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# Characterizing a class of accelerating wormholes with periodic potential

Soham Chatterjee<EMAIL_ADDRESS>Sagnik Roy<EMAIL_ADDRESS>Ratna
Koley<EMAIL_ADDRESS>Department of Physics, Presidency
University, 86/1 College Street, Kolkata 700073, India

###### Abstract

The newly discovered Wormhole C–metric is a solution of Einstein’s field
equation coupled with a phantom scalar field which describes the accelerated
wormholes. In the zero acceleration limit the solution reduces to an
asymptotically flat wormhole. For certain range of parameter space this
solution doesn’t possess any horizon, thus making it a viable candidate of
wormhole. To completely unveil this property we have studied the topological
properties of this spacetime and shown that the throat is marginally
connected. In the aforementioned range of parameters, the spacetime doesn’t
posses any photon orbit confirming the absence of shadow. We further analysed
the stability of this spacetime under scalar perturbation. Under the usual
boundary conditions (outgoing waves at both spatial infinities) there exists a
continuous spectra. On the contrary one may achieve the quantization of the
modes by exploiting a different but physically intuitive boundary condition.
The lowest lying mode behaves as normal mode, and the imaginary part comes
into play for the modes corresponding to first overtone number $(n=1)$ marking
the onset of quasi-nomral modes for all azimuthal quantum number, $L$. We have
also argued that the spacetime has a tendency to hold the excitation in it due
to the external perturbation, rather than a fast de-excitation.

##  I Introduction

The C–metric, a family of vacuum solutions of the Einstein equation,
characterizes a pair of causally disconnected black holes with a uniform
acceleration [1, 2, 3, 4, 5]. The boost-rotation symmetry of the solution
brings it under the general class of Plebański–Demiański spacetimes (and hence
Petrov D) and the Weyl spacetimes. The acceleration of each black hole arises
either from a conical deficit angle along the symmetry axis, akin to a semi-
infinite cosmic string, or from a strut with negative tension developing
between two black holes [6]. In the linear approximation of the acceleration,
the C–metric emerges as a perturbation of the Schwarzschild black hole with a
distributive stringy source, or as a nonlinear superposition of the
Schwarzschild and Rindler spacetimes. Consequently, the geodesics in a
C–metric represent the trajectories of test particles under the gravitational
field of a black hole subject to a uniform constant force – commonly known as
the gravitational Stark effect [7]. Extensive studies have delved into the
causal structures, as well as the physical and geodesic properties [8, 9, 10,
11, 12], of the black hole C–metric. These investigations remain pivotal
within the Physics community, as not only it is useful to model accelerated
black holes, but also it can shed light on the binary black hole mergers,
causing gravitational radiation [13].

Another interesting class of solution of the Einstein field equation is the
wormhole spacetime. The solution describes a topologically non-trivial and
non-singular geometry which can act as a tunnel between different universes or
different parts of the same universe. Traversable wormhole was conceived by
Morris, Thorne and Yurtsever [14, 15] where the geometry is supported by the
energy condition violating exotic matter. Many such traversable wormhole
solutions have been extensively studied over the last few decades [16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26]. Recently, Nozawa and Torii have proposed [27]
the accelerated version of wormhole by the suitable analytic continuation of
one parameter family of the $\mathcal{N}=2$ gauged supergravity – termed as
the wormhole C–metric. This is a solution of Einstein’s gravity coupled with a
phantom scalar field, expressed in terms of a superpotential. The potential
exhibits multiple $AdS$ extrema, with local minima corresponding to critical
points of the superpotential. Through a flipping transformation, the metric
undergoes a transition to a different family of C–metrics and upon approaching
the zero acceleration limit, each solution converges towards two wormhole
solutions. Within a natural coordinate framework, the C–metric solution
approaches $AdS$ at the origin of the potential. Through maximal extension,
the solution seamlessly connects to the other side of the universe, where the
scalar field evolves towards a distinct critical point of the potential. An
interesting characteristic of these C–metrics is their ability to avoid the
conical singularities typically required for inducing acceleration under
appropriate parameter choices. Instead, the acceleration in these scenarios is
solely facilitated by the phantom character associated to the scalar field in
contrast to the black hole C–metric.

In this article, the geodesics and quasinormal behaviour of the wormhole
C–metric have been explored extensively which will be useful to find the
possible observational imprints. At first we study the geodesics and shadow
profile in this geometry for a better understanding of the spacetime. The
shadow of an object is quite intriguing as it is closely related to the
behaviour of geodesic motion of photon in the vicinity of the object [28, 29,
30]. This direction of research has been resurrected with the observation of
black hole M87 shadow by EHT in 2019, followed by the detection of Sagittarius
A* in 2022 [31, 32, 33, 34, 35]. Various wormhole shadows have been studied
over the past decade with the aim of distinguishing them from the shadows of
ordinary black holes, which could potentially aid in probing these exotic
objects. For instance, in the case of a traversable wormhole, the central
region of the shadow is directly illuminated by low angular momentum photons
coming from the other side of the wormhole. Additionally, images of
reflection-asymmetric wormholes [36, 37] may contain a region where photons
emitted on one side of the wormhole travel to the other side and are reflected
back to their side of origin. Conversely, if the wormhole is rotating (see,
e.g., [21]), then for small rotation and small throat size, it can closely
mimic a Kerr black hole with the same rotation parameter and mass. However, as
the spin or throat size increases, the shadow of the Kerr black hole starts to
deform more than that of the rotating wormhole shadow. Using this distinctive
feature, Rahaman et al. [38] constrained the wormhole parameters from the M87*
shadow. A similar feature was also observed in [39]. There is extensive
literature on these topics, and readers may refer to [40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 28] for further details. In the spacetime
under consideration the null energy condition has been shown to be violated
not only near the throat but across the entire spacetime for the associated
parameter space. Surprisingly the spacetime does not possess any photon orbit
resulting in absence of a shadow. This is a very interesting feature in the
sense that previously no spacetime solution was found - to the best of our
knowledge - for which electromagnetic shadow doesn’t form. The detection of
this kind of spacetime may reveal interesting observational aspects as well as
a probe for potential verification of the underlying quantum gravity theory.
Astronomical objects are primarily detected through two main methods:
electromagnetic observations which involve studying phenomena such as
accretion or observation of the shadow profile cast by the spacetime and the
detection of gravitational waves.

It is well established that the merger of binary black holes or compact
objects results in emission of gravitational waves. There are mainly three
phases in this process: inspiral, merger and ringdown. During these phases
enormous amount of energy is released in the form of gravitational waves. In
2016, first direct detection of gravitational wave was made by LIGO and Virgo
[55]. Generally the astrophysical objects are not isolated and they are always
subject to some kind of perturbations. The stability of a given spacetime
against these perturbations is an important question. The black holes exhibit
damped oscillations under perturbation due to the radiation of gravitational
waves [56, 57] and these are described by the quasi-normal modes (QNMs). The
QNM waveform produces a ringdown signal until the two objects are settled to a
stable final object and these late-time tails carry important information
about the final object. The QNM of accelerating black hole was calculated by
Destounis et. al. [58]. In this work we study the QNMs in the given spacetime
and the effect of boundary conditions on the QNMs in greater detail.
Interestingly we have found that although the spacetime is stable it holds the
excitation due to the perturbation rather than getting de-excited and coming
to the ground state. These important features demand attention both from
theoretical and observational perspectives.

The work is organised in the following way. In Section II, the physical
properties and the causal structures of the wormhole C–metric has been
analysed. The null geodesic has been studied for a special type of mass
parameter. Section III deals with the geodesic properties and the absence of
shadow profile in the spacetime. In Section IV, we first discussed the
conformal structure of the spacetime. Then for the standard choice of boundary
conditions (i.e. the waves are outgoing at the special infinities), the scalar
modes of the spacetime have been calculated and it is found to be continuous.
Then for a new choice of the boundary conditions the modes have been shown to
be quantized. Physical interpretation of the results have been given. Finally
we concluded in Section V. In Appendix A the notations, conventions and the
symbols used throughout the article have been mentioned.

##  II Wormhole C–metric spacetime

The $4$-D Einstein gravity minimally coupled with a real scalar field can be
described by the action

$\mathcal{S}=\dfrac{1}{2}\displaystyle\int\boldsymbol{\sqrt{-\tilde{g}}d^{4}x}\left[R-\epsilon_{1}(\partial\Phi)^{2}-2\mathcal{V}(\Phi)\right]$
(1)

where $\epsilon_{1}=+1$ corresponds to the ordinary scalar field, whereas
$\epsilon_{1}=-1$ represents the phantom scalar field mediating the repulsive
force. Our aim is to explore the consequences of the phantom property of the
scalar field throughout the paper. The potential of the phantom scalar field
is denoted by $\mathcal{V}(\Phi)$ that can be written as

$\mathcal{V}(\Phi)=-g^{2}\left(2+\cos{\sqrt{2}\Phi}\right)$ (2)

where $g$ acts as an overall scale factor of the potential that can be chosen
non-negative without any loss of generality, which we will continue to follow
for the rest of the paper. As a special case, the potential can be set to be
zero by letting $g=0$. Here only the periodic and phantom nature of the
potential has been considered for the sake of convenience, but for more
general case, one can refer [27].

