Quadrupole Perturbations of Slowly Spinning Ellis–Bronnikov Wormholes

Abstract

We study the axial and polar perturbations of slowly rotating Ellis–Bronnikov wormholes in General Relativity, applying a perturbative double expansion. In particular, we derive the equations for $l=2$, $M_z=2$ perturbations of these objects, which are parametrized by an asymmetry parameter. The equations constitute an astrophysically interesting sector of the perturbations that contribute dominantly to the gravitational wave radiation. Moreover, calculation of these modes may exhibit potential instabilities in the quadrupole sector.

1. Introduction

The detection of gravitational waves from the inspiral, merger, and ringdown of compact objects provides an excellent tool to learn about both gravitation in the strong gravity regime and the compact objects themselves [1,2,3,4,5,6,7]. Besides black holes and neutron stars, a variety of further compact objects are being discussed as hypothetical astrophysical objects, whose signatures might be observable. Many of these objects feature as black hole mimickers, making their study an interesting endeavor [8].

One type of black hole mimicker is represented by wormholes [9]. In General Relativity (GR), wormholes need exotic matter [10,11,12,13,14,15] or quantum matter [16,17,18] for their support. In contrast, in alternative theories of gravity, the energy conditions may also be violated by the effective stress–energy tensor arising from the modified gravitational interaction [12,19,20,21,22,23].

Numerous potentially observable features of wormholes have already been discussed (see e.g., gravitational lensing by wormholes [24,25,26,27,28,29,30,31,32,33,34,35], shadows of wormholes [33,36,37,38,39,40,41,42], or accretion disks surrounding wormholes [43,44,45,46,47,48,49,50]). Concerning the ringdown of wormholes, so far, mainly quasinormal modes have been studied for static wormholes [51,52,53,54,55,56,57,58,59,60,61,62,63,64,65], and only recently has the exploration of quasinormal modes of rotating wormholes begun [66]. Besides those quasinormal modes, echoes of Kerr-like wormholes have also been addressed [67].

Here we focus on the ringdown of slowly rotating wormholes, choosing Ellis–Bronnikov wormholes [13,14] as the background solutions. Slowly rotating Ellis–Bronnikov wormholes have been constructed perturbatively up to second order in rotation in closed form [68,69,70,71]. For rapid rotation, no closed-form solutions have been obtained so far [72], while, numerically, some sets of solutions are available [73,74]. The analysis of the quasinormal modes in the slow rotation case usually proves rather valuable, since on the one hand, important features of the presence of rotation arise already, while on the other hand, the modes provide a crucial limiting test for the rapidly rotating case (see e.g., [75,76]).

In this paper, we derive the complete set of quadrupole perturbation equations for the quasinormal modes of second-order Ellis–Bronnikov background solutions. In Section 2, we present the theoretical settings together with the background solutions. Section 3 provides the general ansatz for the perturbations and discusses the derivation in the set of perturbation equations. The derivation makes large use of previous work for quasinormal modes of slowly rotating black holes [77], which showed that the exact quasinormal mode spectrum was approximated with rather good precision at least up to 50% of the extremal angular momentum. We conclude in Section 4 and provide some of the lengthy equations in the Appendix A.

2. Theoretical Settings

Ellis–Bronnikov wormholes are based on the action

$$\begin{array}{ccc}\hfill S[g,\Phi ]& =& {\displaystyle \frac{1}{16\pi G}}\int d^{4}x\sqrt{-g}\left[\mathrm{R}+2{\partial }_{\mu }\Phi {\partial }^{\mu }\Phi \right],\hfill \end{array}$$

(1)

where a phantom scalar field $\Phi$ is coupled minimally to GR. Thus, $\mathrm{R}$ is the curvature scalar, G is Newton’s constant, and the kinetic term of the scalar field has the sign reversed as compared to an ordinary scalar field. Variation of the action leads to the field equations of the theory

$$\begin{array}{c}\hfill {\mathrm{R}}_{\mu \nu }=-2{\partial }_{\mu }\Phi {\partial }_{\nu }\Phi ,{\nabla }_{\mu }{\nabla }^{\mu }\Phi =0.\end{array}$$

(2)

The resulting Ellis–Bronnikov wormhole solutions are well-known static spherically symmetric solutions that can be expressed in the form

$$\begin{array}{ccc}\hfill d{s}^2& =& g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }=-e^{f}d{t}^2+e^{-f}\left[d{r}^2+\left(r^2+{\mathit{r}}_{\mathsf{0}}^2\right)\left(d{\theta }^2+{sin}^2\theta d{\phi }^2\right)\right],\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill \Phi \left(r\right)& =& {\displaystyle \frac{Q_{0}f\left(r\right)}{C}}\hfill \end{array}$$

(4)

with

$$f={\displaystyle \frac{C}{{\mathit{r}}_{\mathsf{0}}}}\left[{tan}^{-1}\left({\displaystyle \frac{r}{{\mathit{r}}_{\mathsf{0}}}}\right)-{\displaystyle \frac{\pi }{2}}\right].$$

(5)

