#  Generalised Hayward spacetimes: Geometry, matter and scalar quasinormal
modes

Poulami Dutta Roy, Sayan Kar
poulamiphysics<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology Kharagpur, 721 302, India

###### Abstract

Bardeen’s 1968 idea of a regular black hole spacetime was revived by Hayward
in 2006 through the construction of a new example of such a geometry. Later it
was realised by Neves and Saa, that a wider, two-parameter class exists, with
Bardeen and Hayward spacetimes as special cases. In this article, we revisit
and generalise the Hayward spacetime by applying the Damour-Solodukhin (DS)
prescription. Recalling the DS suggestion of a deformed Schwarzschild
spacetime where $g_{tt}=-\left(1-\frac{2M_{1}}{r}\right)$,
$g_{rr}=\left(1-\frac{2M_{2}}{r}\right)^{-1}$ and $M_{1}\neq M_{2}$, we
propose a similar deformation of the Hayward geometry. The $g_{tt}$ and
$g_{rr}$ in the original Hayward line element remain functionally the same,
albeit mutations introduced via differently valued metric parameters,
following the DS idea. This results in a plethora of spacetime geometries,
known as well as new, and including singular black holes, wormholes or regular
black holes. We first study the geometric features and matter content of each
of such spacetimes in some detail. Subsequently, we find the scalar
quasinormal modes corresponding to scalar wave propagation in these
geometries. We investigate how the real and imaginary parts of the quasinormal
modes depend on the values and ranges of the metric parameters used to
classify the geometries. Finally, we argue how our results on this family of
spacetimes suggest their utility as black hole mimickers.

##  I Introduction

In General Relativity, black holes are largely considered today as acceptable
spacetimes primarily because one has been able to ‘prove’ their existence
theoretically Penrose (1965),Hawking and Penrose (1970) and also ‘see’ them in
numerous observational contexts Abbott et al. (2016a); Akiyama et al. (2019);
Abbott et al. (2020a, 2019, b) including the recent imaging of a supermassive
black hole at the center of the Milky Way Akiyama et al. (2022a, b). As we
know, black holes are characterised by their event horizon(s) and the
singularity. On the other hand, we have naked singularities which are, by
definition horizon-less. The Penrose cosmic censorship hypothesis Penrose
(1965) says that naked singularities cannot be there but neither a proof of
this conjecture in a general setting exists nor are we able to ignore the
numerous counterexamples representing nakedly singular spacetimes with matter
satisfying all the energy conditions. Still, studies on naked singularities
and cosmic censorship continue and seem to lie in a sort of comfort zone among
researchers!

Contrast the above-mentioned types with spacetimes wherein the singularity is
generically absent, i.e. we are in the so-called non-singular domain. Such
spacetimes can still have horizon(s) or they may not–the former leading us to
the regular black hole while the latter suggests wormholes. Thus, within each
category fixed by a spacetime being either singular or non-singular, one finds
further sub-classes defined via the presence or the absence of horizon(s).
Eventually, this leads us to black holes and naked singularities on one side
and regular black holes and wormholes on the other. In this article, we intend
to focus mainly on this non-singular class–with or without horizons, i.e.
regular black holes and wormholes. It remains a fact there appears to be a
good deal of hesitancy in accepting such spacetimes as possibilities in
General Relativity or in any other metric theory of gravity. The reason behind
is the issue of matter–in particular, what is the stress energy that could
make such spacetimes possible? It is known that there are genuine issues about
energy condition violations for wormholes Morris et al. (1988); Morris and
Thorne (1988) and even though we have viable models for the matter required
for regular black holes, they are plagued by the presence of the mass-
inflation instability. The mass parameter exponentially increases, for such
regular spacetimes, with a timescale determined by the surface gravity of the
inner horizon. This instability is well studied for both regular and singular
black holes in the literature Ori (1991); Poisson and Israel (1989); Carballo-
Rubio et al. (2018, 2021). Recently, some authors have suggested possible ways
of taming this instability in regular black holes Carballo-Rubio et al.
(2022); Bonanno et al. (2021) but it still remains an open issue which needs
to be further addressed. These problems with non-singular spacetimes makes one
unsure whether such geometries are indeed worth consideration, until any
observation emerges in support.

Despite the above pros and cons which are associated with any study on regular
black holes or wormholes, we prefer to remain optimistic and try to find paths
and avenues which may perhaps provide new knowledge about their
characteristics and signatures. Let us now take a look at the approach we
follow in constructing model spacetimes which represent regular black holes or
wormholes. Much of this is inspired by the Damour–Solodukhin idea where a line
element of the following form was suggested and studied,

$\displaystyle
ds^{2}=-\left(1-\frac{2M_{1}}{r}\right)dt^{2}+\frac{dr^{2}}{1-\frac{2M_{2}}{r}}+r^{2}d\Omega_{2}^{2}$
(1)

The novelty here is in the $M_{1}$, $M_{2}$ which are different though,
functionally, the $g_{tt}$ and $g_{rr}$ are, as in Schwarzschild spacetime.
However, such a variation changes the geometry quite drastically. Firstly,
there is a nonzero energy-momentum (matter) and secondly, one can avoid a
horizon and live with just a throat (depending on the allowed range of values
of $M_{1}$, $M_{2}$).