Now for the system (1) and (2), the C-metric solution takes the form [27]

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\dfrac{1}{(1+Arx)^{2}}\left[V(x)\left\\{-\Delta_{r}(r)d\tau^{2}+\dfrac{dr^{2}}{\Delta_{r}(r)}\right\\}+r^{2}V_{r}(r)\left\\{\dfrac{dx^{2}}{\Delta_{x}(x)}+\Delta_{x}(x)d\varphi^{2}\right\\}\right]$
(3) $\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\sqrt{2}\left[\tan{{}^{-1}\left(\dfrac{r}{m}\right)}-\tan{{}^{-1}\left(Amx\right)}-\dfrac{\pi}{2}\right]$
(4)

Clearly for $m=0$, the scalar field, $\Phi$ becomes trivial and thus let us
refrain from considering this. In Eq. (3), $A$ denotes the acceleration of the
spacetime and a dimensionless acceleration parameter can be defined out of
this: $\mathcal{A}=Am$. Further defining a dimensionless radial coordinate
$\mathcalligra{r}=r/m$ and a dimensionless scale factor $g$ by replacing $gm$
and considering the global topology of $x-\varphi$ to be $\mathbb{S}^{2}$, the
solution reduces as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\dfrac{1}{(1+\mathcal{A}\mathcalligra{r}x)^{2}}\left[V(x)\left\\{-\Delta{r}(\mathcalligra{r})d\tau^{2}+\dfrac{d\mathcalligra{r}^{2}}{\Delta{r}(\mathcalligra{r})}\right\\}+\mathcalligra{r}^{2}V{r}(\mathcalligra{r})\left\\{\dfrac{dx^{2}}{\Delta_{x}(x)}+\Delta_{x}(x)d\varphi^{2}\right\\}\right]$
(5) $\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\sqrt{2}\tan{{}^{-1}\dfrac{1+\mathcal{A}\mathcalligra{r}x}{\mathcal{A}x-\mathcalligra{r}}}$
(6)

where $V(x)$, $\Delta{r}(\mathcalligra{r})$, $V{r}(\mathcalligra{r})$,
$V{r}(\mathcalligra{r})$ must be positive in order to maintain the Lorentzian
signature of the metric and thus can be defined as

$\displaystyle V(x)$ $\displaystyle=$ $\displaystyle
1+\mathcal{A}^{2}x^{2},~{}~{}\Delta{r}(\mathcalligra{r})=1+g^{2}+\left(g^{2}-\mathcal{A}^{2}\right)\mathcalligra{r}^{2},~{}~{}V{r}(\mathcalligra{r})=1+\dfrac{1}{\mathcalligra{r}^{2}},~{}~{}\Delta_{x}(x)=1-x^{2}$
(7)

As mentioned in [27], $x$ represents the direction cosine on $\mathbb{S}^{2}$
and hence can be taken to be proportional to $\cos{\theta}$. But unlike the
traditional C–metrics, the two dimensional surface spanned by $x-\varphi$ of
this spacetime, being free from conical singularity, $x$ can be considered as
$\cos{\theta}$ only. It is also worth mentioning that one do not need
distributional sources to maintain the acceleration of wormholes, which stands
in stark contrast to the vacuum case.

### Infinity

This spacetime possesses two killing vectors: $\partial_{\tau}$ and
$\partial_{x}$, which correspond to the two conserved quantities, energy $(E)$
and angular momentum $(L)$ respectively. The affine parameter along a null
curve with $(\tau,x,\phi)=$ constant can therefore be written as [59]

$\displaystyle\lambda\sim\left|\dfrac{\mathcal{A}\mathcalligra{r}\left(1+\mathcal{A}^{2}x^{2}\right)}{E(1+\mathcal{A}\mathcalligra{r}x)}\right|^{1/2}$
(8)

which clearly indicates that it takes an infinite amount of “time” to reach
any point on the surface $\mathcalligra{r}=-\frac{1}{\mathcal{A}x}$, whereas
$\mathcalligra{r}\to\infty$ surface can be reached with a finite affine
parameter. Hence unlike the standard Morris–Thorne wormhole [14] (or
Lorentzian wormholes [60]), $\mathcalligra{r}\to-\frac{1}{\mathcal{A}x}$ will
be treated as infinity for this spacetime. Obviously at the equatorial plane,
the infinity lies at $\mathcalligra{r}\to\infty$, but not in general.

### Curvature properties and energy conditions

The existence of curvature singularities in a Lorentzian manifold is usually
determined by the divergence of the Kretschmann scalar
$(K=R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta})$ as this remains
invariant under coordinate transformation. Therefore, if $K$ diverges in one
frame of reference, it will also be divergent in all other coordinate frames,
removing the ambiguity associated with the co–ordinate singularity. For the
spacetime metric in Eq. (5), the analytical expression of this quadratic
scalar invariant is too cumbersome. That is why instead of writing the
analytical form explicitly, the variation of $K$ over $x-\varphi$ surface has
been depicted pictorially in Fig. (1).

(a)

(b)

(c)

(d)

Figure 1: Variation of Kretschmann scalar for $\mathcal{A}=5\times 10^{-3}$,
$\mathcal{A}=5\times 10^{-2}$, $\mathcal{A}=0.5$, $\mathcal{A}=1.0$ (from
upper left to lower right) for a fixed value of scaling factor, $g=1.5$.

From the figure, it is evident that the isocurvature surfaces of the spacetime
exhibit a degree of symmetry for very low acceleration, which is expected
since the spacetime itself becomes spherically symmetric under such
conditions. As the acceleration parameter increases the assymetrical
deformation of these surfaces starts on one side of the wormhole. Additionally
the curvature of the spacetime does not monotonically increase or decrease
with radial distance; rather there exists a plateau region (indicated by the
black region in Fig. 1). The Kretschmann scalar (and hence the spacetime
curvature as a whole) does not diverge anywhere in the domain, as indicated by
the analytic expression of $K$. Therefore the spacetime does not exhibit any
curvature singularity. Furthermore, it is noteworthy that the spacetime lacks
asymptotic flatness near infinity, as observed in Fig. 1, a feature
corroborated by the line element (5) Considering the future oriented null
vectors $k^{\mu}=(\sqrt{-g_{00}},-\sqrt{g_{11}},0,0)$ and
$(\sqrt{-g_{00}},0,-\sqrt{g_{22}},0)$, the violation of null energy condition
requires only the positivity of $\Delta{r}(\mathcalligra{r})$ and
$\Delta_{x}(x)$ over the entire spacetime, respectively, which is always
satisfied. Therefore the energy condition is always being violated in this
case, resulting in the spacetime being classified as a wormhole spacetime.
Although the violation of null energy condition alone can’t justify the
spacetime being a wormhole, it requires rigorous topological analysis to
conclude this, which has been discussed briefly in Appendix B.

### Causal structure

With the previous rescaling of coordinates, $r\to m\mathcalligra{r}$,
$A\to\mathcal{A}/m$ and $g\to g/m$, the $3$ parameters of the metric have
reduced effectively to $2$ parameters. Now considering the fact that the line
element without conformal factor admits the mirror symmetry,
$\mathcal{A}\to-\mathcal{A}$ and considering $\mathcal{A}$ to be non–negative
without losing any generality, an analytic classification of the ranges of
values of $\mathcal{A}$ is possible: $0\leq\mathcal{A}<g$, $\mathcal{A}=g$ and
$g<\mathcal{A}<\infty$. Among these cases, $\mathcal{A}=g=0$ corresponds to
the well-known Ellis wormhole with drain-hole parameter $=1$ and for $g=0$, it
takes the form of a generalized Ellis–Bronnikov wormhole. However, we will
consider the case where $\mathcal{A}>g$. Under these constraints, there will
be two horizons located on both sides of the wormhole at $\mathcalligra{r}=\pm
1/\mathcal{A}$. For detailed discussions on the characterization of these
horizons and the corresponding Penrose diagrams, one may look into the
original paper of Wormhole C–metric [27]. Now in case of the presence of these
horizons, as evident from the line element (5), the outer region of the
horizon is spacelike, whereas the inner part is timelike. So if we admit that
the observer is in a timelike region of the spacetime, then the observer must
live inside the horizon, which is bizarre. Also if it is considered that the
observer is inside the horizons, then it will not get access to the spatial
infinity (on the equatorial plane, spatial infinity can _marginally_ be
accessed sitting inside the horizons though), which is non-intuitive as well.
Therefore it is safe to exclude the parameter space of $\mathcal{A}$ and $g$,
for which the spacetime admits horizons. For the case of $\mathcal{A}=g$, the
spacetime reduces to the well known Ellis–Bronnikov wormhole spacetime by the
slight redefinitions of $\tau$ and $\mathcalligra{r}$ coordinates. The only
non–trivial case left is $0\leq\mathcal{A}<g$, which we will assume to be
valid in the subsequent analysis.

For $\mathcal{A}<g$, it is possible to extend Visser’s proposal of defining
the wormhole throat [60, Chapter 11] in a well defined way. The proper radial
distance $l$ in this spacetime is given by

$l(\mathcalligra{r})=\displaystyle\int_{0}{r}\dfrac{d\mathcalligra{r}^{\prime}}{\Delta{r}^{\prime}(\mathcalligra{r})^{\prime
1/2}}=\dfrac{1}{\sqrt{g^{2}-\mathcal{A}^{2}}}\tanh{{}^{-1}\left[\dfrac{\mathcalligra{r}\sqrt{g^{2}-\mathcal{A}^{2}}}{\Delta{r}(\mathcalligra{r})}\right]}$
(9)

Therefore after some manipulations, the coefficient of $d\Omega^{2}$ in the
metric (5) can be written as

$1+r(l)^{2}=1+\dfrac{1+g^{2}}{g^{2}-\mathcal{A}^{2}}\sinh{{}^{2}\left(l\sqrt{g^{2}-\mathcal{A}^{2}}\right)}$
(10)

for which, the minima of the function $\sqrt{1+r(l)^{2}}$, and thus the
wormhole throat is located at $l=0$ or $r=0$. Note that $r=0$ is a point in
the spacetime, not a surface and hence two regions of the wormhole is
marginally connected via a _single point_. So it will not be a traversable
wormhole even though there is no horizon for our choice of acceleration
parameter.

##  III Geodesic Motion and Shadow Profile analysis

The trajectory of a particle in this spacetime can be described by the curve
$q^{\mu}(\lambda)=(\tau(\lambda),r(\lambda),x(\lambda),\varphi(\lambda))$,
parameterized by $\lambda$ and the Lagrangian will be
$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\dot{q}^{\mu}\dot{q}^{\nu}$ (where overdot
denotes the derivative with respect to the affine parameter). For the metric
in Eq. (5), we can write

$\displaystyle
2\mathcal{L}=\dfrac{1}{(1+\mathcal{A}\mathcalligra{r}x)^{2}}\left[(1+\mathcal{A}\mathcalligra{r}^{2}x^{2})\left(-\Delta{r}(\mathcalligra{r})\dot{\tau}^{2}+\dfrac{\dot{\mathcalligra{r}}^{2}}{\Delta{r}(\mathcalligra{r})}\right)+(1+\mathcalligra{r}^{2})\left(\dfrac{\dot{x}^{2}}{\Delta_{x}(x)}+\Delta_{x}(x)\dot{\phi}^{2}\right)\right]=\epsilon$
(11)

where $\epsilon=-1$ and $\epsilon=0$ correspond to timelike and lightlike
geodesics respectively. Then the conjugate momenta
$p_{\mu}=\frac{\partial{\mathcal{L}}}{\partial\dot{q}^{\mu}}$ will be

$\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle-\dfrac{1+\mathcal{A}^{2}x^{2}}{(1+\mathcal{A}\mathcalligra{r}x)^{2}}\Delta{r}(\mathcalligra{r})\dot{\tau}\equiv-E$
(12) $\displaystyle p{r}$ $\displaystyle=$
$\displaystyle\dfrac{1+\mathcal{A}^{2}x^{2}}{(1+\mathcal{A}\mathcalligra{r}x)^{2}}\dfrac{\dot{\mathcalligra{r}}}{\Delta{r}(\mathcalligra{r})}$
(13) $\displaystyle p_{x}$ $\displaystyle=$
$\displaystyle\dfrac{1+\mathcal{A}^{2}x^{2}}{(1+\mathcal{A}\mathcalligra{r}x)^{2}}\dfrac{\dot{x}}{\Delta_{x}(x)}$
(14) $\displaystyle p_{\varphi}$ $\displaystyle=$
$\displaystyle\dfrac{1+\mathcalligra{r}^{2}}{(1+\mathcal{A}\mathcalligra{r}x)^{2}}-\Delta_{x}(x)\dot{\varphi}\equiv\ell$
(15)