This solution contains several parameters: C is related to the symmetry of the wormholes with respect to reflections $r\to -r$. Since only $C=0$ leads to a symmetric wormhole, C is referred to as the asymmetry parameter. The mass $M_0$ at (radial) plus infinity of the static wormholes is also given in terms of the asymmetry parameter C, $M_0=C/2$; the symmetric wormholes thus have vanishing mass, $M_0=0$. The corresponding scalar charge $Q_0$ of the static wormholes is given by $Q_0=\sqrt{C^2/4+{\mathit{r}}_{\mathsf{0}}^2}$. Here, ${\mathit{r}}_{\mathsf{0}}$ is a free parameter that determines the size of the throat of symmetric wormholes. Note that for $C=0$, the scalar charge is simply given by this free parameter. The circumferential radius $\mathcal{R}$ is obtained from $g_{\varphi \varphi }$,

$${\mathcal{R}}^2=e^{-f}(r^2+{\mathit{r}}_{\mathsf{0}}^2).$$

(6)

Clearly, for symmetric wormholes, it reaches its minimum for $r=0$. However, for asymmetric wormholes, the minimum, and thus the throat, is located at $r=C/2$. The area A of the throat is therefore given by

$${\left.A=4\pi {\mathcal{R}}^2\right|}_{r=C/2}.$$

(7)

Note that this expression corrects the expression for the area given in [70].

3. Perturbative Background Solutions

We now briefly recall the background solutions which are given up to a second order in rotation by [70,71,78,79]

$$\begin{array}{ccc}\hfill d{s}^2& =& -e^f\left[1+{ϵ}_r^{2}2\left(h_0\left(r\right)+h_2\left(r\right)P_2\left(\theta \right)\right)\right]d{t}^2+e^{-f}\left[1+{ϵ}_r^{2}2\left(b_0\left(r\right)+b_2\left(r\right)P_2\left(\theta \right)\right)\right]d{r}^2\hfill \\ & +& e^{-f}{R}^2\left[1+{ϵ}_r^{2}2\left(k_0\left(r\right)+k_2\left(r\right)P_2\left(\theta \right)\right)\right]\times \left[d{\theta }^2+{sin}^2\left(\theta \right){\left[d\phi -{ϵ}_{r}w\left(r\right)dt\right]}^2\right],\hfill \end{array}$$

(8)

where $R^2=(r^2+{\mathit{r}}_{\mathsf{0}}^2)$, and $P_2\left(\theta \right)=\left(3{cos}^2\left(\theta \right)-1\right)/2$ denotes the Legendre polynomial. With the parameter ${ϵ}_r\ll 1$, we keep track of the order of the slow rotation perturbation contributions. Up to second order we need to introduce seven radial perturbation functions, $h_0,h_2,b_0,b_2,k_0,k_2,w$, that can, in general, be reduced to six by choosing a gauge $k_0\left(r\right)=0$ and redefining $k_2=h_2-{\nu }_2$. At the same time, the phantom field is given by

$$\begin{array}{c}\hfill \Phi =\varphi \left(r\right)+{ϵ}_r^2\left({\varphi }_{20}\left(r\right)+{\varphi }_{22}\left(r\right)P_2\left(\theta \right)\right),\end{array}$$

(9)

for the second order in rotation, with the two perturbation functions ${\varphi }_{20},{\varphi }_{22}$. $\varphi \left(r\right)$ is the static background scalar given by Equation (4). Thus, together there are eight unknown functions, $w,h_0,h_2,b_0,b_2,{\nu }_2,{\varphi }_{20},{\varphi }_{22}$, that have to be determined by solving the field equations to obtain the desired background metric.

For the first order we obtain the function $w\left(r\right)$ for the slowly rotating background [70,71],

$$\begin{array}{c}\hfill w\left(r\right)={\displaystyle \frac{3J}{2C(C^2+{\mathit{r}}_{\mathsf{0}}^2)}}\left[1-\left(1+2C{\displaystyle \frac{C+r}{R^2}}\right)e^{2f}\right],\end{array}$$

(10)

where J denotes the angular momentum of the wormhole. The remaining seven functions $h_0,h_2,b_0,b_2,{\nu }_2,{\varphi }_{20},{\varphi }_{22}$ decouple into two sets of functions, ${\mathcal{P}}_0=\{h_0,b_0,{\varphi }_{20}\}$ and ${\mathcal{P}}_2=\{h_2,{\nu }_2,b_2,{\varphi }_{22}\}$. Closed-form expressions for these functions can be found in [70,71]. We exhibit all eight functions in Figure 1 for several values of the asymmetry parameter C.

The mass of the wormholes at plus infinity is then given in the second order in rotation by $M=M_0+\Delta M$, where the mass correction $\Delta M$ is extracted from the solution for $b_0\left(r\right)$,

$$\begin{array}{ccc}\hfill \Delta M& =& {\displaystyle \frac{3J^2}{\left({\mathit{C}}^2+4{\mathit{r}}_{\mathsf{0}}^2\right){\left({\mathit{C}}^2+{\mathit{r}}_{\mathsf{0}}^2\right)}^2\left(\left({\mathit{C}}^2+4{\mathit{r}}_{\mathsf{0}}^2\right){\cot }^{-1}\left(\frac{\mathit{C}}{2{\mathit{r}}_{\mathsf{0}}}\right)-2\mathit{C}{\mathit{r}}_{\mathsf{0}}\right){\mathit{C}}^2}}\times \hfill \\ & & (\left(17{\mathit{C}}^4{\mathit{r}}_{\mathsf{0}}^3-2{\mathit{C}}^2{\mathit{r}}_{\mathsf{0}}^5+8{\mathit{r}}_{\mathsf{0}}^7\right){\mathrm{e}}^{-{\displaystyle \frac{2\mathit{C}{\cot }^{-1}\left(\frac{\mathit{C}}{2{\mathit{r}}_{\mathsf{0}}}\right)}{{\mathit{r}}_{\mathsf{0}}}}}+\left({\mathit{C}}^2+4{\mathit{r}}_{\mathsf{0}}^2\right)\times \hfill \\ & & \left(\left({\mathit{C}}^5+5{\mathit{C}}^3{\mathit{r}}_{\mathsf{0}}^2+4\mathit{C}{\mathit{r}}_{\mathsf{0}}^4\right){\cot }^{-1}\left({\displaystyle \frac{\mathit{C}}{2{\mathit{r}}_{\mathsf{0}}}}\right)-2{\mathit{C}}^4{\mathit{r}}_{\mathsf{0}}-7{\mathit{r}}_{\mathsf{0}}^3{\mathit{C}}^2-2{\mathit{r}}_{\mathsf{0}}^5\right)),\hfill \end{array}$$