The takeaway from the Damour–Solodukhin case is this freedom of varying the
constants in the metric functions and looking into its consequences. We choose
to do this for the regular black hole known as Hayward spacetime which is
within a broader class of spacetimes of which the Bardeen geometry appears to
be another special case. As we will see below, our generalisation (following
the Damour–Solodukhin prescription) results in a wide variety of possible
spacetimes. We have not bothered much about ‘matter content’ in the sense of a
Lagrangian description, while proposing these generalisations. One may
consider this as a deficiency but, at the same time, we proceed and try to
find different features/characteristics pertaining to the geometry, matter and
perturbations associated with them. It goes without saying that it would
certainly be of interest to precisely know about matter sources for these
spacetimes and we hope to address such issues in future.

Our work is organised as follows. In the next section (Section II) and also in
Section III we discuss the variety of spacetimes in full detail. Sections IV
and V deal with scalar waves, quasinormal modes and echoes. In Section VI we
higlight how we can distinguish between different types of spacetimes through
the quasinormal modes. Finally, Section VII is a conclusion.

Unless otherwise stated we have used geometrised units $G=c=1$.

##  II The spacetime geometries: early work and a proposal

In a pioneering conference paper of 1968, the concept of a regular black hole
was first introduced by Bardeen Bardeen (1968) following the ideas of Sakharov
Sakharov (1966) and Gilner Gliner (1966) who discussed a possible end state of
gravitational collapse without a final singularity (for a review see Ansoldi
(2008) and references therein). These spherically-symmetric singularity free
solutions, known today as the Bardeen regular black holes, opened a new window
in the class of regular solutions in GR. The Bardeen regular black hole metric
has the form,

$ds^{2}=-\Big{(}1-\frac{2Mr^{2}}{(r^{2}+g^{2})^{3/2}}\Big{)}dt^{2}+\frac{dr^{2}}{\Big{(}1-\frac{2Mr^{2}}{(r^{2}+g^{2})^{3/2}}\Big{)}}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2})$ (2)

with $g$ being a constant and $M$ as the mass parameter. Setting $g\rightarrow
0$ reduces the solution to the standard Schwarzschild black hole. The above
metric was later found as a magnetic solution to Einstein’s equations coupled
to nonlinear electrodynamics Ayon-Beato and Garcia (2000). The parameter $g$
there, is physically interpreted as the monopole charge of the self-
gravitating magnetic field.
Later, in Neves and Saa (2014) a generalization of the Bardeen metric was
introduced with a line element of the form

$ds^{2}=-\Big{(}1-\frac{2m(r)}{r}\Big{)}dt^{2}+\frac{dr^{2}}{1-\frac{2m(r)}{r}}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2})$ (3)

with

$m(r)=\frac{M_{0}}{\Big{(}1+(\frac{r_{0}}{r})^{q}\Big{)}^{p/q}}$ (4)

The $M_{0}$ and $r_{0}$ here are mass and length parameters respectively, with
p and q being constants. This spacetime reduces to the Bardeen metric with a
choice of $p=3,q=2$ and proper identification of $M_{0}$ and $r_{0}$.

Another kind of regular black hole known today as the Hayward metric, first
appeared as an independent class of regular solutions in Hayward (2006). One
can check that the Hayward spacetime may be obtained from the broader class
mentioned above Neves and Saa (2014), by choosing $p=q=3$. Hence, both the
Bardeen and Hayward geometries arise as special cases of the generalised
$(p,q)$ spacetimes suggested in Neves and Saa (2014). We will concentrate
mostly on the Hayward variety in this article.

The line element for the Hayward spacetime is explicitly written as

$ds^{2}=-F(r)dt^{2}+\frac{dr^{2}}{F(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (5)

with

$F(r)=1-\frac{2Mr^{2}}{r^{3}+2M\ell^{2}}$ (6)

where $(\ell,M)$ are positive constants. The function $F(r)$ has the property
that it reduces to the Schwarzschild form asymptotically

$F(r)\sim 1-\frac{2M}{r}\hskip 14.45377ptas\hskip
14.45377ptr\rightarrow\infty$

and being a regular spacetime, it behaves like a de-Sitter space for small
values of $r$ , i.e.

$F(r)\sim 1-\frac{r^{2}}{\ell^{2}}\hskip 14.45377ptas\hskip
14.45377ptr\rightarrow 0$

The cosmological constant is identified as $\Lambda=3/\ell^{2}$ in the small
$r$ limit. This regular black hole can also be sourced by nonlinear
electrodynamics though in the weak field limit, standard Maxwellian
electrodynamics is not recovered Lemy (2018).

Several articles in the literature have proposed generalized mass functions
that result in different kinds of regular black hole solutions Fan and Wang
(2016); Neves and Saa (2014); Balart and Vagenas (2014); Ayon-Beato and Garcia
(2005), one of which is mentioned in eq.(3). Through our work, we propose
another generalisation of the Hayward regular black hole. In our construction
we retain the functional forms of the metric functions, but tweak the
parameters therein to generate a wide range of diverse spacetimes that
includes regular and singular black holes as well as wormholes.
In more concrete terms, we consider a spherically symmetric, static metric

$ds^{2}=-f_{1}(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (7)

where $f_{1}(r)$ and $f(r)$ are chosen as

$f(r)=1-\frac{2Mr^{2}}{r^{3}+2M\ell^{2}};\hskip
14.45377ptf_{1}(r)=1-\frac{2M_{1}r^{2}}{r^{3}+2M_{1}\ell^{2}}$ (8)

The novelty in the metric above is that it yields a generalization of the
Hayward regular black hole mentioned in eq.(5) through the presence of a new
parameter $M_{1}$ in $f_{1}(r)$ which is different from the $M$ in $f(r)$. The
introduction of the parameter $M_{1}$ is in the same vein as was done in the
Damour-Solodukhin (DS) constructionDamour and Solodukhin (2007). In the DS
wormhole, alternatively known as the black hole foil, the metric deviates from
the Schwarzschild black hole through the appearance of a parameter $\lambda$
in the $g_{tt}$ component of the metric. Effectively, for DS wormhole we have
$g_{tt}=-\left(1-\frac{2M}{r}\right)$ and
$g_{rr}=\left(1-\frac{2M_{1}}{r}\right)^{-1}$ with $M\neq M_{1}$.

