QNMs or the small region around the level-crossing point. In order to discuss higher modes or the larger region around the crossing point, we need to increase the order

*N*since the numerical accuracy decreases as the mode number is increased generally. It would be interesting to check whether other unstable branches exist for higher modes but this is beyond the scope of this paper. - 10.
For the level crossings in black strings, where the effective masses due to Kaluza-Klein reduction is naturally introduced, see [52]. It seems this phenomena is universal in massive QNMs.

- 11.
Since the quantity \({f(r_0)}/{r_0}\), which is given by \( -{\Lambda }r_0 +{1}/{r_0}-{Q^2}/{r_0^3}+{Q^4}/{4 \beta ^2 r_0^7}+ \mathcal{O}(r_0^{-11}), \) corresponds to the surface gravity for black hole case, \(2{\kappa }=4 \pi T_H=(df(r)/dr)|_{r_+}\), as noted in footnote No. 7, the wormhole’s QNM frequencies (59) may also be fitted to \({\omega }_I \approx d_0-d_1 {\Lambda }^{-1} f(r_0)/r_0,~ {\omega }_R \approx e_0-e_1 {\Lambda }^{-1} f(r_0)/r_0\) for the leading order of large \(r_0\).

- 12.
The scaling for the (abbreviated) wormhole mass

*M*, which differs from that of scalar field mass*m*, is due to the omitted Newton’s constant*G*:*G*has the dimension \([M]^{-1} [L]^3 [T]^{-2}\) and hence transforms as \(G\rightarrow \alpha ^2 G\) so that the ADM mass, \(M_{ADM}\sim GM\), transforms as \(M_{ADM}\rightarrow \alpha M_{ADM}\) as in (60). - 13.
Expanding near the critical mass squared \(m^2_*\) and comparing with (58), one can obtain the relations \({\bar{\zeta }} {\Lambda }r_0 + {\bar{\eta }} \beta \approx 1/2-r_0 \mathcal{M}(m^2_*) ,~ c_1-i b_1 \approx \mathcal{M}'(m^2_*), ~ c_2-i b_2 \approx \mathcal{M}''(m^2_*)/2\), where \(\mathcal{M}(m^2)\equiv \left( {\bar{\delta }} m^2 +{\bar{\gamma }}{\Lambda }^{-1}m^4 +\cdots \right) \) .

- 14.
For the case of \(\beta Q>1/2\), the numerical accuracy decreases as one increases the mass

*m*beyond the plots shown in Fig. 12 and we did not include those cases in this paper. - 15.
For \(3 \Lambda /4 (\approx -0.15) \le m^2 \le 2 \Lambda /3 (\approx -0.13333)\), the vanishing Dirichlet boundary condition (\(\varphi (0)=0\)) may not uniquely determine the desired solution with \(C_2=0\) in (56), in contrast to the normalizability condition in Sec. III. But for a truncated summation, \(\varphi _N (x)=\sum _{n=0}^{N}a_{n}(-x_0)^n\), it seems that only the more-rapidly decaying solution of \(C_1\) part may be obtained from that boundary condition.

- 16.
The boundary term vanishes as \({\widetilde{r}}_*^{\sqrt{1-4 {\lambda }}}\) for the desired solution. But it diverges as \({\widetilde{r}}_*^{-\sqrt{1-4 {\lambda }}}\) for the other solution \({\widetilde{\varphi }}(x) \sim {\widetilde{r}}_*^{\frac{1}{2}(1-\sqrt{1-4 {\lambda }})}\) with \(B_1=0\) in (37), which means the infinite amount of flux \(\sim {\widetilde{\varphi }}^* d {\widetilde{\varphi }}/dr_*\) at infinity. This can be considered as an alternative criterion for the desired solution [20, 21, 22, 23, 30].

- 17.
For the first case, the situation looks more complicated since, as noted in Sec. III, the solution is not light-like generally at the throat due to \(\widetilde{V}(r_0)\ne 0\) and \(k_R\) is not a simple function of \({\omega }_R\) alone. But since the

*sign*of \(k_R\) coincides with that of \({\omega }_R\), the argument is basically the same. - 18.

## Notes

### Acknowledgements

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2018R1D1A1B07049451 (COL), 2016R1A2B401304 (MIP)).

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