Here the $\partial_{t}$ and $\partial_{\varphi}$ being the boost and
rotational Killing vectors respectively, the conjugate momenta $p_{t}$ and
$p_{\varphi}$ will be conserved and we define them as the energy $E$ and the
angular momentum $\ell$ of the particle. The Euler-Lagrange equation
$\dfrac{d}{d\lambda}\left(\dfrac{\partial\mathcal{L}}{\partial\dot{q}^{\mu}}\right)-\dfrac{\partial\mathcal{L}}{\partial
q^{\mu}}=0$ corresponding to the above Lagrangian (11) will give the equations
of motion. However, without directly deriving the geodesic equations,
analyzing the effective potential can unveil certain properties regarding the
behavior of a particle in this spacetime. Using the expressions of $\dot{t}$
and $\dot{\phi}$, we can rewrite eqn. (11) as

$\displaystyle
E^{2}-\dfrac{(1+\mathcal{A}^{2}x^{2})^{2}}{(1+\mathcal{A}\mathcalligra{r}x)^{2}}\dot{\mathcalligra{r}}^{2}-\dfrac{(1+\mathcalligra{r}^{2})(1+\mathcal{A}^{2}x^{2})\Delta{r}(\mathcalligra{r})}{(1+\mathcal{A}\mathcalligra{r}x)^{4}\Delta_{x}(x)}\dot{x}^{2}=V_{eff}$
(16)

where

$\displaystyle
V_{eff}(\mathcalligra{r})=\dfrac{\ell^{2}(1+\mathcal{A}^{2}x^{2})\Delta{r}(\mathcalligra{r})}{(1+\mathcalligra{r}^{2})\Delta_{x}(x)}-\dfrac{\epsilon(1+\mathcal{A}^{2}x^{2})\Delta{r}(\mathcalligra{r})}{(1+\mathcal{A}\mathcalligra{r}x)^{2}}$
(17)

We will analyse the near-throat behaviour of the effective potential
$(V_{eff})$ as the astrophysically interesting phenomena takes place near the
throat of the wormhole. For massive particles, the effective potential reaches
an asymptotic value
$(g^{2}-\mathcal{A}^{2})(1+\mathcal{A}^{2}x^{2})\left[\frac{\ell^{2}}{1-x^{2}}+\frac{1}{\mathcal{A}^{2}x^{2}}\right]$,
or in other words it will reach a constant value at large $r$ for timelike
geodesic except when $\theta=0$ and $\theta=\pi/2$. In Fig. 2, the plots
depict the variation of the effective potentials, a massive particle will
experience, for different acceleration parameters $\mathcal{A}$. For
$\theta=0$ and $\theta=\pi/2$, there is an unstable equilibrium point at
$r=0$, but as for this spacetime, $r=0$ is merely a point, there will be no
closed timelike geodesic at north pole and at equatorial plane. In between
these two regions, there can exist an unstable equilibrium for massive
particles that reaches close to $r=0$ at slow acceleration limit. All of these
analysis is also in agreement with the expression of the first derivative of
effective potential at $r=0$ for $\epsilon=-1$, i.e.,
$2\mathcal{A}x(1-x^{2}-g^{2}(x^{2}-1))$. Furthermore, a minimum exists in the
effective potential for all values of $\mathcal{A}$ except when $\theta\approx
0$. Therefore, for massive particles, a bound state is present everywhere
except near the north pole. For photons $\epsilon=0$,
$V_{eff}(\mathcalligra{r})$ possess only one maxima at throat,
$\mathcalligra{r}=0$, independent of $x$ as evident from Fig. 3. So, there is
no stable or unstable orbit for the photon in contrast to the timelike case.
In both of the cases, the maximum value of the effective potential increase
from equatorial plane to north pole gradually. It is noteworthy that for both
of the cases, in small acceleration limit $(\mathcal{A}\rightarrow 0)$, the
behaviour of effective potential is nearly same for all values of
$\mathcal{A}$ for a particular azimuth.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Variation of the effective potential with acceleration parameters
for massless particles in different azimuthal angles $(\theta)$ for the
following set of parameters: $g=1.5$, $\ell=7.0$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Variation of the effective potential with acceleration parameters
for massless particles in different azimuthal angles $(\theta)$ for the
following set of parameters: $g=1.0$, $\ell=5.0$.

Now, using the Hamiltonian of the particle, obtained by the Legendre
transformation of the Lagrangian $\mathcal{L}$, we can also write the
Hamilton-Jacobi equation
$\mathcal{H}\left(q,\frac{\partial\mathcal{S}}{\partial
q}\right)+\frac{\partial\mathcal{S}}{\partial\lambda}=0$ and substituting the
following ansatz of Hamilton’s principle function for a photon,
$\mathcal{S}=-E\tau+\mathcal{S}{r}(\mathcalligra{r})+\mathcal{S}_{x}(x)+\ell\varphi$,
the equation separates as

$\displaystyle\dfrac{\partial\mathcal{S}_{x}}{\partial x}$ $\displaystyle=$
$\displaystyle\pm\left[\dfrac{\kappa}{\Delta_{x}(x)(1+\mathcal{A}^{2}x^{2})}-\dfrac{\ell^{2}}{\Delta^{2}_{x}(x)}\right]^{1/2}$
(18) $\displaystyle\dfrac{\partial\mathcal{S}{r}}{\partial\mathcalligra{r}}$
$\displaystyle=$
$\displaystyle\pm\left[\dfrac{E^{2}}{\Delta^{2}{r}(\mathcalligra{r})}-\dfrac{\kappa}{\Delta{r}(\mathcalligra{r})(1+\mathcalligra{r}^{2})}\right]^{1/2}$
(19)

and this yields a Carter–like separation constant $\kappa$. Now using
$p_{\mu}=\frac{\partial\mathcal{S}}{\partial q^{\mu}}$ along with the
expressions of conjugate momenta from Lagrangian, we obtain

$\displaystyle\dot{\tau}$ $\displaystyle=$
$\displaystyle\dfrac{E(1+\mathcal{A}\mathcalligra{r}x)^{2}}{(1+\mathcal{A}^{2}x^{2})\Delta{r}(\mathcalligra{r})}$
(20) $\displaystyle\dot{\mathcalligra{r}}$ $\displaystyle=$
$\displaystyle\pm\dfrac{(1+\mathcal{A}\mathcalligra{r}x)^{2}}{(1+\mathcal{A}^{2}x^{2})}\left[E^{2}-\dfrac{\kappa\Delta{r}(\mathcalligra{r})}{1+\mathcalligra{r}^{2}}\right]^{1/2}$
(21) $\displaystyle\dot{x}$ $\displaystyle=$
$\displaystyle\pm\dfrac{(1+\mathcal{A}\mathcalligra{r}x)^{2}}{(1+\mathcalligra{r}^{2})}\left[\dfrac{\kappa\Delta_{x}(x)}{1+\mathcal{A}^{2}x^{2}}-\ell^{2}\right]^{1/2}$
(22) $\displaystyle\dot{\varphi}$ $\displaystyle=$
$\displaystyle\dfrac{\ell(1+\mathcal{A}\mathcalligra{r}x)^{2}}{(1+\mathcalligra{r}^{2})\Delta_{x}(x)}$
(23)

The existence of the geodesics require the positiveness of the terms under the
square roots in Eqs. (21) and (22) which in turn imply $\kappa\geq 0$ and
$\kappa>\ell^{2}$. Now these conditions indicate that $\dot{\mathcalligra{r}}$
and $\ddot{\mathcalligra{r}}$ cannot vanish simultaneously and same goes for
$\dot{x}$ and $\ddot{x}$ also. So there neither exists aspherical photon
surface nor any photon cone in this spacetime. Therefore _no_ shadow will be
cast by the spacetime due to the non-existence of any special photon surface.
We can also understand this as the effect of the absence of any photon orbit
in this spacetime.

From the analysis presented in this section, two key points emerge: Firstly,
for slow acceleration, the nature of the effective potential remains “almost”
indistinguishable for different values of acceleration for both massive and
massless particles near the astrophysically most enriched region of the
wormhole, i.e., near the throat. Secondly, the spacetime lacks the ability to
cast shadow, making it challenging to observationally discern this class of
spacetime by studying particle behavior. Hence we will now turn ourselves to
the study of (normal and) quasinormal modes of this spacetime, which may lead
to phenomenologically interesting possibilities.

##  IV Scalar waves

Let us consider a free massless scalar field non–minimally coupled to this
geometry, while also not interacting with the background scalar field $\Phi$.
Additionally, assume that this field does not contribute to the stress-energy
tensor of the background system. Thus, it will merely act as a perturbation to
the system, allowing us to analyze the effect of the background geometry on
this free scalar field. The conformally invariant Klein–Gordon equation for
this field is given by

$\dfrac{1}{\sqrt{-\bar{g}}}\partial_{\mu}\left(\sqrt{-\bar{g}}\bar{g}^{\mu\nu}\partial_{\nu}\bar{\phi}\right)+\alpha\bar{R}\bar{\phi}=0$
(24)

where $\alpha$ is generally viewed either as a coupling factor of
$\bar{\phi}\bar{R}$ type coupling or it takes value $-\frac{1}{6}$ when the
metric is used without its conformal factor [61]. In that case,
$\bar{g}^{\mu\nu}$ and $\bar{R}$ are the conformal metric and its Ricci scalar
respectively. Now at this point, this is worth emphasizing that the line
element (5) doesn’t admit any rank-$2$ conformal Killing–Yano tensor, which
makes it impossible to perform separation of variable of the Klein–Gordon
equation (24) leading to enormous difficulties in solving it. In this
situation, the entire line element (5) can be redefined by scaling through the
conformal factors of the $2$D line element
$\left(-\Delta{r}(\mathcalligra{r})d\tau^{2}+\Delta{r}(\mathcalligra{r})^{-1}dr^{2}\right)$
as

$d\tilde{s}^{2}=-\left(1+g^{2}+(g^{2}-\mathcal{A}^{2})\mathcalligra{r}^{2}\right)d\tau^{2}+\dfrac{d\mathcalligra{r}^{2}}{1+g^{2}+(g^{2}-\mathcal{A}^{2})\mathcalligra{r}^{2}}+\dfrac{1+\mathcalligra{r}^{2}}{1+\mathcal{A}^{2}x^{2}}\left(\dfrac{dx^{2}}{1-x^{2}}+(1-x^{2})d\varphi^{2}\right)$
(25)