The mass parameter $M_{1}$, in the generalised Hayward case, extends the
metric to a broader family that generates diverse types of spacetimes which
includes the regular Hayward black hole. Further simplification can be
achieved by writing the parameters $M,M_{1},\ell$ in dimensionless form using
$\sigma=\frac{M_{1}}{M}$, $\kappa=\frac{\ell}{M}$ with the radial coordinate
$x=\frac{r}{M}$. The line element then becomes

$ds^{2}=-\Big{(}1-\frac{2\sigma
x^{2}}{x^{3}+2\sigma\kappa^{2}}\Big{)}dt^{2}+\frac{dr^{2}}{\Big{(}1-\frac{2x^{2}}{x^{3}+2\kappa^{2}}\Big{)}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (9)

The benefit of writing the metric in this fashion (i.e. via dimensionless
parameters) will be evident in the following sections where we discuss
numerous special cases. Thus, the final form which we use in this article is
the one given in eq.(9). Henceforth, we will use this line element in all
future discussions and special characteristics will be linked to different
values and ranges of $(\sigma,\kappa)$. Below, we indicate the different
aspects we elaborate upon in subsequent sections.

###  II.1 Matter energy-momentum

Obtaining the Einstein tensor components (in the frame basis) for the above-
stated line element and using them in the Einstein field equations
$G_{ij}=8\pi T_{ij}$, we write down $\rho$, $\tau$ and $p$ which comprise the
diagonal elements in the energy-momentum tensor $T_{ij}$, in the frame basis.
In order to get simplified expressions for the energy-momentum tensor
components, we define the following quantities,

$\displaystyle\begin{split}&h_{1\sigma}=x^{3}+2\sigma\kappa^{2}-2\sigma
x^{2};\hskip 36.135pth_{\sigma}=x^{3}+2\sigma\kappa^{2}\\\
&h_{1}=x^{3}+2\kappa^{2}-2x^{2};\hskip 45.53044pth=x^{3}+2\kappa^{2}\\\
&A=\frac{d}{dx}log\Big{(}\frac{h_{1\sigma}}{h_{\sigma}}\Big{)}\hskip
57.81621ptB=\frac{d}{dx}log\Big{(}\frac{h_{1}}{h}\Big{)}\end{split}$ (10)

Now we are in a position to write the $T_{ij}$ components (in the frame basis)
making use of the newly defined quantities
$h_{1\sigma},\,h_{\sigma},\,h_{1},\,h$ along with $A,B$. We obtain,

$\displaystyle\begin{split}&\rho=\frac{1}{8\pi
x^{2}M^{2}}\Big{(}1-\frac{h_{1}}{h}\Big{)}-\frac{1}{8\pi
M^{2}}\frac{h_{1}}{h}\frac{B}{x}\\\ &\tau=\frac{-1}{8\pi
x^{2}M^{2}}\Big{(}1-\frac{h_{1}}{h}\Big{)}+\frac{1}{8\pi
M^{2}}\frac{h_{1}}{h}\frac{A}{x}\\\ &p=\frac{1}{8\pi
x^{2}M^{2}}\frac{h_{1}}{h}\Big{[}\frac{1}{2}Ax+\frac{1}{4}A(A+B)x^{2}+\frac{1}{2}x^{2}\frac{dA}{dx}+\frac{1}{2}Bx\Big{]}\end{split}$
(11)

The expressions above will hold true for any $h_{1\sigma}$,$h_{\sigma}$,
$h_{1}$ and $h$, irrespective of their functional forms. Note also that
$f_{1}=\frac{h_{1\sigma}}{h_{\sigma}}$ and $f=\frac{h_{1}}{h}$.

It is easy to see that $h_{1}$, $h$ and hence $B$ and $\rho$ are independent
of the parameter $\sigma$. On the other hand, the $\sigma$ dependence arises
in $\tau$ and $p$ through the second term in $\tau$ and first three terms in
$p$.
Some observations about the energy-momentum tensor for different spacetimes
are in order. When we set $\sigma=1\rightarrow A=B$, we obtain the condition
for the Hayward regular black hole for which $\rho=-\tau$. The regular black
hole is characterized with a de-Sitter core which corresponds to
$\rho=-\tau=-p=\frac{3}{8\pi\kappa^{2}M^{2}}$ at $x=0$. As expected, at the
center of the regular black hole, only the parameter $\kappa$ remains which is
nothing but the redefined cosmological constant. The above condition satisfied
by $\rho,\tau$ and $p$ at $x=0$ holds true even for spacetimes with $A\neq
B\neq 0$. To further explore the status of the energy conditions, specific
parameter values have to be chosen. The above complicated expressions
(especially that for $p$) are much simplified in most of the cases we deal
with. This will be done for each spacetime, separately, in the following
section on classification.