(11)

which reduces for $C=0$ to

$$\begin{array}{c}\hfill \Delta M=3J^2({\pi }^2-8)/\left(2\pi {\mathit{r}}_{\mathsf{0}}^3\right),\end{array}$$

(12)

and is read off at plus infinity. Thus the mass M of the rotating wormhole is non-vanishing for $C=0$, i.e., ${M|}_{C=0}=\Delta M$ (see also [69]). Analogously, to the mass M, the scalar charge Q also consists of the static charge $Q_0$ and the correction term $\Delta Q$,

$$Q=Q_0+{ϵ}_r^2\Delta Q,$$

(13)

where $\Delta Q=-\Delta M$.

4. Quadrupole Perturbations

Having established the background solutions $\{g^{\left(sr\right)},{\Phi }^{\left(sr\right)}\}$ up to second order in rotation, we now perturb up to the first order in ${ϵ}_q$ the metric field

$$\begin{array}{ccc}\hfill g_{\mu \nu }& =& g_{\mu \nu }^{\left(sr\right)}+{ϵ}_q\delta h_{\mu \nu }(t,r,\theta ,\phi )\hfill \\ & =& g_{\mu \nu }^{\left(sr\right)}+{ϵ}_q\left(\delta h_{\mu \nu }^{\left(A\right)}(t,r,\theta ,\phi )+\delta h_{\mu \nu }^{\left(P\right)}(t,r,\theta ,\phi )\right),\hfill \end{array}$$

(14)

and the phantom scalar field

$$\begin{array}{ccc}\hfill \Phi & =& {\Phi }^{\left(sr\right)}+{ϵ}_q\delta {\varphi }^{\left(P\right)}(t,r,\theta ,\phi ).\hfill \end{array}$$

(15)

The metric field perturbations consist of both the axial-led (A) and polar-led (P) perturbations, while the scalar field contributes solely to the polar perturbations.

In the static limit, the metric perturbations are given by

$$\begin{array}{c}\hfill \delta h_{\mu \nu }=e^{i\omega t}\left(\begin{array}{cccc}-e^{f}NY& -\mathit{H}\mathsf{1}Y& -\mathit{h}\mathsf{0}{\partial }_{\phi }Y/s_{\theta }& \mathit{h}\mathsf{0}{s}_{\theta }{\partial }_{\theta }Y\\ -\mathit{H}\mathsf{1}Y& -e^{-f}LY& -\mathit{h}\mathsf{1}{\partial }_{\phi }Y/s_{\theta }& \mathit{h}\mathsf{1}{s}_{\theta }{\partial }_{\theta }Y\\ -\mathit{h}\mathsf{0}{\partial }_{\phi }Y/s_{\theta }& -\mathit{h}\mathsf{1}{\partial }_{\phi }Y/s_{\theta }& R^{2}TY& 0\\ \mathit{h}\mathsf{0}{s}_{\theta }{\partial }_{\theta }Y& \mathit{h}\mathsf{1}{s}_{\theta }{\partial }_{\theta }Y& 0& R^{2}T{s}_{\theta }^{2}Y\end{array}\right),\end{array}$$

(16)

where Y denotes the spherical harmonics, $s_{\theta }=sin\theta$, $R^2=r^2+{\mathit{r}}_{\mathsf{0}}^2$, and $\omega$ is an eigenvalue. The axial metric perturbations are given by $\mathit{h}\mathsf{0}\left(r\right)$ and $\mathit{h}\mathsf{1}\left(r\right)$, while the polar metric perturbations are $T\left(r\right),L\left(r\right),N\left(r\right)$ and $\mathit{H}\mathsf{1}\left(r\right)$. The scalar perturbations are decomposed as

$$\begin{array}{c}\hfill \delta \varphi (t,r,\theta ,\phi )=e^{i\omega t}\varphi \mathsf{1}\left(r\right)Y(\theta ,\phi ).\end{array}$$

(17)

Besides the spherical harmonics $Y(\theta ,\phi )$, all the r-dependent perturbation functions also carry two quantum numbers, l and $M_z$.