Now we perform another coordinate transformation,
$t\to\frac{\zeta}{\sqrt{1+g^{2}}}$ and $\mathcalligra{r}\to\sqrt{1+g^{2}}\rho$
so that the metric takes the form

$d\tilde{s}^{2}=-\left(1+\varepsilon\rho^{2}\right)d\zeta^{2}+\dfrac{d\rho^{2}}{1+\varepsilon\rho^{2}}+\dfrac{1+a^{2}\rho^{2}}{1+\mathcal{A}^{2}x^{2}}\left(\dfrac{dx^{2}}{1-x^{2}}+(1-x^{2})d\varphi^{2}\right)$
(26)

where $\varepsilon=g^{2}-\mathcal{A}^{2}$ and $a^{2}=1+g^{2}$. This metric
will now admit a Killing–Yano $2$-form
$\frac{(1+a^{2}\rho^{2})^{3/2}}{1+\mathcal{A}^{2}x^{2}}dx\wedge d\varphi$,
along with five other rank-$2$ Killing tensors, which make it possible to
perform the following decomposition of the scalar field,

$\bar{\phi}(\zeta,\rho,x,\varphi)=\dfrac{(1+a\mathcal{A}\rho
x)^{2}}{1+\mathcal{A}^{2}x^{2}}\phi(\zeta,\rho,x,\varphi)\sim\dfrac{(1+a\mathcal{A}\rho
x)^{2}}{1+\mathcal{A}^{2}x^{2}}\displaystyle\int
d\omega~{}e^{-i\omega\zeta+iM\varphi}\dfrac{\psi_{LM}(\rho)\Theta_{LM}(x)}{\sqrt{1+\varepsilon\rho^{2}}}$
(27)

where $L$ denotes the excitation level of the scalar field and $M$ is a
quantum number representing the direction of the orientation of that field in
spacetime. From now on, the subscript $LM$ will be dropped for the sake of
brevity.

The ansatz (27) will now lead to the separation of the Eq. (24) into a radial
part and an angular part. In the limit of extremely low acceleration
$(\mathcal{A}\rightarrow 0)$, the angular component simplifies to the well-
known associated Legendre polynomial $Y_{LM}(x)$ due to the system’s spherical
symmetry. We will now analyse these two separated equations below by confining
ourselves mostly in the low acceleration limit mainly for two reasons:

  1. 1.

The experimental bound on the acceleration parameter of an accelerating
Kerr–Newman black hole, from the shadow of M87* [62], suggests that celestial
bodies either have extremely small acceleration or zero acceleration.

  2. 2.

For the original spacetime (5), as mentioned earlier, the position of the
spatial infinities inversely varies with the acceleration parameter. Thus it
is convenient and physically intuitive to take small acceleration limit, so
that the spatial infinities gets mapped to the mathematical infinities
$\pm\infty$.

### Angular part

The decomposition (27) along with Eqs. (24) and (26) will give the angular
equation as follows.

$\left(1-x^{2}\right)\dfrac{d^{2}\Theta}{dx^{2}}-2x\dfrac{d\Theta}{dx}+\left[\dfrac{\xi}{\left(1+\mathcal{A}^{2}x^{2}\right)}-\dfrac{M^{2}}{1-x^{2}}-\dfrac{\left(1+\mathcal{A}^{2}\right)\left(1-\mathcal{A}^{2}x^{2}\right)}{3\left(1+\mathcal{A}^{2}x^{2}\right)^{2}}\right]\Theta=0$
(28)

where $6\xi$ is the separation constant of angular and radial part equation.
This equation is almost like the equation of an associated Legendre polynomial
with eigenvalue $6\xi$, except for the fact that there is an extra parameter
for acceleration here. At this point, one might ask the question how to solve
this equation, as there is no standard technique to find the exact analytical
solution of this. Similar (but not exactly same) angular equation arises while
doing the QNM analysis of ordinary C–metrics. It is popular in the literature
[63, 64, 65, 66, 67, 68, 69, 70, 71, 72] to find the numerical solution of
that equation by imposing the appropriate boundary conditions: $\Theta(x)\sim
e^{\pm Mx}$ as $x\to\pm 1$. But in our case, pseudospectral method [73] along
with these boundary conditions producing poor and sometimes, null results.
Therefore let us turn our attention to an approximate analytical result, which
is not only physically more intuitive, but also give accurate results in
proper physical limit(s).

In the low acceleration limit, a perturbative series expansion of the solution
of Eq. (28) can be performed and the eigenvalue $\xi$ for a given $M$ and
$\mathcal{A}$ can be calculated upto $2$nd order. We will mostly confine
ourselves to the expression of $\xi$ only, rather than the solution itself;
because, the solution $\Theta(x)$ can be affected by the source terms, but the
eigenvalue can’t and thus it carries the inherent properties of the system.
Besides these, $\xi$ is the only term that will be needed to calculate the
final (normal and) quasinormal modes for the system. The detailed perturbative
analysis of the equation has been given in Appendix C.

Second order expansion of $\xi$ reads as

$\xi=L(L+1)+\dfrac{1}{3}+\mathcal{A}^{2}\left[2+\dfrac{(2L+1)(L-M)!}{2(L+M)!}\left(L(L+1)-\dfrac{2}{3}\right)\displaystyle\int_{-1}^{+1}\chi^{\prime
2}P_{LM}(\chi^{\prime})^{2}d\chi^{\prime}\right]+\mathcal{O}(\mathcal{A}^{4})$
(29)

As the rescaled line element (25) and (26) only contains $\mathcal{A}^{2}$,
the even power of $\mathcal{A}$, so Eq. (29) also contains only the even
powers of $\mathcal{A}$. Now the definite integral in Eq. (29) can be
evaluated in terms of $L$ and $M$ in general using Gaunt’s formula [74, 75,
76], but that will make the expression unnecessarily messy. Rather it is
easier to compute the integral for $M=0$. For $L\geq 2$, the integral reduces
to

$\displaystyle\int_{-1}^{+1}\chi^{\prime
2}P_{LM}(\chi^{\prime})^{2}d\chi^{\prime}=\dfrac{2}{3(2L+1)}\left[1+\dfrac{2L(L+1)}{(2L+3)}\right]$
(30)

and using this, in eikonal limit, $\xi$ for $M=0$ can be written as

$\xi\simeq
L^{2}\left[1+\dfrac{\mathcal{A}^{2}L}{9}\right]+\mathcal{O}(\mathcal{A}^{4})$
(31)

### Radial part

Substitution of the ansatz (27) in Eq. (24) decouples the radial equation as

$\dfrac{d^{2}\psi}{d\rho_{*}^{2}}+\left(\varpi^{2}-\dfrac{1}{\varepsilon}V_{eff}^{s=0}\right)\psi=0$
(32)

where $\varpi$ is the reduced frequency, defined by
$\varpi=\omega/\sqrt{\varepsilon}$ and $\rho_{*}$ is the tortoise coordinate,
defined by

$\rho_{*}=\sqrt{\varepsilon}\displaystyle\int_{0}^{\rho}\dfrac{d\rho^{\prime}}{1+\varepsilon\rho^{\prime
2}}=\tan{{}^{-1}\left(\sqrt{\varepsilon}\rho\right)}$ (33)

and $V_{eff}^{s=0}(\rho)$ is the scalar (corresponds to $s=0$) effective
potential, given by

$V_{eff}^{s=0}(\rho)=\dfrac{1+\varepsilon\rho^{2}}{3\left(1+a^{2}\rho^{2}\right)^{2}}\left[a^{2}\rho^{2}(\varepsilon+3\xi)+3\xi+a^{2}-\varepsilon-a^{4}\rho^{2}\right]$
(34)

where the expression for $\xi$ is given in Eq. (29). This potential has a
global maxima at the throat of the wormhole. Besides this, for
$\xi<\frac{1}{3}a^{2}-\varepsilon=\mathcal{A}^{2}-\frac{2}{3}g^{2}+\frac{1}{3}$,
it will also have one minima on each side of the wormhole. But as the
acceleration (and possibly $g$ also) is considered to be small, resulting
$\mathcal{A}^{2}-\frac{2}{3}g^{2}+\frac{1}{3}$ to be small (or maybe even
negative), so in _eikonal limit_ , the potential will have only one stationary
point at $\rho=0$ and that is maxima. Also this is worth noting that the
potential (34) reaches an asymptotic value of
$\frac{\varepsilon(-a^{2}+\varepsilon+3\xi)}{3a^{2}}$ as $\rho\to\infty$,
which is an artifact of the masslessness of perturbing scalar field. The
variation of the scalar effective potential over the radial distance for
different choices of acceleration and for a fixed $g$ has been shown in Fig.
4.

(a) Figure 4: $V_{eff}^{s=0}(\rho)$ for different acceleration parameters and
for a fixed $g=1.0$ with $M=0$.

The boundary conditions for the master equation (32) can be chosen
conventionally: the waves are outgoing at the spatial infinities on both sides
of the wormhole or no waves come from either the left or the right infinity.
However, since the tortoise coordinate doesn’t extend from $-\infty$ to
$+\infty$, the WKB approximation, even with Padé approximations [77], cannot
be applied. Alternatively, the use of _Mathematica_ package QNMspectral [73]
produces extremely poor and insignificant results. Therefore it is better to
seek for an approximated analytic result to have some physical insight of the
system.