###  II.2 Embedding diagrams

The embedding of two dimensional sections of the given spacetime, in a
Euclidean background, helps us visualise the shape of each geometry, as
associated with different parameter ranges. We begin by considering the
$t=constant$, $\theta=\pi/2$ spatial 2- D slice of the spacetime and embed it
in 3-D Euclidean space written in cylindrical coordinates. The 2-D spatial
slice of the generalized metric from eq.(7) gives

$ds^{2}=\frac{dr^{2}}{f(r)}+r^{2}d\phi^{2}$ (12)

On the other hand, in cylindrical coordinates the 3-D flat space metric
becomes

$\displaystyle ds^{2}=d\zeta^{2}+\zeta^{2}d\psi^{2}+dz^{2}$ (13)

Comparing the two metrics (with $z=z(r)$ in eq.(13), $\psi=\phi$, $\zeta=r$),
we arrive at the embedding functions as

$\displaystyle\zeta=r$ (14)
$\displaystyle\frac{dz}{dr}=\sqrt{\frac{1}{f(r)}-1}$ (15)

$z=z(r)$ needs to be integrated numerically for different values of
$(\sigma,\kappa)$ which will be discussed in the forthcoming sections. It is
to be noted that the embedding function depends only on the metric parameter
$\kappa$ and all geometries having same value of $\kappa$ will have similar
embedding diagram even though $\sigma$ values may be different.

###  II.3 Defining the tortoise coordinate

The radial coordinate ideally ranges from $[0,\infty)$ depending on the nature
of the spacetime. But in most scenarios, it is convenient to work with
tortoise coordinate $r_{*}$ defined as

$dr_{*}^{2}=\frac{dr^{2}}{f_{1}(r)f(r)}$ (16)

which ranges from $(-\infty,\infty)$. For wormholes, this mapping is
especially helpful as it covers the two universes on both sides of the throat
Visser (1995). Hence, in case of wormholes, $r=r_{0}$ (throat radius) is
mapped to $r_{*}=0$. On the other hand, for black holes, the point
$r_{*}=-\infty$ denotes the position of the horizon at some $r=$ constant
value. For most values of parameter $(\sigma,\kappa)$, eq.(16) is not
integrable analytically . The tortoise coordinate is thus calculated
numerically as an interpolating function using Mathematica. The conditions
imposed for the numerical computation of the tortoise coordinate are
elaborated in the Appendix.

###  II.4 Classifying spacetimes: black holes and wormholes

In our study we have been able to use a single line element which, for
different parameter ranges, may represent either black holes or wormholes. The
essential criteria for classification into either of the two broad categories
mentioned above will therefore have to be spelt out in full detail. In simple
terms, how do we check, for some parameter values, whether the geometry is a
black hole or a wormhole?
To this end, we begin by finding the roots of the metric components $g_{tt}$
and $g^{rr}$. A black hole will have atleast one $r=constant$ null
hypersurface which represents its event horizon. It may or may not harbour a
singularity inside the horizon–the former being a singular black hole, while
the latter represents a regular one. On the other hand, for a wormhole, we
need a timelike hypersurface at some $r=r_{0}$ which will act as the throat
and, around which, violation of the energy conditions will be unavoidable.
Simultaneously, if the spacetime is regular for the entire range of $r$ namely
$[r_{0},\infty)$ then it can be classified as a wormhole. The violation of
energy condition ensures the divergence of the congruence of geodesics which
is the typical behavior associated with wormhole spacetimes.
Certain parameter choices might result in the spacetime having both a null and
a timelike hypersurface. Their relative positions and the behavior of the
other conditions will then be used to set up the classifying criteria.

##  III Classification: cases for various parameter ranges and features

Figure 1: Parameter space ($\sigma$, $\kappa$) showing different spacetimes
for specific, different ranges of the parameters.

We will now explore the distinct geometries arising from specific ranges of
the parameters $\sigma$ and $\kappa$. The Fig.(1) shows the six different
spacetime classes obtained from our generalized metric. Along with checking
the energy conditions satisfied by each spacetime, we will also study their
geometric properties and embedding diagrams.

###  III.1 Case I: $\sigma=1$, $\kappa=0$ (Schwarzschild black hole)

$ds^{2}=-\Big{(}1-\frac{2M}{r}\Big{)}dt^{2}+\frac{dr^{2}}{\Big{(}1-\frac{2M}{r}\Big{)}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (17)

$\bullet$ Nature of spacetime: This choice simplifies the parameters as
$M=M_{1}$ and $\ell=0$. The case then corresponds to the most trivial
scenario, that of the Schwarzschild black hole with
$f_{1}(r)=f(r)=1-\frac{2M}{r}$. The event horizon is at $r=2M$ which is a null
hypersurface as well as the infinite redshift surface.
$\bullet$ Energy conditions: The components of the energy-momentum tensor
simply become zero since the Schwarzschild black hole is a vacuum solution in
GR. Hence, one need not worry about the energy conditions as they are
trivially satisfied.