In the presence of rotation, the perturbations need to be summed over all possible values of l and $M_z$. While the axial symmetry of the background configurations still leads to a decoupling of the different values of $M_z$, the l values are now coupled and one obtains a tower of equations, when the ansatz is inserted in the field equations for the metric and phantom scalar components,

$$\begin{array}{c}\hfill {\mathcal{G}}_{\mu \nu }={\mathcal{G}}_{\mu \nu }^{\left(sr\right)}+{ϵ}_q\delta {\mathcal{G}}_{\mu \nu }{e}^{-i\omega t}=0,\end{array}$$

(18)

$$\begin{array}{c}\hfill \mathcal{S}={\mathcal{S}}^{\left(sr\right)}+{ϵ}_q\delta \mathcal{S}{e}^{-i\omega t}=0,\end{array}$$

(19)

where the slowly rotating background solution ensures ${\mathcal{G}}_{\mu \nu }^{\left(sr\right)}=0$, and ${\mathcal{S}}^{\left(sr\right)}=0$.

Here, we follow [77,80] and decompose the field equations in terms of spherical harmonics, and then truncate the tower of equations in the slow rotation approximation, mixing different l. The resulting modes can be identified by the value of l (as well as by $M_z$), to which the modes reduce in the static limit of the background solutions, referring to the modes then as l-led modes. Moreover, in the perturbative scheme, the equations decouple into two sets, the polar-led and the axial-led perturbations [77].

The resulting system of equations then has the generic structure

$$\begin{array}{c}\hfill {{\overrightarrow{z}}_i}^{\prime }={\mathbf{M}}_i{\overrightarrow{z}}_i,\end{array}$$

(20)

where ${\overrightarrow{z}}_i$ denotes a vector of perturbation functions or their derivatives; ${\mathbf{M}}_i$ is a matrix containing the background functions, which depend on the radial coordinate r, the free parameter ${\mathit{r}}_{\mathsf{0}}$, the asymmetry parameter C, and the eigenvalue $\omega$; and $i=p,a$ denotes the polar-led and axial-led equations, respectively. In the following, we focus on quadrupole, i.e., $l=2$-led, perturbations, and select $M_z=2$.

The vector ${\overrightarrow{z}}_a$ for the axial-led perturbations reads

$$\begin{array}{ccc}\hfill {\overrightarrow{z}}_a=& & {\left[\mathit{h}{\mathsf{0}}_{2,2},\mathit{h}{\mathsf{1}}_{2,2},\mathit{H}{\mathsf{1}}_{3,2},T_{3,2},L_{3,2},N_{3,2},\varphi {\mathsf{1}}_{3,2},\varphi {\mathsf{1}}_{3,2}^{\prime }\right]}^T,\hfill \end{array}$$

(21)

where ${\mathbf{M}}_a$ is a $8\times 8$ matrix whose entries depend on the functions of the slowly rotating background and the eigenvalue $\omega$. Analogously, the vector ${\overrightarrow{z}}_p$ for the polar-led perturbations reads

$$\begin{array}{ccc}\hfill {\overrightarrow{z}}_p=& & {\left[\mathit{h}{\mathsf{0}}_{3,2},\mathit{h}{\mathsf{1}}_{3,2},\mathit{H}{\mathsf{1}}_{2,2},T_{2,2},L_{2,2},N_{2,2},\varphi {\mathsf{1}}_{2,2},\varphi {\mathsf{1}}_{2,2}^{\prime }\right]}^T.\hfill \end{array}$$

(22)

Thus, the $l=2,M_z=2$ axial-led perturbation functions $\mathcal{A}$ consist of the axial $\mathit{h}{\mathsf{0}}_{2,2},\mathit{h}{\mathsf{1}}_{2,2}$ functions and the polar $\mathit{H}{\mathsf{1}}_{3,2},T_{3,2},L_{3,2},N_{3,2},\varphi {\mathsf{1}}_{3,2}$ functions, while the $l=2,M_z=2$ polar-led perturbation functions $\mathcal{P}$ consist of the polar $\mathit{H}{\mathsf{1}}_{2,2},T_{2,2},L_{2,2},N_{2,2},\varphi {\mathsf{1}}_{2,2}$ functions and the axial $\mathit{h}{\mathsf{0}}_{3,2},\mathit{h}{\mathsf{1}}_{3,2}$ functions.

Consequently, there are, for each of the axial-led and polar-led $l=M_z=2$ perturbations in total, seven perturbation equations that have to be determined. We present the explicit expression for these perturbation equations for the symmetric wormholes, $C=0$, in the Appendix A, since in this case they already lead to rather lengthy expressions.

To obtain the solutions for the modes, one has to numerically solve the resulting system of equations as an eigenvalue problem, subject to the appropriate set of boundary conditions. For the wormholes, suitable boundary conditions are purely outgoing boundary conditions at both radial infinities.

To obtain the appropriate behavior at the boundaries, we consider the following parametrization of the perturbation functions at plus infinity,

$$\begin{array}{ccc}\hfill \mathit{h}{\mathsf{0}}_{l,2}\left(r\right)& =& \sqrt{r^2+{\mathit{r}}_{\mathsf{0}}^2}\mathit{h}\mathsf{0}\left(r\right)e^{\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill \mathit{h}{\mathsf{1}}_{l,2}\left(r\right)& =& \sqrt{r^2+{\mathit{r}}_{\mathsf{0}}^2}\mathit{h}\mathsf{1}\left(r\right)e^{\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill \mathit{H}{\mathsf{1}}_{l,2}\left(r\right)& =& \sqrt{r^2+{\mathit{r}}_{\mathsf{0}}^2}\mathit{H}\mathsf{1}\left(r\right)e^{\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill T_{l,2}\left(r\right)& =& T\left(r\right)e^{\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill {\mathit{L}}_{l,2}\left(r\right)& =& \sqrt{r^2+{\mathit{r}}_{\mathsf{0}}^2}\mathit{L}\left(r\right)e^{\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill {\mathit{N}}_{l,2}\left(r\right)& =& \sqrt{r^2+{\mathit{r}}_{\mathsf{0}}^2}\mathit{N}\left(r\right)e^{\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill \varphi {\mathsf{1}}_{l,2}\left(r\right)& =& \left(1/\sqrt{r^2+{\mathit{r}}_{\mathsf{0}}^2}\right)\varphi \mathsf{1}\left(r\right)e^{\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(29)