Let us consider the eikonal limit (large $\xi$) of the system, such that
$\xi>>a^{2}$, which immediately follows the fact that $\varepsilon+3\xi$ and
$3\xi+a^{2}-\varepsilon$ can safely be approximated as $3\xi$. With solely
this minimal amount of realistic consideration, the scalar effective potential
can be written as

$\displaystyle\dfrac{1}{\varepsilon}V_{eff}^{s=0}(\rho)$ $\displaystyle\simeq$
$\displaystyle\dfrac{\xi\left(1+\varepsilon\rho^{2}\right)}{\varepsilon\left(1+a^{2}\rho^{2}\right)}$
(35) $\displaystyle=$
$\displaystyle\dfrac{\xi}{a^{2}}+\left(\dfrac{1}{\varepsilon}-\dfrac{1}{a^{2}}\right)\dfrac{\xi}{1+a^{2}\rho^{2}}$
(36)

Now using Eq. (33) and the transformation rules, the effective potential
reduces after some algebraic and trigonometric manipulations as follows:

$\dfrac{1}{\varepsilon}V_{eff}^{s=0}(\rho_{*})\simeq\dfrac{\xi}{\left(1+g^{2}\right)-\frac{1}{2}\left(1+\mathcal{A}^{2}\right)\left(1+\cos{2\rho_{*}}\right)}$
(37)

Using this potential, the solution of the master equation (32) takes the form

$\displaystyle\psi(\rho_{*})=$ $\displaystyle c_{1}$
$\displaystyle\operatorname{H\ell}\left[\dfrac{1+g^{2}}{1+\mathcal{A}^{2}},\dfrac{\xi-\left(1+g^{2}\right)\varpi^{2}}{4\left(1+\mathcal{A}^{2}\right)};-\dfrac{\varpi}{2},\dfrac{\varpi}{2},\dfrac{1}{2},\dfrac{1}{2};\cos{{}^{2}\rho_{*}}\right]+$
(38) $\displaystyle c_{2}$
$\displaystyle\operatorname{H\ell}\left[\dfrac{1+g^{2}}{1+\mathcal{A}^{2}},\dfrac{\xi-\left(1+g^{2}\right)\left(\varpi^{2}-1\right)}{4\left(1+\mathcal{A}^{2}\right)};\frac{1-\varpi}{2},\dfrac{1+\varpi}{2},\dfrac{3}{2},\dfrac{1}{2};\cos{{}^{2}\rho_{*}}\right]\cos{\rho_{*}}$

where $\operatorname{H\ell}$ is the local Heun function (see Appendix D),
which is regular at the spatial infinity $(\rho_{*}\to\pm\pi/2)$, but not at
the throat $(\rho_{*}=0)$ of the wormhole. It is intriguing to note that
although the scalar effective potentials (34) and (37) exhibit symmetry about
the throat, the system only permits even solutions, ruling out odd solutions
regardless of the value of $\varpi$. Additionally, observing that
$\psi(\rho_{*})=c_{1}$ at spatial infinities is a consequence of the
normalization of the Heun function, as detailed in Appendix D. It is worth
emphasizing that the solution (38) is sufficiently generalized, and we have
not employed any approximation other than the eikonal limit, which is both
justified and self-explanatory. Now to ensure non-zero finiteness of the
solutions at the throat, it necessitates the regularization of the local Heun
function at that point.

For this purpose, we use Sleeman–Meixner–Schäfke theorem [78, 79]. It states
that for the local Heun function
$\operatorname{H\ell}(\tilde{a},\tilde{q};\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta};\tilde{z})$
to be regular at $\tilde{z}=1$, it requires

$\mu_{0}+\mathop{\raisebox{-5.0pt}{\huge
K}}_{j=1}^{\infty}\dfrac{\nu_{j}\lambda_{j}}{\mu_{j}}=0$ (39)

where $\operatornamewithlimits{K}$ is the notation for standard continued
fraction and the rest of the symbols explained below:

$\displaystyle\mu_{j}$ $\displaystyle=$
$\displaystyle-\left[j^{2}(1+\tilde{a})+j\\{\tilde{a}(\tilde{\gamma}+\tilde{\delta}-1)+(\tilde{\alpha}+\tilde{\beta}-\tilde{\delta})\\}+\tilde{q}\right],$
(40) $\displaystyle\nu_{j}$ $\displaystyle=$
$\displaystyle\tilde{a}j(j+\tilde{\gamma}-1),$ (41) $\displaystyle\lambda_{j}$
$\displaystyle=$ $\displaystyle-(j+\tilde{\alpha}-1)(j+\tilde{\beta}-1)$ (42)

We have used this theorem (39) for each of the local Heun functions in Eq.
(38) by truncating the continued fraction upto a value (say $N$) and found the
root for $\varpi$. Truncation of the continued fraction makes it a rational
polynomial and the number of roots are obviously directly proportional to $N$.
The value of $N$ can be chosen such that after which the rate of change of
desired number of roots go below the tolerance one wants (we chose $N=15$ and
got sufficiently accurate roots). Thus one finds two sets of quantized
$\varpi$ corresponding to each of the solutions in Eq. (38).

Requiring $\psi(\rho_{*})$ to be finite at the throat necessitates both
solutions of Eq. (32) to be regular at $\rho_{*}=0$. Therefore only the common
values of $\varpi$ from the aforementioned two sets are permissible. Hence
under the previously outlined boundary conditions, no common modes exist in
the system, leading to divergence of the field at the throat and disconnection
between the field in the two halves of the wormhole. Also $\varpi$ remains
unquantized, indicating that these boundary conditions yield continuous
spectra.

  * •

Possibilities with a different boundary condition: As the spacetime doesn’t
have asymptotic flatness and we also don’t get interesting result from
perturbation theory, there is an avenue to explore other possibilities of the
boundary conditions; such as, the field derivative may decay so much, such
that it vanishes at the spatial infinity, which is also supported from the
asymptotic behaviour of the scalar effective potential.

In that case, the integration coefficient $c_{2}$ in Eq. (38) must vanish
[80]. Then the resultant first few (normal and quasinormal) frequencies for
different mode values (with $M=0$) obtained from the Sleeman–Meixner–Schäfke
theorem are tabulated in Table 1-4. (The potential (35) doesn’t seem to result
in growing modes making the system unstable. So from $\pm\omega_{R}\pm
i\omega_{I}$ and $\pm\omega_{R}\mp i\omega_{I}$ type of modes, we have only
chosen the positive real and negative imaginary ones for the sake of
conveniences). As evident from the following tables, the real part of the
frequency consistently surpasses the imaginary part, regardless of the value
of $\mathcal{A}$. Moreover, both components of the frequency show an
increasing trend as the acceleration parameter rises. The increase in
imaginary part indicates that the system is damping due to the leakage of
waves at two extreme ends, though we have not explicitly considered the
outgoing nature of waves at the two spatial infinities in our boundary
conditions. This implies that our current boundary conditions in turn imply
the outgoing nature of some part of the waves at excited states (i.e. $n>0$)
at the spatial infinities. With the increase in overtone number,
$\Re{(\omega)}$ increases much faster than $\Re{(\omega)}$, that means that
the field holds the excitation more and more when it goes to the higher
excited states.

Table 1: For $\mathcal{A}=0.1$ and $g=1.0$ n | $L=400$ | $L=500$ | $L=700$ | $L=850$ | $L=1000$
---|---|---|---|---|---
$0$ | $(22.7072,0.0000)$ | $(22.7147,0.0000)$ | $(22.7202,0.0000)$ | $(22.7219,0.0000)$ | $(22.7228,0.0000)$
$1$ | $(25.3524,-10.7430)$ | $(28.1353,-16.2397)$ | $(36.3661,-28.1010)$ | $(44.0305,-37.4159)$ | $(52.5687,-47.0839)$
$2$ | $(28.6404,-17.4866)$ | $(33.9600,-25.1759)$ | $(47.0434,-40.9291)$ | $(58.3024,-53.2943)$ | $(70.5162,-66.2251)$
$3$ | $(31.4040,-24.5303)$ | $(39.0125,-33.4793)$ | $(56.2685,-52.1527)$ | $(70.6284,-67.0301)$ | $(86.0250,-82.7051)$
$4$ | $(37.5818,-33.7568)$ | $(47.9323,-44.5644)$ | $(70.7409,-67.6400)$ | $(89.4471,-86.2645)$ | $(109.3890,-105.9930)$
$5$ | $(50.8534,-47.6931)$ | $(65.6663,-62.1306)$ | $(97.9158,-93.3085)$ | $(124.2040,-118.6200)$ | $(152.1610,-145.4940)$
$6$ | $(84.1677,-75.0804)$ | $(109.1120,-97.4371)$ | $(163.2190,-145.8840)$ | $(207.2450,-185.2830)$ | $(254.0300,-227.1440)$
Table 2: For $\mathcal{A}=0.02$ and $g=1.0$ n | $L=400$ | $L=500$ | $L=700$ | $L=850$ | $L=1000$
---|---|---|---|---|---
$0$ | $(22.8067,0.0000)$ | $(22.8190,0.0000)$ | $(22.8269,0.0000)$ | $(22.8294,0.0000)$ | $(22.8308,0.0000)$
$1$ | $(24.4459,-7.9773)$ | $(25.8862,-11.7363)$ | $(30.3779,-19.7167)$ | $(34.5869,-25.6826)$ | $(39.1919,-31.5608)$
$2$ | $(26.3106,-13.1878)$ | $(29.6462,-18.9416)$ | $(37.6806,-29.8498)$ | $(44.3097,-37.7495)$ | $(51.2140,-45.5290)$
$3$ | $(27.6234,-19.6083)$ | $(32.8536,-26.2352)$ | $(44.0091,-38.9942)$ | $(52.7147,-48.3691)$ | $(61.5973,-57.6848)$
$4$ | $(32.2453,-27.9533)$ | $(39.5535,-35.8175)$ | $(54.5812,-51.3411)$ | $(66.0692,-62.9423)$ | $(77.6873,-74.5612)$
$5$ | $(43.0946,-40.0503)$ | $(53.6761,-50.4451)$ | $(75.0940,-71.2623)$ | $(91.3257,-86.9473)$ | $(107.6810,-102.7120)$
$6$ | $(71.0397,-63.2988)$ | $(88.9217,-79.3415)$ | $(124.9430,-111.6150)$ | $(152.1710,-135.9930)$ | $(179.5740,-160.5200)$
Table 3: For $\mathcal{A}=0.005$ and $g=1.0$ n | $L=400$ | $L=500$ | $L=700$ | $L=850$ | $L=1000$
---|---|---|---|---|---
$0$ | $(22.8103,0.0000)$ | $(22.8229,0.0000)$ | $(22.8309,0.0000)$ | $(22.8335,0.0000)$ | $(22.8350,0.0000)$
$1$ | $(24.4158,-7.8635)$ | $(25.8022,-11.5408)$ | $(30.1272,-19.3268)$ | $(34.1697,-25.1195)$ | $(38.5766,-30.7976)$
$2$ | $(26.2188,-12.9970)$ | $(29.4679,-18.6599)$ | $(37.2667,-29.3315)$ | $(43.6687,-37.0059)$ | $(50.3030,-44.5182)$
$3$ | $(27.4642,-19.3914)$ | $(32.5874,-25.9097)$ | $(43.4550,-38.3835)$ | $(51.8816,-47.4833)$ | $(60.4316,-56.4711)$
$4$ | $(32.0166,-27.6997)$ | $(39.1862,-35.4283)$ | $(53.8443,-50.5905)$ | $(64.9745,-61.8420)$ | $(76.1663,-73.0441)$
$5$ | $(42.7595,-39.7186)$ | $(53.1474,-49.9279)$ | $(74.0493,-70.2506)$ | $(89.7817,-85.4571)$ | $(105.5420,-100.6520)$
$6$ | $(70.4714,-62.7885)$ | $(88.0300,-78.5419)$ | $(123.1890,-110.0440)$ | $(149.5820,-133.6760)$ | $(175.9910,-157.3130)$
Table 4: For $\mathcal{A}=0.002$ and $g=1.0$ | $L=400$ | $L=500$ | $L=700$ | $L=850$ | $L=1000$
---|---|---|---|---|---
$0$ | $(22.8105,0.0000)$ | $(22.8231,0.0000)$ | $(22.8312,0.0000)$ | $(22.8337,0.0000)$ | $(22.8352,0.0000)$
$1$ | $(24.4142,-7.8571)$ | $(25.7975,-11.5298)$ | $(30.1132,-19.3048)$ | $(34.1462,-25.0876)$ | $(38.5419,-30.7544)$
$2$ | $(26.2136,-12.9862)$ | $(29.4579,-18.6441)$ | $(37.2434,-29.3022)$ | $(43.6325,-36.9638)$ | $(50.2515,-44.4609)$
$3$ | $(27.4552,-19.3792)$ | $(32.5724,-25.8913)$ | $(43.4238,-38.3490)$ | $(51.8345,-47.4332)$ | $(60.3656,-56.4023)$
$4$ | $(32.0038,-27.6855)$ | $(39.1655,-35.4063)$ | $(53.8027,-50.5481)$ | $(64.9127,-61.7797)$ | $(76.0801,-72.9582)$
$5$ | $(42.7407,-39.6999)$ | $(53.1177,-49.8987)$ | $(73.9903,-70.1935)$ | $(89.6945,-85.3729)$ | $(105.4200,-100.5350)$
$6$ | $(70.4395,-62.7598)$ | $(87.9797,-78.4968)$ | $(123.0900,-109.9560)$ | $(149.4360,-133.5450)$ | $(175.7880,-157.1320)$