###  III.2 Case II: $\sigma=\kappa=0$ (Schwarzschild wormhole)

$ds^{2}=-dt^{2}+\frac{dr^{2}}{\Big{(}1-\frac{2M}{r}\Big{)}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (18)

$\bullet$ Nature of spacetime: The component $g_{tt}=-1$ makes the spacetime
ultra-static and devoid of any horizon. The root of the metric function
$g^{rr}$ gives $r=2M$ as a timelike hypersurface which acts as a throat to the
corresponding traversable wormhole. This results in the first wormhole
geometry we encounter in the parameter space. The radial coordinate $r$ ranges
from $[2M,\infty)$ which denotes one copy of the universe while the tortoise
coordinate $r_{*}$ covers both sides of the throat ranging from
$(-\infty,\infty)$. The Ricci scalar for this wormhole is trivially 0 while
the Kretschmann scalar remains finite for the entire range of $r$ indicating
the absence of any singularity in the spacetime.
$\bullet$ Energy conditions: The energy-momentum tensor components mentioned
in eq.(11) become, at the throat $r=2M$, $\rho(r=2M)=0$,
$\tau(r=2M)=-1/\left(8\pi(4M^{2})\right)$ and
$p(r=2M)=1/\left(8\pi(8M^{2})\right)$. Hence, the weak energy condition is
violated at the throat, indicating the divergence of a congruence of
geodesics.
$\bullet$ Embedding diagram: The embedding function $(z(r),\zeta(r))$ is
obtained by integrating eq.(15). This is the well-known Flamm’s paraboloid.
$z(r)$ ($z(\zeta)$) is plotted in Fig. 2.

Figure 2: Plot of $z(\zeta)$ showing embedding of the $t=constant$,
$\theta=\pi/2$ slice of the Schwarzschild wormhole.

###  III.3 Case III: $0<\sigma<1,\kappa=0$ (Damour–Solodukhin wormhole)

$ds^{2}=-\Big{(}1-\frac{2\sigma}{x}\Big{)}dt^{2}+\frac{dr^{2}}{\Big{(}1-\frac{2}{x}\Big{)}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (19)

$\bullet$ Nature of spacetime: Proceeding as before, we first determine the
roots of the metric components $g_{tt}$ and $g^{rr}$ which gives $r=2\sigma M$
and $r=2M$ respectively. While the $r=2M$ surface is timelike, the
hypersurface corresponding to $r=2\sigma M$ is null. In order to ascertain if
the spacetime harbours any singularity, we calculate the curvature scalars and
check their behaviour. It is observed that the Ricci and Kretschmann scalars
remain finite at $r=2M$ but diverge only at $r=2\sigma M<2M$.
$\bullet$ Energy conditions: It is found that the $\rho+\tau<0$ at $r=2M$
which indicates violation of energy condition. The spacetime is thus a
wormhole. The radial coordinate ranges from $[2M,\infty)$. The spacetime is
regular everywhere in this domain of $r$. With proper choice of coordinates,
this wormhole metric can be identified as the Damour-Solodukhin wormhole which
is well-studied in the literature Damour and Solodukhin (2007).
$\bullet$ Embedding diagram:  Since the embedding will be independent of value
of $\sigma$, the embedding diagram will be identical to the one shown in
Fig.(2) for the previous case (the value of $\kappa$ here, is still 0).

###  III.4 Case IV: $\sigma=0,\kappa\neq 0$ (Hayward wormhole)

$ds^{2}=-dt^{2}+\frac{dr^{2}}{\Big{(}1-\frac{2x^{2}}{x^{3}+2\kappa^{2}}\Big{)}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (20)

$\bullet$ Nature of spacetime: The metric tensor component $g_{tt}=-1$
indicates the ultra-static nature of this spacetime. Thus, the spacetime does
not harbour any null surface and hence no horizon. The possibility of the
spacetime being a black hole is thus ruled out completely. We then find the
roots associated to $g^{rr}=0$ which exist provided
$\kappa\leq\frac{4}{3\sqrt{3}}$. The demand for the $g^{rr}$ component to have
atleast one real, positive root sets the range for the parameter $\kappa$.
Although calculating the discriminant of the cubic equation and setting it
greater than zero gives the required restriction on $\kappa$, this condition
simply refers to the equation having three distinct real roots. If the
discriminant is less than 0, then the equation being cubic, will have two
complex conjugate and one real root. The single real root might be positive or
negative. So a better way to formulate the limit on $\kappa$ is by studying
the behaviour of the cubic function in the range $x=[0,\infty)$. To perform
this for any general $\kappa$, we first find its behaviour near $x=0$ and
$x\rightarrow\infty$. Both $f(0)$ and $f(\infty)$ come out to be positive
quantities. Now, taking derivative we get the critical points as $x=0,4/3$. At
$x=0$, the function is a maximum and at $x=4/3$, it is a minimum. If the value
of the function $f(4/3)$ is negative, then the function is bound to intersect
the x-axis at two points. Also, from Descartes rule of sign change we know the
function can have two or zero real positive roots. Therefore, if $f(4/3)<0$,
the function must have two positive real roots as desired. Simple algebra
shows that the above condition reduces to $\kappa\leq\frac{4}{3\sqrt{3}}$ in
order for the function to have atleast one real positive root.
The larger root of $g^{rr}$ say $r=r_{0}$, which is also a timelike
hypersurface, gives the position of the throat making the spacetime a
wormhole. The scalars, Ricci and Kretschmann, remain finite for the entire
range of $[r_{0},\infty)$ making the spacetime regular everywhere. This
wormhole geometry can be called a ‘Hayward wormhole’ because of its similarity
with the Hayward regular black hole metric with the only difference being its
ultra-static nature.
$\bullet$ Energy conditions: In order to ensure the wormhole nature of the
spacetime, divergence of a geodesic congruence is essential, at least at the
throat, for all values of $\kappa$. The energy-momentum tensor components of
eq.(11) for this case becomes