for $l=2,3$. At minus infinity, for axial-led perturbations, we consider the following parametrization,

$$\begin{array}{ccc}\hfill \mathit{h}{\mathsf{0}}_{2,2}\left(r\right)& =& {(r^2+{\mathit{r}}_{\mathsf{0}}^2)}^{3/2}\mathit{h}\mathsf{0}\left(r\right)e^{-\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill \mathit{h}{\mathsf{1}}_{2,2}\left(r\right)& =& {(r^2+{\mathit{r}}_{\mathsf{0}}^2)}^{3/2}\mathit{h}\mathsf{1}\left(r\right)e^{-\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill \mathit{H}{\mathsf{1}}_{3,2}\left(r\right)& =& (r^2+{\mathit{r}}_{\mathsf{0}}^2)\mathit{H}\mathsf{1}\left(r\right)e^{-\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill T_{3,2}\left(r\right)& =& \sqrt{r^2+{\mathit{r}}_{\mathsf{0}}^2}T\left(r\right)e^{-\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill {\mathit{L}}_{3,2}\left(r\right)& =& \mathit{L}\left(r\right)e^{-\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill {\mathit{N}}_{3,2}\left(r\right)& =& \mathit{N}\left(r\right)e^{-\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill \varphi {\mathsf{1}}_{3,2}\left(r\right)& =& \varphi \mathsf{1}\left(r\right)e^{-\mathrm{i}\omega R^{*}}.\hfill \end{array}$$

(36)

For polar-led perturbations at minus infinity, we have

$$\begin{array}{ccc}\hfill \mathit{h}{\mathsf{0}}_{3,2}\left(r\right)& =& (r^2+{\mathit{r}}_{\mathsf{0}}^2)\mathit{h}\mathsf{0}\left(r\right)e^{-\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill \mathit{h}{\mathsf{1}}_{3,2}\left(r\right)& =& (r^2+{\mathit{r}}_{\mathsf{0}}^2)\mathit{h}\mathsf{1}\left(r\right)e^{-\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(38)

$$\begin{array}{ccc}\hfill \mathit{H}{\mathsf{1}}_{2,2}\left(r\right)& =& {(r^2+{\mathit{r}}_{\mathsf{0}}^2)}^{3/2}\mathit{H}\mathsf{1}\left(r\right)e^{-\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(39)

$$\begin{array}{ccc}\hfill T_{2,2}\left(r\right)& =& (r^2+{\mathit{r}}_{\mathsf{0}}^2)T\left(r\right)e^{-\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(40)

$$\begin{array}{ccc}\hfill {\mathit{L}}_{2,2}\left(r\right)& =& \mathit{L}\left(r\right)e^{-\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill {\mathit{N}}_{2,2}\left(r\right)& =& \mathit{N}\left(r\right)e^{-\mathrm{i}\omega R^{*}},\hfill \end{array}$$

(42)

$$\begin{array}{ccc}\hfill \varphi {\mathsf{1}}_{2,2}\left(r\right)& =& \sqrt{r^2+{\mathit{r}}_{\mathsf{0}}^2}\varphi \mathsf{1}\left(r\right)e^{-\mathrm{i}\omega R^{*}}.\hfill \end{array}$$

(43)

The tortoise coordinate $R^{*}$ is given by

$$\begin{array}{c}\hfill {\displaystyle \frac{d{R}^{*}}{dr}}=e^{-f\left(r\right)}-{\displaystyle \frac{e^{-2f\left(r\right)}}{r}}\left(r{e}^{f\left(r\right)}{h}_0\left(r\right)-b_0\left(r\right)\right),\end{array}$$

(44)

where $f\left(r\right),h_0\left(r\right),b_0\left(r\right)$ are the background functions.

At plus infinity, an outgoing solution implies the following behavior for axial-led perturbations,

$$\begin{array}{ccc}\hfill \mathit{h}\mathsf{0}\left(r\right)& \approx & -C_{\mathit{h}{\mathsf{1}}_0}-{\displaystyle \frac{2\mathrm{i}}{r\omega }}{C}_{\mathit{h}{\mathsf{1}}_0},\hfill \end{array}$$

(45)

$$\begin{array}{ccc}\hfill \mathit{h}\mathsf{1}\left(r\right)& \approx & C_{\mathit{h}{\mathsf{1}}_0}+{\displaystyle \frac{3}{r\pi \omega {\mathit{r}}_{\mathsf{0}}^3}}(({\pi }^2-8)J^2\omega +\mathrm{i}\pi {\mathit{r}}_{\mathsf{0}}^3)C_{\mathit{h}{\mathsf{1}}_0},\hfill \end{array}$$

(46)

$$\begin{array}{ccc}\hfill \mathit{H}\mathsf{1}\left(r\right)& \approx & \mathrm{i}\omega C_{T_0}-{\displaystyle \frac{5C_{T_0}}{r}},\hfill \end{array}$$