(a)

(b)

(c)

(d)

Figure 5: $\Re{(\omega)}$ vs $\Im{(\omega)}$ plot for different modes $(L)$
with different acceleration parameter and a fixed $g=1.0$.

##  V Concluding Remarks

In this paper, we present a thorough analysis of the physical properties of a
newly constructed wormhole C–metric, which arises as an axisymmetric solution
of Einstein’s field equations with a phantom scalar field source. We have
argued that the acceleration parameter of the wormhole should be bounded by
the amplitude of the potential of the underlying scalar field to make the
solution physically viable. Also we have found that the wormhole will not be
traversable at all as its throat is located at the origin. Still this
spacetime possesses diverse interesting properties. Our investigation reveals
intriguing characteristics of this spacetime, including its lack of asymptotic
flatness despite the absence of curvature singularities. Furthermore, the
study of energy conditions as well as the topological analysis confirms its
wormhole nature, emphasizing the importance of distinguishing it
observationally from black holes. Notably, the absence of photon orbits poses
a challenge for electromagnetic observation, complicating efforts to discern
this spacetime from the black holes. But this spacetime can possess the
_massive particle orbit_ , which may carry the observational interesting
features if the detection is carried out using neutrino or alike particles,
which is massive as well as can travel long distances without much more
interaction with the surroundings. Additionally, we have explored the behavior
of quasinormal modes, finding that the space time will result in a continuous
spectra if one of the boundary condition is chosen as the field to be outgoing
at both infinities. In that case, the field also getting divergent at the
throat causing the field to behave in a disconnected manner in two halves of
the wormhole. Considering another set of boundary conditions, the radial
derivative of effective potential have an asymptotic behavior at the spatial
infinities alongwith the finiteness of the field at the throat, we found a
continuous spectra of quasinormal modes. We observe a transition from purely
oscillatory modes for $n=0$ to damped oscillations with increasing overtone
number for $M=0$, which immediately implies that the boundary conditions
result in the leakage of field at the infinities for excited modes causing the
dissipation in the signal at the detector. Also the relative increment in
$\Re{(\omega)}$ and $\Im{(\omega)}$ with the increase in the overtone number
$(n)$ indicates that spacetime tries to holds the excitation rather than
getting de-excited and coming to the ground state. The reason behind why this
phenomena happens and what is its physical interpretation, it requires a
thorough analysis of gravitational perturbation of the spacetime. As this is a
_non–vacuum axistationary_ spacetime, so it is judicious to carry out that
gravitational perturbation via Newman–Penrose formalism (citations), which is
not only a very recent concept for a non–vacuum spacetime (citations)
(gravitational QNM of vacuum black hole C–metric has been studied very
recently (citation)), but also it involves many non-intuitive difficulties.
Therefore that analysis is left for our future studies. We believe that these
intriguing features for this kind of accelerated wormholes spacetimes in a
periodic potential will result in phenomenologically and observationally
interesting prospects.

##  VI Acknowledgement

SC gratefully acknowledges the support and resources provided by the
Presidency University library and online facilities. The authors collectively
express their gratitude to the authors of the QNMspectral [73] and WKB.m [77]
packages for making their codes publicly available, which immensely benefited
our research efforts. Furthermore, we express our gratitude to Emanuele Berti
(JHU) for making his notebooks publicly available. These resources greatly
facilitated the dissemination of our research findings.

##  Appendix A Notations and Conventions

Throughout the article, we have chosen the natural unit system $(c=\hbar=8\pi
G=1)$ and the mostly positive metric signature $(-,+,+,+)$ has been adopted.
The list of symbols and notations used in the article has been given below:

  1. 1.

$\boldsymbol{\sqrt{-\tilde{g}}~{}d^{4}x}$: $4$-dimensional invariant volume
element, first appeared in Eq. (1).

  2. 2.

$\boldsymbol{x}$: spacetime coordinates $(\tau,r,x,\varphi)$ or
$(\tau,\mathcalligra{r},x,\varphi)$ or $(\zeta,\rho,x,\varphi)$.

  3. 3.

$\Phi(\boldsymbol{x})$: scalar field coupled with, first appeared in Eq. (1).

  4. 4.

$\mathcal{A}$: rescaled acceleration of wormhole, first appeared in Eq. (6).

  5. 5.

$\epsilon$: categorizes the null $(0)$, timelike $(-1)$ and spacelike $(+1)$
geodesics, first appeared in Eq. (11).

  6. 6.

$E$: energy of a particle or photon moving in this spacetime, first appeared
in Eq. (12).

  7. 7.

$\ell$: angular momentum of particle of photon in this spacetime, first
appeared in Eq. (15).

  8. 8.

$\bar{\phi}$: test scalar field, first appeared in Eq. (24).

  9. 9.

$\varepsilon$: $g^{2}-\mathcal{A}^{2}$, first appeard in Eq. (26).

  10. 10.

$\xi$: separation of constant corresponding to angular equation in
Klein–Gordon equation, first appeared in Eq. (28).

  11. 11.

$L,M$: azimuthal quantum number and magnetic quantum number respectively,
associated with spherical harmonics, first appeared in Eq. (29).

  12. 12.

$\psi$: radial part of the test scalar field, first appeared in Eq. (32).

##  Appendix B Topological properties of the spacetime

By definition [60], in a Lorentzian spacetime, if there exists a compact
region denoted by $\Omega$, and if $\Omega$ exhibits a topology resembling
$\mathbb{R}\times\Sigma$, where $\Sigma$ represents a three–manifold with non-
trivial topology and its boundary $\partial\Sigma$ homeomorphic to
$\mathbb{S}^{2}$, and additionally, if all hypersurfaces $\Sigma$ are
spacelike, then it follows that the region $\Omega$ contains a _quasipermanent
intra–universe wormhole_. Here, “quasipermanent” refers that the wormhole
exists for a finite non–zero duration of time.

For the case of line element (5), the topology of spacelike hypersurface
$\Sigma$ can be determined by the constant $\varphi$ hypersurfaces (let,
$\Sigma^{\prime}$) due to the symmetry. Then the induced metric over
$\Sigma^{\prime}$ can be written as

$ds_{\Sigma^{\prime}}^{2}=\dfrac{1}{(1+\mathcal{A}\mathcalligra{r}x)^{2}}\left[\dfrac{1+\mathcal{A}^{2}x^{2}}{1+g^{2}+(g^{2}-\mathcal{A}^{2})\mathcalligra{r}^{2}}d\mathcalligra{r}^{2}+\dfrac{1+\mathcalligra{r}^{2}}{1-x^{2}}dx^{2}\right]\equiv
h_{\mu\nu}dx^{\mu}dx^{\nu}~{}\text{(let)}$ (43)

where $\Omega$ to $\Sigma^{\prime}$ and $\Sigma$ to $\Sigma^{\prime}$
projectors are essentially the Kronecker deltas and the normal to
$\Sigma^{\prime}$ is also the same. The triviality or non-triviality of
topology of the manifold $\Sigma^{\prime}$ can now be determined by its Euler
characteristic $(\chi)$ from the _Gauss–Bonnet formula_. If the Gaussian
curvature of this $2$ dimensional manifold is being denoted by $K_{G}$, then
by _Gauss’s Theorema Egregium_ [81], we have

$K_{G}=-\dfrac{1}{2\sqrt{h_{11}h_{22}}}\left[\partial_{x}\left(\dfrac{\partial_{x}h_{11}}{\sqrt{h_{11}h_{22}}}\right)+\partial{r}\left(\dfrac{\partial{r}h_{22}}{\sqrt{h_{11}h_{22}}}\right)\right]$
(44)

Considering $\Sigma^{\prime}$ to be orientable compact $2$ dimensional
Riemannian manifold, Gauss–Bonnet theorem reads

$\chi=\dfrac{1}{2\pi}\displaystyle\int_{\Sigma^{\prime}}K_{G}\sqrt{h}~{}d\mathcalligra{r}~{}dx+\dfrac{1}{2\pi}\displaystyle\int_{\partial\Sigma^{\prime}}k_{g}d\mathcalligra{s}$
(45)

where $k_{g}$ is the geodesic curvature of the boundary
$\partial\Sigma^{\prime}$ and $d\mathcalligra{s}$ be its line element. In this
case, $k_{g}=0$. Therefore using Eq. (44), rest of the part of Eq. (45)
reduces to

$\chi=-\dfrac{1}{4\pi}\left[\displaystyle\int_{\Sigma^{\prime}}d\mathcalligra{r}\left.\dfrac{\partial_{x}h_{11}}{\sqrt{h_{11}h_{22}}}\right|_{x\to
1}-\displaystyle\int_{\Sigma^{\prime}}d\mathcalligra{r}\left.\dfrac{\partial_{x}h_{11}}{\sqrt{h_{11}h_{22}}}\right|_{x\to-1}+\displaystyle\int_{\Sigma^{\prime}}dx\left.\dfrac{\partial{r}h_{22}}{\sqrt{h_{11}h_{22}}}\right|_{r\to\infty}-\displaystyle\int_{\Sigma^{\prime}}dx\left.\dfrac{\partial{r}h_{22}}{\sqrt{h_{11}h_{22}}}\right|_{r\to-\infty}\right]$
(46)

All the limiting terms in Eq. (46) identically vanishes resulting $\chi=0$,
resulting this to be a quasipermanent intra–universe wormhole spacetime.