$\begin{split}\rho(x)=\frac{12\kappa^{2}}{8\pi
M^{2}(x^{3}+2\kappa^{2})^{2}}\\\
\rho(x)+\tau(x)=\frac{-2(x^{3}-4\kappa^{2})}{8\pi
M^{2}(x^{3}+2\kappa^{2})^{2}}\\\ \rho(x)+p(x)=\frac{(x^{3}+8\kappa^{2})}{8\pi
M^{2}(x^{3}+2\kappa^{2})^{2}}\end{split}$ (21)

where we use the dimensionless radial coordinate $x=r/M$. The WEC will be
satisfied provided the quantity $\rho(x)+\tau(x)$ is always positive. At the
throat of the wormhole $x=x_{0}$, $\rho+\tau=-\frac{(3x_{0}-4)}{8\pi
M^{2}(x_{0}^{3}+2\kappa^{2})}$. Now from the previous discussion about finding
the roots of $f(r)$, we know that the larger root of $f(r)$ (i.e. the throat)
to be greater than $x=4/3$ which indicates $\rho+\tau$ to be negative for all
$\kappa$. A specific case arises when $\kappa=\frac{4}{3\sqrt{3}}$ so that at
throat $x_{0}=4/3$, $\rho+\tau$ is 0. The violation then occurs a bit away
from throat where $\rho+\tau$ is negative. Hence, WEC is always violated
atleast at the location of the throat or near it, as is expected in a wormhole
geometry.

$\bullet$ Embedding diagram: Once again, to construct the embedding diagram of
the $t=$ constant hypersurface for this wormhole geometry, we integrate
eq.(15) for different values of $\kappa$ and plot them as shown in Fig.(3).
The throat radius of the wormholes also change with $\kappa$.

Figure 3: Plot of $z(\zeta)$ showing embedding of the $t=constant$,
$\theta=\pi/2$ slice of the Hayward wormhole for different $\kappa$ values.

One might wonder about the case when $\kappa>\frac{4}{3\sqrt{3}}$. In the
absence of any real positive root to the metric component $g^{rr}$, the
spacetime will be regular and can be smoothly extended to $r=0$. The
corresponding embedding diagram obtained by integrating eq.(15) for
$\kappa>\frac{4}{3\sqrt{3}}$ indicates a non-trivial behaviour as shown in
Fig.(4). The embedding function $z(\zeta)$, for such parameter ranges,
undergoes a change in gradient which is unique and needs to be explored
through detailed future studies (see Fig.(4)). At present, we will only be
interested in cases with $\kappa\leq\frac{4}{3\sqrt{3}}$ where there is a
well-defined horizon or a throat.

Figure 4: The embedding diagram for Case IV spacetime with
$\kappa>\frac{4}{3\sqrt{3}}$ where the function can be smoothly extended to
$r=0$.

###  III.5 Case V: $\sigma=1,\kappa\neq 0$ (regular Hayward black hole)

$ds^{2}=-\Big{(}1-\frac{2x^{2}}{x^{3}+2\kappa^{2}}\Big{)}dt^{2}+\frac{dr^{2}}{\Big{(}1-\frac{2x^{2}}{x^{3}+2\kappa^{2}}\Big{)}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (22)

$\bullet$ Nature of spacetime: We notice that here $f_{1}(r)=f(r)$ i.e. the
two mass functions are equal. For $f(r)$ to have roots,
$\kappa\leq\frac{4}{3\sqrt{3}}$. Equating $g_{tt}=0$ we get the larger root
$r=r_{h}(x=x_{h})$ as a null hypersurface as well as the infinite redshift
surface. The point $r=r_{h}$ gives the location of the horizon in this
spacetime –thus it is a black hole. Extending the radial coordinate $r$ beyond
the event horizon to $r=0$ shows that the curvature scalars remain finite even
at $r=0$ indicating the spacetime to be a regular black hole. One notes that
this spacetime is just the well-known Hayward regular black hole proposed in
eq.(5).
$\bullet$ Energy conditions: Calculation of the energy-momentum tensor
components show the energy conditions to be satisfied for this regular black
hole geometry.
$\bullet$ Embedding diagram: We once again integrate eq.(15) for these
parameter values and plot the embedding function for different values of
$\kappa$ as shown in Fig.(5). Similar to the previous case, the position of
horizon changes with $\kappa$.

Figure 5: Plot of $z(\zeta)$ showing embedding of the $t=constant$,
$\theta=\pi/2$ slice of the Hayward regular black hole.

###  III.6 Case VI: $0<\sigma<1$, $0<\kappa\leq\frac{4}{3\sqrt{3}}$
(Hayward–Damour–Solodukhin wormhole)

$ds^{2}=-\Big{(}1-\frac{2\sigma
x^{2}}{x^{3}+2\sigma\kappa^{2}}\Big{)}dt^{2}+\frac{dr^{2}}{\Big{(}1-\frac{2x^{2}}{x^{3}+2\kappa^{2}}\Big{)}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (23)