(47)

$$\begin{array}{ccc}\hfill T\left(r\right)& \approx & C_{T_0}+{\displaystyle \frac{C_{T_0}}{2r^2{\omega }^2}}(2\mathrm{i}{\omega }^{2}J-{\left({\mathit{r}}_{\mathsf{0}}\omega \right)}^2+30),\hfill \end{array}$$

(48)

$$\begin{array}{ccc}\hfill L\left(r\right)& \approx & -\mathrm{i}\omega C_{T_0}+{\displaystyle \frac{5C_{T_0}}{r}},\hfill \end{array}$$

(49)

$$\begin{array}{ccc}\hfill N\left(r\right)& \approx & -\mathrm{i}\omega C_{T_0}+{\displaystyle \frac{5C_{T_0}}{r}},\hfill \end{array}$$

(50)

$$\begin{array}{ccc}\hfill \varphi \mathsf{1}\left(r\right)& \approx & C_{\varphi {\mathsf{1}}_0}+{\displaystyle \frac{6\mathrm{i}}{r\omega }}{C}_{\varphi {\mathsf{1}}_0},\hfill \end{array}$$

(51)

and for polar-led perturbations,

$$\begin{array}{ccc}\hfill \mathit{h}\mathsf{0}\left(r\right)& \approx & -C_{\mathit{h}{\mathsf{1}}_0}-{\displaystyle \frac{5\mathrm{i}}{r\omega }}{C}_{\mathit{h}{\mathsf{1}}_0},\hfill \end{array}$$

(52)

$$\begin{array}{ccc}\hfill \mathit{h}\mathsf{1}\left(r\right)& \approx & C_{\mathit{h}{\mathsf{1}}_0}+{\displaystyle \frac{6\mathrm{i}}{r\omega }}{C}_{\mathit{h}{\mathsf{1}}_0},\hfill \end{array}$$

(53)

$$\begin{array}{ccc}\hfill \mathit{H}\mathsf{1}\left(r\right)& \approx & \mathrm{i}\omega C_{T_0}+{\displaystyle \frac{3C_{T_0}}{r\pi {\mathit{r}}_{\mathsf{0}}^3}}(\mathrm{i}{J}^2{\pi }^2\omega -8\mathrm{i}{J}^2\omega -2\pi {\mathit{r}}_{\mathsf{0}}^3/3),\hfill \end{array}$$

(54)

$$\begin{array}{ccc}\hfill T\left(r\right)& \approx & C_{T_0}+{\displaystyle \frac{1}{140\pi {\omega }^2{\mathit{r}}_{\mathsf{0}}^3{r}^2}}(-80\sqrt{7}{J}^2{\omega }^3{\pi }^2{C}_{\mathit{h}{\mathsf{1}}_0}{\mathit{r}}_{\mathsf{0}}^2\hfill \\ & +& 32{\mathit{C}}_{{\mathit{T}}_{\mathsf{0}}}\left(\mathrm{i}\left({\mathit{r}}_{\mathsf{0}}^2{\omega }^2-{\displaystyle \frac{315}{32}}\right)\omega J^2{\pi }^2-{\displaystyle \frac{35\pi {\mathit{r}}_{\mathsf{0}}^3({\mathit{r}}_{\mathsf{0}}^2{\omega }^2-6)}{16}}+{\displaystyle \frac{315\mathrm{i}{J}^2\omega }{4}}\right)),\hfill \end{array}$$

(55)

$$\begin{array}{ccc}\hfill L\left(r\right)& \approx & -\mathrm{i}\omega C_{T_0}-{\displaystyle \frac{6C_{T_0}}{r{\pi }^2{\mathit{r}}_{\mathsf{0}}^6}}(\mathrm{i}\omega J^2{\pi }^3{\mathit{r}}_{\mathsf{0}}^3-{\mathit{r}}_{\mathsf{0}}^6{\pi }^2/3-8\mathrm{i}\omega {\mathit{r}}_{\mathsf{0}}^3{J}^2\pi ),\hfill \end{array}$$

(56)

$$\begin{array}{ccc}\hfill N\left(r\right)& \approx & -\mathrm{i}\omega C_{T_0}+{\displaystyle \frac{2C_{T_0}}{r}},\hfill \end{array}$$

(57)

$$\begin{array}{ccc}\hfill \varphi \mathsf{1}\left(r\right)& \approx & C_{\varphi {\mathsf{1}}_0}+{\displaystyle \frac{1}{2r\omega \pi {\mathit{r}}_{\mathsf{0}}^2}}(3({\pi }^2-8)\omega J^2{C}_{T_0}+6\mathrm{i}\pi {\mathit{r}}_{\mathsf{0}}^2{C}_{\varphi {\mathsf{1}}_0}).\hfill \end{array}$$

(58)

At minus infinity, the axial-led perturbation functions behave as

$$\begin{array}{ccc}\hfill \mathit{h}\mathsf{0}\left(r\right)& \approx & -{\displaystyle \frac{6\mathrm{i}\sqrt{7}{J}^2{\pi }^2}{7{\mathit{r}}_{\mathsf{0}}^6}}{C}_{T_1}-{\displaystyle \frac{J\pi }{7r\omega {\mathit{r}}_{\mathsf{0}}^6}}\left(\sqrt{7}\omega {\mathit{r}}_{\mathsf{0}}^3{C}_{T_1}\right),\hfill \end{array}$$