##  Appendix C Perturbation series analysis of scalar field angular equation

In the zero acceleration limit, the angular equation (28) of the scalar
perturbation can be written as

$\hat{\mathbb{L}}(x)\Theta_{0}(x)+\left[\xi_{0}-\dfrac{1}{3}-\dfrac{M^{2}}{1-x^{2}}\right]\Theta_{0}(x)=0$
(47)

where $\hat{\mathbb{L}}$ is a Strum–Liouville operator defined in range
$x\in[-1,+1]$ as

$\hat{\mathbb{L}}(x)\equiv\left(1-x^{2}\right)\dfrac{d^{2}}{dx^{2}}-2x\dfrac{d}{dx}$
(48)

This is self-adjoint as well as Hermitian operator i.e. it has complete set of
orthonormal eigenvectors alongwith real eigenvalues. Eq. (47) can readily be
solved by means of associated Legendre polynomials $P_{LM}(x)$, which will be
finite over the entire range and to guarantee the necessary convergence of the
associated Frobenius series, it needs $\xi_{0}=L(L+1)+\frac{1}{3}$. This is
the zeroth order approximation of the eigenvalue and the eigenvector. For the
higher order corrections, the eigenvalue $\xi$ and the eigenfunction $\Theta$
can be expanded in a perturbation series as

$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\xi_{0}+\mathcal{A}^{2}\xi_{2}+\mathcal{A}^{4}\Theta_{4}(x)+\mathcal{O}(\mathcal{A}^{6})$
(49) $\displaystyle\Theta(x)$ $\displaystyle=$
$\displaystyle\Theta_{0}(x)+\mathcal{A}^{2}\Theta_{2}(x)+\mathcal{A}^{4}\Theta_{4}(x)+\mathcal{O}(\mathcal{A}^{6})$
(50)

where $\Theta_{0}(x)=P_{LM}(x)$. Substituting Eqs. (49) and (50) in Eq. (28)
and expanding the terms in Taylor series, we have after comparing the
coefficients of $\mathcal{A}^{2}$ on both sides,

$\left[\hat{\mathbb{L}}(x)+\xi_{0}-\dfrac{1}{3}\right]\Theta_{1}(x)+\xi_{2}\Theta_{0}(x)=0$
(51)

Multiplying both sides of Eq. (51) by $\Theta_{0}^{*}(x)$ (complex conjugated
function of $\Theta_{0}(x)$) and using the fact that $\hat{\mathbb{L}}(x)$ is
a self-adjoint operator being its Strum–Liouville nature, we arrive at Eq.
(29). This is the $2^{\text{nd}}$ order perturbative approximation of the
eigenvalue, where the $1^{\text{st}}$ and other odd order corrections are
zero.

##  Appendix D Heun function

The general Heun equation is given by the following equation [82]:

$\dfrac{d^{2}u}{d\tilde{z}^{2}}+\left(\dfrac{\tilde{\gamma}}{\tilde{z}}+\dfrac{\tilde{\delta}}{\tilde{z}-1}+\dfrac{\tilde{\epsilon}}{\tilde{a}}\right)\dfrac{du}{d\tilde{z}}+\dfrac{\tilde{\alpha}\tilde{\beta}\tilde{z}-\tilde{q}}{\tilde{z}(\tilde{z}-1)(\tilde{z}-\tilde{a})}u=0$
(52)

where the parameters satisfy the Fuchsian relation
$\tilde{\epsilon}=1+\tilde{\alpha}+\tilde{\beta}-\tilde{\gamma}-\tilde{\delta}$.
This equation has four regular singular points at $0,1,\tilde{a},\infty$ and
any differential equation with four regular singular points (such as
Hypergeometric differential equations or Lamé equation) can be transformed to
this equation by a change of variables [83]. For each singular points, it is
possible to construct Frobenius series solution, but one cannot get a
recursion relation between two consecutive coefficients. We have a relation at
least between three coefficients [82]. The solutions are convergent inside a
circle, whose radius is generically determined by the distance from the
nearest singularity.

It turns out that for infinite set of values of the parameter $\tilde{q}$,
there are solutions which are analytic at $0$ and at $1$. These are called
Heun functions, whereas those which are analytic only at one point are called
local Heun functions, denoted by
$\operatorname{H\ell}(\tilde{a},\tilde{q};\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta};\tilde{z})$.
Local Heun function that is also regular at $\tilde{z}=1$ is called Heun
function, denoted by
$\operatorname{Hf}(\tilde{a},\tilde{q};\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta};\tilde{z})$.
For integer values of one of $\tilde{\alpha}$,
$\tilde{\gamma}-\tilde{\alpha}$, $\tilde{\delta}-\tilde{\alpha}$,
$\tilde{\epsilon}-\tilde{\alpha}$, and for special finite values of
$\tilde{q}$, solutions analytic at three singularities exist, the so-called
Heun polynomials, denoted by
$\operatorname{Hp}(\tilde{a},\tilde{q};\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta};\tilde{z})$
[84]. So if a Local Heun function is tried to make regular at $\tilde{z}=1$,
then the accessory parameter $\tilde{q}$ have to be quantized [78] and in that
case, it should satisfy Eq. (39).

The local Heun function can be normalized to unity at $\tilde{z}=0$ [85] as

$\operatorname{H\ell}(\tilde{a},\tilde{q};\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta};0)=1$
(53)

and its derivative with respect to $\tilde{z}$ at $\tilde{z}=0$ takes the
value $\frac{\tilde{q}}{\tilde{a}\tilde{\gamma}}$ and diverges at
$\tilde{z}=1$.

In Mathematica [86], the local Heun function and its first derivative are
implemented by the library functions as
HeunG$[\tilde{a},\tilde{q},\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta},\tilde{z}]$
and
HeunGPrime$[\tilde{a},\tilde{q},\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta},\tilde{z}]$
respectively. These have the same normalization and other properties discussed
above. Globally HeunG gives analytic continuation of $\operatorname{H\ell}$.
Though sometimes multi-valuedness causes a problem in numerical computations,
all our computations are done inside the convergence circle. Hence there is no
multi-value problem.