$\bullet$ Nature of spacetime: This is the most general version of our
spacetime with $\sigma$ and $\kappa$ taking all possible combination of values
in the specified range of parameter space. We begin by finding the roots of
$f(r)$ (which we denote as $r=r_{0}$) and $f_{1}(r)$ (denoted as $r=r_{h}$).
As discussed earlier, we will classify the spacetime depending on the position
of $r_{h}$, $r_{0}$ along with the behaviour of energy conditions and
curvature scalars. If $r_{0}>r_{h}$, then at $r=r_{0}$, $g_{tt}$ remains
positive indicating the hypersurface to be timelike.
$\bullet$ Energy conditions: In eq.(11), we find the quantities $\tau$ and $p$
to be inversely dependent on
$h_{1\sigma}=x^{3}-2x^{2}\sigma+2\kappa^{2}\sigma$, which is related to the
$g_{tt}$ component of the metric. Hence, they will not be well-behaved at the
roots of $f_{1}(r)$ i.e. at $r=r_{h}$, which implies that $r=r_{h}$ cannot be
considered as a part of the spacetime. Thus, the possibility of the spacetime
being a black hole is eliminated.
Then we proceed to investigate the behaviour of the energy conditions for
different parameter ranges. The energy density $\rho(x)$ is always positive
and hence will satisfy the condition $\rho\geq 0$. Next, we probe the
behaviour of $\rho+\tau$ which has a form

$\rho+\tau=\frac{1}{8\pi M^{2}}\frac{h_{1}}{h}\left(\frac{A-B}{x}\right)$ (24)

The location of the throat is $h_{1}=0$. One can easily show (using the
expressions for A and B given earlier) that, at the throat,
$\rho+\tau=\frac{-1}{8\pi
M^{2}}\frac{{h_{1}}^{\prime}}{x_{0}h}=\frac{-3(x_{0}-4/3)}{8\pi
M^{2}(x_{0}^{3}+2\kappa^{2})}$. Now, following the discussion in Case IV on
finding the condition $\kappa\leq\frac{4}{3\sqrt{3}}$, we see the two positive
roots of $f(r)$ to lie on both sides of $x=4/3$. The throat $x=x_{0}$
corresponds always to the larger root of $f(r)$ making $x_{0}>\frac{4}{3}$.
Although for $\kappa=\frac{4}{3\sqrt{3}}$, at $x_{0}=4/3$, $\rho+\tau$ becomes
0, it stays negative away from the throat. Thus $\rho+\tau$ is negative for
all parameters leading to violation of energy conditions for the wormhole, as
expected.

$\bullet$ Embedding diagram: The embedding diagram for this class of wormholes
will be similar to the ones shown in Fig.(3) as they only depend on $\kappa$.
The curvature scalars, Ricci and Kretschmann, have also been found to be well-
behaved in the entire range of the radial coordinate.

For this general case, another possibility opens up, that with
$\kappa>\frac{4}{3\sqrt{3}}$. As discussed earlier, $g^{rr}$ will not have any
roots and hence the possibility of the existence of a throat is absent. But
what about the roots of $g_{tt}=0$? We have explicitly checked that $f_{1}(r)$
too does not have any real, positive root for any value of $\sigma$
corresponding to $\kappa>\frac{4}{3\sqrt{3}}$. Analytically, we can calculate
the condition for $f_{1}(r)$ to have real, positive roots. The condition one
will arrive at is- $\sigma\leq 0$ or $\kappa\leq\frac{4\sigma}{3\sqrt{3}}$.
Since both the conditions are not satisfied for this parameter range,
$f_{1}(r)$ will not have any real, positive root for
$\kappa>\frac{4}{3\sqrt{3}}$. Hence, in that parameter range, the spacetime is
simply regular everywhere and beyond our area of interest. The embedding
diagrams will be similar to Fig.(4) irrespective of the value of $\sigma$.

We can further extend the parameter space and consider spacetimes with
$\sigma>1,\kappa\geq 0$ which will result in a naked singularity at $r=r_{h}$
(root of $f_{1}(r)$). Interestingly, the energy conditions also get violated
near $r=r_{h}$ providing an example of a scenario with singularity and energy
condition violating matter. This corroborates the fact that spacetime
singularities are indeed focal points of geodesic congruence but focusing does
not necessarily imply a spacetime singularity. One can also consider
$\sigma<0$ parameter values which corresponds to negative values of parameter
$M_{1}$. Such $\sigma<0$ spacetimes are wormhole geometries irrespective of
the value of $\kappa$. The $\sigma>1$ and $\sigma<0$ parameter cases will not
be considered further in this paper. We hope to discuss the related issues to
such spacetimes in our future explorations.

##  IV Scalar wave propagation and effective potentials

We have obtained six physically relevant spacetimes that range from singular
and regular black holes to various wormhole geometries, as discussed in the
previous section. We will focus on this broad class of six metrics with
positive parameter ($\sigma$, $\kappa$) values. The stability of these
spacetimes will be probed by calculating the quasi-normal modes associated
with the propagation of a massless test scalar field. Quasi-normal modes are
complex characteristic frequencies excited in a perturbed object, through
which the object looses energy to attain equilibrium. These modes were first
studied for black holes by Vishveshwara in the 1970s Vishveshwara (1970a, b).
Since then they have been investigated extensively in the literature Kokkotas
and Schmidt (1999); Maggiore (2018); Cardoso (2004); Konoplya and Zhidenko
(2011); Berti et al. (2009) for various spacetimes and are especially relevant
in the context of gravitational wave observations Ferrari and Gualtieri
(2008); Berti et al. (2006); Abbott et al. (2016b); Barausse et al. (2014);
Ghosh et al. (2021). Apart from the stability analysis, the QNMs are of use in
parameter estimation of diverse astrophysical objects which arise as remnants
in various scenariosBerti et al. (2007a); Gossan et al. (2012); Carullo et al.
(2018); Isi et al. (2019); Christensen and Meyer (2022). They also help us in
testing the validity of the underlying theory of gravity in which the geometry
appears as a solutionAbbott et al. (2016b); Konoplya and Zhidenko (2016);
Bhattacharyya and Shankaranarayanan (2017); Dreyer et al. (2004) . QNMs are
relevant in proving the existence of event horizons in black hole spacetimes
Cardoso et al. (2016a); Cardoso and Pani (2019) and in determining the shape
of various wormholes Konoplya (2018); Dutta Roy et al. (2020); Roy (2022). In
the same way, for our general metric, we aim to utilize the scalar QNMs to be
found in each of the above background spacetimes, as a tool for characterising
the associated geometry.
We begin by recalling the Klein-Gordon equation in a given curved geometry,
i.e.