(59)

$$\begin{array}{ccc}\hfill \mathit{h}\mathsf{1}\left(r\right)& \approx & -{\displaystyle \frac{18}{{\mathit{r}}_{\mathsf{0}}^6}}{J}^2{\pi }^2{C}_{\mathit{h}{\mathsf{1}}_2}-{\displaystyle \frac{54\mathrm{i}}{r\omega {\mathit{r}}_{\mathsf{0}}^6}}J\pi C_{\mathit{h}{\mathsf{1}}_2}(J\pi -\omega {\mathit{r}}_{\mathsf{0}}^3/9),\hfill \end{array}$$

(60)

$$\begin{array}{ccc}\hfill \mathit{H}\mathsf{1}\left(r\right)& \approx & {\displaystyle \frac{11J\pi \omega C_{T_1}}{2{\mathit{r}}_{\mathsf{0}}^3}}+{\displaystyle \frac{C_{T_1}}{7r{\mathit{r}}_{\mathsf{0}}^3}}(-7\mathrm{i}\omega {\mathit{r}}_{\mathsf{0}}^3-175\mathrm{i}J\pi ),\hfill \end{array}$$

(61)

$$\begin{array}{ccc}\hfill T\left(r\right)& \approx & {\displaystyle \frac{6\mathrm{i}J\pi C_{T_1}}{{\mathit{r}}_{\mathsf{0}}^3}}+{\displaystyle \frac{C_{T_1}}{r}},\hfill \end{array}$$

(62)

$$\begin{array}{ccc}\hfill L\left(r\right)& \approx & {\displaystyle \frac{6C_{T_1}}{{\mathit{r}}_{\mathsf{0}}^3}}J\omega \pi r^2-{\displaystyle \frac{\mathrm{i}{C}_{T_1}}{{\mathit{r}}_{\mathsf{0}}^3}}(\omega {\mathit{r}}_{\mathsf{0}}^3+24J\pi )r-5C_{T_1}+{\displaystyle \frac{3C_{T_1}J\pi }{\omega {\mathit{r}}_{\mathsf{0}}^3}}({\omega }^2{\mathit{r}}_{\mathsf{0}}^2-30),\hfill \end{array}$$

(63)

$$\begin{array}{ccc}\hfill N\left(r\right)& \approx & {\displaystyle \frac{5C_{T_1}}{{\mathit{r}}_{\mathsf{0}}^3}}J\omega \pi r^2-{\displaystyle \frac{\mathrm{i}{C}_{T_1}}{7{\mathit{r}}_{\mathsf{0}}^3}}(7\omega {\mathit{r}}_{\mathsf{0}}^3+168J\pi )r-5C_{T_1},\hfill \end{array}$$

(64)

$$\begin{array}{ccc}\hfill \varphi \mathsf{1}\left(r\right)& \approx & {\displaystyle \frac{6\mathrm{i}J\pi C_{\varphi {\mathsf{1}}_1}}{{\mathit{r}}_{\mathsf{0}}^3}}+{\displaystyle \frac{C_{\varphi {\mathsf{1}}_1}}{r}},\hfill \end{array}$$

(65)

and for polar-led perturbations, they are

$$\begin{array}{ccc}\hfill \mathit{h}\mathsf{0}\left(r\right)& \approx & {\displaystyle \frac{\sqrt{7}J\pi C_{T_2}}{7{\mathit{r}}_{\mathsf{0}}^3}}+{\displaystyle \frac{6\mathrm{i}J\pi C_{\mathit{h}{\mathsf{1}}_1}}{{\mathit{r}}_{\mathsf{0}}^3}},\hfill \end{array}$$

(66)

$$\begin{array}{ccc}\hfill \mathit{h}\mathsf{1}\left(r\right)& \approx & {\displaystyle \frac{6\mathrm{i}J\pi C_{\mathit{h}{\mathsf{1}}_1}}{{\mathit{r}}_{\mathsf{0}}^3}}+{\displaystyle \frac{C_{\mathit{h}{\mathsf{1}}_1}}{r}},\hfill \end{array}$$

(67)

$$\begin{array}{ccc}\hfill \mathit{H}\mathsf{1}\left(r\right)& \approx & {\displaystyle \frac{12\mathrm{i}{J}^2{\pi }^2\omega C_{T_2}}{{\mathit{r}}_{\mathsf{0}}^6}}+{\displaystyle \frac{J\pi C_{T_2}}{7r{\mathit{r}}_{\mathsf{0}}^6}}(35\omega {\mathit{r}}_{\mathsf{0}}^3+126J\pi ),\hfill \end{array}$$

(68)

$$\begin{array}{ccc}\hfill T\left(r\right)& \approx & -{\displaystyle \frac{18J^2{\pi }^2{C}_{T_2}}{{\mathit{r}}_{\mathsf{0}}^6}}+{\displaystyle \frac{6\mathrm{i}J\pi C_{T_2}}{r{\mathit{r}}_{\mathsf{0}}^3}},\hfill \end{array}$$

(69)

$$\begin{array}{ccc}\hfill L\left(r\right)& \approx & -{\displaystyle \frac{18\mathrm{i}\omega J^2{\pi }^2{C}_{T_2}{r}^3}{{\mathit{r}}_{\mathsf{0}}^6}}-{\displaystyle \frac{6\pi J{C}_{T_2}\omega r^2}{{\mathit{r}}_{\mathsf{0}}^3}},\hfill \end{array}$$