## References

  * Kinnersley and Walker [1970] W. Kinnersley and M. Walker, Phys. Rev. D 2, 1359 (1970).
  * Bonnor [1983] W. B. Bonnor, General Relativity and Gravitation 15, 535 (1983).
  * Krtouš [2005] P. Krtouš, Physical Review D 72, 10.1103/physrevd.72.124019 (2005).
  * Dias and Lemos [2003] O. J. C. Dias and J. P. S. Lemos, Physical Review D 67, 10.1103/physrevd.67.064001 (2003).
  * Letelier and Oliveira [2001] P. S. Letelier and S. R. Oliveira, Physical Review D 64, 10.1103/physrevd.64.064005 (2001).
  * Griffiths  _et al._ [2006] J. B. Griffiths, P. Krtouš, and J. Podolský, Classical and Quantum Gravity 23, 6745–6766 (2006).
  * Bini  _et al._ [2004] D. Bini, C. Cherubini, and B. Mashhoon, Physical Review D 70, 10.1103/physrevd.70.044020 (2004).
  * Lim [2014] Y.-K. Lim, Physical Review D 89, 10.1103/physrevd.89.104016 (2014).
  * Lim [2021] Y.-K. Lim, Physical Review D 103, 10.1103/physrevd.103.024007 (2021).
  * Frost and Perlick [2021] T. C. Frost and V. Perlick, Classical and Quantum Gravity 38, 085016 (2021).
  * Pravda and Pravdova [2001] V. Pravda and A. Pravdova, Class. Quant. Grav. 18, 1205 (2001).
  * Pravda and Pravdova [2002] V. Pravda and A. Pravdova, Gravitation: Following the Prague Inspiration 10.1142/97898127769380008 (2002).
  * Podolský  _et al._ [2003] J. Podolský, M. Ortaggio, and P. Krtouš, Physical Review D 68, 10.1103/physrevd.68.124004 (2003).
  * Morris and Thorne [1988] M. S. Morris and K. S. Thorne, American Journal of Physics 56, 10.1119/1.15620 (1988).
  * Morris  _et al._ [1988] M. S. Morris, K. S. Thorne, and U. Yurtsever, Phys. Rev. Lett. 61, 1446 (1988).
  * Visser [1989] M. Visser, Phys. Rev. D 39, 3182 (1989).
  * Lobo [2005] F. S. N. Lobo, Phys. Rev. D 71, 084011 (2005).
  * Maldacena and Milekhin [2021] J. Maldacena and A. Milekhin, Phys. Rev. D 103, 066007 (2021).
  * Kar [2022a] S. Kar, Phys. Rev. D 105, 024013 (2022a).
  * Konoplya and Zhidenko [2022] R. A. Konoplya and A. Zhidenko, Phys. Rev. Lett. 128, 091104 (2022).
  * Teo [1998] E. Teo, Phys. Rev. D 58, 024014 (1998).
  * Kar [2022b] S. Kar, Gen. Rel. Grav. 54, 66 (2022b).
  * Lobo [2006] F. S. N. Lobo, Phys. Rev. D 73, 064028 (2006).
  * Maldacena  _et al._ [2023] J. Maldacena, A. Milekhin, and F. Popov, Classical and Quantum Gravity 40, 155016 (2023).
  * Biswas  _et al._ [2024] S. Biswas, C. Singha, and S. Chakraborty, Phys. Rev. D 109, 064043 (2024).
  * Dutta Roy  _et al._ [2020] P. Dutta Roy, S. Aneesh, and S. Kar, Eur. Phys. J. C 80, 850 (2020).
  * Nozawa and Torii [2023a] M. Nozawa and T. Torii, Phys. Rev. D 108, 10.1103/physrevd.108.064036 (2023a).
  * Perlick and Tsupko [2022] V. Perlick and O. Y. Tsupko, Physics Reports 947, 1–39 (2022).
  * Ohgami and Sakai [2015a] T. Ohgami and N. Sakai, Physical Review D 91, 10.1103/physrevd.91.124020 (2015a).
  * Rahaman  _et al._ [2021a] F. Rahaman, K. N. Singh, R. Shaikh, T. Manna, and S. Aktar, Classical and Quantum Gravity 38, 215007 (2021a).
  * EHT1 [2019] EHT1, Astrophys. J. Lett. 875, L1 (2019).
  * Akiyama  _et al._ [2019a] K. Akiyama _et al._ (Event Horizon Telescope), Astrophys. J. Lett. 875, L3 (2019a).
  * 3 [2019] T. E. H. T. C. 3, Astrophys. J. Lett. 875, L4 (2019).
  * Akiyama  _et al._ [2019b] K. Akiyama _et al._ (Event Horizon Telescope), Astrophys. J. Lett. 875, L5 (2019b).
  * 5 [2019] T. E. H. T. C. 5, Astrophys. J. Lett. 875, L6 (2019).
  * Wielgus  _et al._ [2020] M. Wielgus, J. c. v. Horák, F. Vincent, and M. Abramowicz, Phys. Rev. D 102, 084044 (2020).
  * Tsukamoto [2021] N. Tsukamoto, Phys. Rev. D 103, 064031 (2021).
  * Rahaman  _et al._ [2021b] F. Rahaman, K. N. Singh, R. Shaikh, T. Manna, and S. Aktar, Classical and Quantum Gravity 38, 215007 (2021b).
  * Shaikh [2018] R. Shaikh, Phys. Rev. D 98, 024044 (2018).
  * Bugaev  _et al._ [2021] M. A. Bugaev, I. D. Novikov, S. V. Repin, and A. A. Shelkovnikova, Astronomy Reports 65, 1185–1193 (2021).
  * Ohgami and Sakai [2015b] T. Ohgami and N. Sakai, Phys. Rev. D 91, 124020 (2015b).
  * Alloqulov  _et al._ [2024] M. Alloqulov, F. Atamurotov, A. Abdujabbarov, B. Ahmedov, and V. Khamidov, Chin. Phys. C 48, 025104 (2024).
  * Ishkaeva and Sushkov [2023] V. A. Ishkaeva and S. V. Sushkov, Phys. Rev. D 108, 084054 (2023).
  * Neto  _et al._ [2023] M. R. Neto, D. Pérez, and J. Pelle, Int. J. Mod. Phys. D 32, 2250137 (2023).
  * Kumar  _et al._ [2024] S. Kumar, A. Uniyal, and S. Chakrabarti, Phys. Rev. D 109, 104012 (2024).
  * Amir  _et al._ [2019] M. Amir, A. Banerjee, and S. D. Maharaj, Annals Phys. 400, 198 (2019).
  * Molla  _et al._ [2024] N. U. Molla, H. Chaudhary, U. Debnath, G. Mustafa, and S. K. Maurya, (2024), arXiv:2406.09492 [gr-qc] .
  * Shaikh  _et al._ [2019] R. Shaikh, P. Banerjee, S. Paul, and T. Sarkar, Phys. Lett. B 789, 270 (2019), [Erratum: Phys.Lett.B 791, 422–423 (2019)], arXiv:1811.08245 [gr-qc] .
  * Jha and Rahaman [2023] S. K. Jha and A. Rahaman, Physics of the Dark Universe 42, 101327 (2023).
  * Bronnikov  _et al._ [2021] K. A. Bronnikov, R. A. Konoplya, and T. D. Pappas, Phys. Rev. D 103, 124062 (2021).
  * Guerrero  _et al._ [2021] M. Guerrero, G. J. Olmo, D. Rubiera-Garcia, and D. S.-C. Gómez, JCAP 2021 (08), 036.
  * Guo and Miao [2022] Y. Guo and Y.-G. Miao, Nucl. Phys. B 983, 115938 (2022).
  * Jha and Rahaman [2022] S. K. Jha and A. Rahaman, (2022), arXiv:2208.13176 [gr-qc] .
  * Jha [2023] S. K. Jha, Eur. Phys. J. Plus 138, 757 (2023).
  * Abbott  _et al._ [2016] B. P. Abbott, R. Abbott, and A. et. al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016).
  * Berti  _et al._ [2009] E. Berti, V. Cardoso, and A. O. Starinets, Classical and Quantum Gravity 26, 163001 (2009).
  * Konoplya and Zhidenko [2011] R. A. Konoplya and A. Zhidenko, Reviews of Modern Physics 83, 793–836 (2011).
  * Destounis  _et al._ [2020a] K. Destounis, R. D. Fontana, and F. C. Mena, Physical Review D 102, 10.1103/physrevd.102.044005 (2020a).
  * Visser [2023] M. Visser, Universe 9, 10.3390/universe9120521 (2023).
  * Visser [1995] M. Visser,  _Lorentzian wormholes: From Einstein to Hawking_ (Am. Inst. of Phys. Melville, New York, 1995).
  * Destounis  _et al._ [2022a] K. Destounis, G. Mascher, and K. D. Kokkotas, Phys. Rev. D 105, 10.1103/PhysRevD.105.124058 (2022a).
  * Sui  _et al._ [2023] T.-T. Sui, Q.-M. Fu, and W.-D. Guo, Physics Letters B 845, 138135 (2023).
  * Lei  _et al._ [2024a] Y. Lei, H. Shu, K. Zhang, and R.-D. Zhu, Journal of High Energy Physics 2024, 140 (2024a).
  * Gwak [2023a] B. Gwak, Eur. Phys. J. Plus 138, 582 (2023a).
  * Destounis  _et al._ [2022b] K. Destounis, G. Mascher, and K. D. Kokkotas, Phys. Rev. D 105, 124058 (2022b).
  * Lin  _et al._ [2015] H. Lin, K. Saifullah, and S.-T. Yau, Modern Physics Letters A 30, 1550044 (2015).
  * Xiong and Li [2023a] W. Xiong and P.-C. Li, Phys. Rev. D 108, 044064 (2023a).
  * Destounis  _et al._ [2020b] K. Destounis, R. D. B. Fontana, and F. C. Mena, Phys. Rev. D 102, 044005 (2020b).
  * Fontana and Mena [2022a] R. D. Fontana and F. C. Mena, Journal of High Energy Physics 2022, 47 (2022a).
  * Chen  _et al._ [2024a] T. Chen, R.-G. Cai, and B. Hu, Phys. Rev. D 109, 084049 (2024a).
  * Fontana and Rincon [2024] R. D. B. Fontana and A. Rincon, (2024), arXiv:2404.09936 [gr-qc] .
  * Barragán Amado and Gwak [2024] J. Barragán Amado and B. Gwak, JHEP 2024 (02), 189.
  * Jansen [2017] A. Jansen, Eur. Phys. J. Plus 132, 546 (2017).
  * Gaunt J. A. [1929] F. R. H. Gaunt J. A., Philosophical Transactions of the Royal Society of London. Series A 228, 10.1098/rsta.1929.0004 (1929).
  * hai Dong and Lemus [2002] S. hai Dong and R. Lemus, Applied Mathematics Letters 15, 10.1016/S0893-9659(02)80004-0 (2002).
  * Slater [1960] J. C. Slater,  _Quantum Theory of Atomic Structure (Volume 1)_ , International series in pure and applied physics (McGraw Hill, 1960) pp. xii+502.
  * Konoplya  _et al._ [2019] R. A. Konoplya, A. Zhidenko, and A. F. Zinhailo, Class. Quant. Grav. 36, 10.1088/1361-6382/ab2e25 (2019).
  * Sleeman [1969] B. D. Sleeman, Mathematical Proceedings of the Cambridge Philosophical Society 65, 447–459 (1969).
  * Meixner and Schäfke [2013] J. Meixner and F. Schäfke,  _Mathieusche Funktionen und Sphäroidfunktionen: Mit Anwendungen auf Physikalische und Technische Probleme_ , Grundlehren der mathematischen Wissenschaften (Springer Berlin Heidelberg, 2013).
  * Filipuk  _et al._ [2020] G. Filipuk, A. Ishkhanyan, and J. Dereziński, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) 55, 200–207 (2020).
  * Gray [1996] A. Gray,  _Modern Differential Geometry of Curves and Surfaces with Mathematica_ , 1st ed. (CRC Press, Inc., USA, 1996).
  * Heun [1888] K. Heun, Mathematische Annalen 33, 161 (1888).
  * Ronveaux [1995] A. Ronveaux,  _Heun’s Differential Equations_ (Oxford University Press, Oxford, UK, 1995).
  * Hortaçsu [2018] M. Hortaçsu, Advances in High Energy Physics 2018, 1–14 (2018).
  * Hatsuda [2020] Y. Hatsuda, Classical and Quantum Gravity 38, 025015 (2020).
  * Wolfram Research, Inc. [2024] Wolfram Research, Inc., Mathematica 14.0.0 (2024).
  * Nozawa [2021] M. Nozawa, Phys. Rev. D 103, 10.1103/physrevd.103.024005 (2021).
  * Nozawa and Torii [2023b] M. Nozawa and T. Torii, Phys. Rev. D 107, 10.1103/physrevd.107.064064 (2023b).
  * Gwak [2023b] B. Gwak, The European Physical Journal Plus 138, 10.1140/epjp/s13360-023-04215-7 (2023b).
  * Fontana and Mena [2022b] R. D. Fontana and F. C. Mena, Journal of High Energy Physics 2022, 10.1007/jhep10(2022)047 (2022b).
  * Xiong and Li [2023b] W. Xiong and P.-C. Li, Quasinormal modes of rotating accelerating black holes (2023b), arXiv:2305.04040 [gr-qc] .
  * Chen  _et al._ [2024b] T. Chen, R.-G. Cai, and B. Hu, Quasinormal modes of gravitational perturbation for uniformly accelerated black holes (2024b), arXiv:2402.02027 [gr-qc] .
  * Lei  _et al._ [2024b] Y. Lei, H. Shu, K. Zhang, and R.-D. Zhu, Quasinormal modes of c-metric from scfts (2024b), arXiv:2308.16677 [hep-th] .
  * Fiziev [2006] P. P. Fiziev, Class. Quant. Grav. 23, 2447 (2006).
  * Plebański and Demiański [1976] J. Plebański and M. Demiański, Annals of Physics 98, 98 (1976).
  * Stephani  _et al._ [2003] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt,  _Exact Solutions of Einstein’s Field Equations_ , Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2003).
  * Griffiths and Podolsky [2009] J. B. Griffiths and J. Podolsky,  _Exact Space-Times in Einstein’s General Relativity_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 2009).
  * Bini [2005] D. Bini, in  _AIP Conference Proceedings_ (AIP, 2005).
  * Berglund and Uzer [2001] N. Berglund and T. Uzer, Foundations of Physics , 283–326 (2001).
  * Namouni and Guzzo [2007] F. Namouni and M. Guzzo, Celestial Mech. 99, 31 (2007), arXiv:0705.3219 [astro-ph] .
  * Belyaev and Rafikov [2010] M. A. Belyaev and R. R. Rafikov, The Astrophysical Journal 723, 1718–1735 (2010).