(70)

$$\begin{array}{ccc}\hfill N\left(r\right)& \approx & -{\displaystyle \frac{6\mathrm{i}\omega J^2{\pi }^2{C}_{T_2}{r}^3}{{\mathit{r}}_{\mathsf{0}}^6}}-{\displaystyle \frac{2\pi J{r}^2{C}_{T_2}}{7{\mathit{r}}_{\mathsf{0}}^6}}(14\omega {\mathit{r}}_{\mathsf{0}}^3+27J\pi ),\hfill \end{array}$$

(71)

$$\begin{array}{ccc}\hfill \varphi \mathsf{1}\left(r\right)& \approx & -{\displaystyle \frac{18J^2{\pi }^2{C}_{\varphi {\mathsf{1}}_2}}{{\mathit{r}}_{\mathsf{0}}^6}}-{\displaystyle \frac{54\mathrm{i}{C}_{\varphi {\mathsf{1}}_2}\left(-{\mathit{r}}_{\mathsf{0}}^3\omega /9+\pi J\right)J\pi }{{\mathit{r}}_{\mathsf{0}}^6\omega r}}.\hfill \end{array}$$

(72)

In general, $C_{X_k}$ is a constant obtained from a specific order k of an expansion of the respective perturbation function $X=T,\mathit{H}\mathsf{1},L,N,\mathit{h}\mathsf{0},\mathit{h}\mathsf{1},\varphi \mathsf{1}$, in $\mathcal{O}\left({\displaystyle \frac{1}{r^k}}\right)$ for $k=0,1,2,..$. Constants for $k>0$ can all be expressed in terms of the three free constants at the zeroth order, i.e., $C_{T_0},C_{\mathit{h}{\mathsf{1}}_0},C_{\varphi {\mathsf{1}}_0}$. All these behaviors of the functions ensure an outgoing wave at the infinities.

Since the modes are determined in perturbation theory, one can also consider the eigenvalue $\omega$ as consisting of the static eigenvalue ${\omega }_0$ and the higher-order contributions. Thus, in the second order in rotation, the eigenvalue will read

$${\omega }_{2,2}={\omega }_{2,2}^{\left(0\right)}+{ϵ}_r\delta {\omega }_{2,2}^{\left(1\right)}+{ϵ}_r^2\delta {\omega }_{2,2}^{\left(2\right)},$$

(73)

where, for clarity, we have omitted further classifications like the harmonic number n or indices for axial/polar. The modes can be obtained by integrating the perturbation equations, and they should connect smoothly to the static limit. In general, the scaled modes display quadratic behavior with respect to the scaled angular momentum, such that

$$M\omega =M{\omega }^{\left(0\right)}+M\delta {\omega }^{\left(1\right)}\left({\displaystyle \frac{J}{M^2}}\right)+M\delta {\omega }^{\left(2\right)}{\left({\displaystyle \frac{J}{M^2}}\right)}^2,$$

(74)

as reported in our previous studies in slow rotation up to the second order (see e.g., [77]).

5. Further Remarks

By performing a double expansion, we have derived the coupled sets of ordinary differential equations that are needed to obtain the quasinormal modes of slowly rotating Ellis–Bronnikov wormholes. In particular, we have focused on $l=2,M_z=2$-led perturbations, which are expected to be accounted for (or at least among) the most prominent modes during a ringdown. The detectability of these modes will depend on a high signal-to-noise ratio from observations. The numerical implementation of the developed scheme, however, still presents challenges in the case of the quadrupole modes of Ellis–Bronnikov wormholes.

The quasinormal modes of the static Ellis–Bronnikov wormholes exhibit a threefold isospectrality in the symmetric case ($C=0)$ [64]. However, this isospectrality is broken for finite values of the asymmetry parameter C. Likewise, we expect rotation to break isospectrality for the slowly rotating wormholes addressed here. Note that for a set of rapidly rotating wormholes, isospectrality is broken, as well [66].

The developed scheme has already been successfully applied in the case of radial modes of the Ellis–Bronnikov wormholes [70,71]. As shown in the left plot of Figure 2, the well-known purely imaginary unstable mode of the wormholes [58,81,82,83,84] was seen to become more stable with increasing rotation, as the absolute value of the scaled imaginary eigenvalue is decreasing towards zero. As the asymmetry parameter C increases up to about $C=0.5$, the critical value of the scaled angular momentum where the imaginary eigenvalue vanishes decreases. However, for $C>0.5$, the critical angular momentum value grows. While in the right plot of Figure 2, a second branch of an unstable mode was seen to emerge from a zero mode in the static limit, this becomes unstable with increasing rotation. Nonetheless, this branch of the mode merges with the previously discussed unstable mode, and, moreover, at an even smaller value of the scaled angular momentum. Based on this evidence, therefore, the radial instability is conjectured to disappear.

Unfortunately, perturbation theory cannot resolve the question of radial stability completely, and (in rotation) non-perturbative calculations along the lines of [66] are needed. A possible scenario for the further evolution of the modes would be one where both purely imaginary modes merge, develop a real part, and slowly become more stable, until the instability disappears when approaching the final extremal configuration.

Wormholes may also suffer from non-radial instabilities, of course. Since the study of the quasinormal mode spectrum will reveal instabilities, determination of the spectrum represents an essential aspect concerning the viability of wormhole solutions as potentially observable compact objects. In recent years, studies in numerical relativity (see e.g., [85]) are also starting to address potentially observational aspects of wormholes. Finally, perturbations of wormholes in alternative theories of gravity, and of other multipolar nature, are left for future studies.