$\Box\Phi=0$ (25)

As our background spacetime is spherically symmetric and static (Eqn. (7)), we
use the following ansatz to decompose $\Phi$ in terms of spherical harmonics,
where the indices of $Y(\theta,\phi)$ have been suppressed for simplicity. For
stability analysis we use the radial coordinate $r$ and assume $M=1$. Thus,
with

$\displaystyle\Phi(t,r,\theta,\phi)=Y(\theta,\phi)\frac{u(r)e^{-i\omega
t}}{r}.$ (26)

and the general metric ansatz (eq.(7)) the Klein-Gordon equation separates.
The radial equation turns out to be a Schrödinger-like equation, written using
the tortoise coordinate $r_{*}$, i.e.,

$\displaystyle\frac{d^{2}u}{dr_{*}^{2}}+[\omega^{2}-V_{eff}]u=0$ (27)

where

$\displaystyle
V_{eff}(r)=\frac{m(m+1)f_{1}(r)}{r^{2}}+\frac{f(r)f_{1}^{\prime}(r)+f^{\prime}(r)f_{1}(r)}{2r}$
(28)

is the effective potential and $m$ is the azimuthal number arising from the
separation of variables. The characteristics of each member spacetime in our
family of six, will be reflected in the corresponding effective potentials. In
this section, we will try to analyse $V_{eff}$ and plot it as a function of
$r$ for the black hole cases and as a function of $r_{*}$, for the wormhole
geometries. Use of the tortoise coordinate for wormholes helps in visualizing
the wormhole as a two-sheeted geometry with a connecting throat.

(a) $\kappa=0,\sigma=1$ (Case I)

(b) $\kappa=0,\sigma=0$ (Case II)

(c) $\kappa=0,\sigma=0.1$ (Case III)

(d) $\kappa=0,\sigma=0.7$ (Case III)

(e) $\kappa=0,\sigma=0.9$ (Case III)

(f) $\kappa=0,\sigma=0.99$ (Case III)

(g) $\kappa=0.1,\sigma=0$ (Case IV)

(h) $\kappa=0.7,\sigma=0$ (Case IV)

Figure 6: Variation of $V_{eff}$ for different parameter values of
$(\sigma,\kappa)$ and angular momentum number $m$ (Case I-IV). We observe that
the double barrier nature of the potential depends on the value of $\sigma$.
Same $\kappa$ wormholes will have same throat radius but the tortoise
coordinate will be different because of its dependence on $\sigma$.

(a) $\kappa=0.1,\sigma=1$ (Case V)

(b) $\kappa=0.7,\sigma=1$ (Case V)

(c) $\kappa=0.1,\sigma=0.1$ (Case VI)

(d) $\kappa=0.1,\sigma=0.9$ (Case VI)

(e) $\kappa=0.7,\sigma=0.1$ (Case VI)

(f) $\kappa=0.7,\sigma=0.75$ (Case VI)

Figure 7: Plots of effective potential $V_{eff}$ for Case V and VI. Once again
the double barrier potential is observed for values of $\sigma$ close to 1.

From the multiple plots shown in Fig.(6) and (7), it is evident that the
generalized metric harbours both single and double barrier potentials
depending on the choice of the parameters $(\sigma,\kappa)$. The double peak
potential is observed in wormholes belonging to Case III and VI as the value
of parameter $\sigma$ approaches (but is never equal to) one. For the Damour-
Solodukhin wormhole, $\sigma\rightarrow 1$, and it can be thought as two
copies of the Schwarzschild black hole potential. This feature has been
explored in other works as well Bueno et al. (2018).
To get a better understanding of the nature of potential especially for the
wormhole geometries, we test whether the potential has a maximum or a minimum
at the throat i.e. $r_{*}=0$. If $r_{*}=0$ is a minimum, then the potential is
atleast a double-barrier. We have, at $r_{*}=0(f(r)=0)$, and with $x=r/M$,

$\frac{d^{2}V_{eff}}{dr_{*}^{2}}\Big{|}_{f=0}=\frac{f^{\prime}(x)f_{1}(x)}{2M^{2}}\Big{[}f_{1}(x)\Big{(}\frac{-f^{\prime}(x)}{2x^{2}}+\frac{f^{\prime\prime}(x)}{2x}-\frac{2m(m+1)}{x^{3}}\Big{)}+f_{1}^{\prime}(x)\Big{(}\frac{f^{\prime}(x)}{x}+\frac{m(m+1)}{x^{2}}\Big{)}\Big{]}$
(29)

For the Case III wormhole with $\kappa=0$, the throat lies at $x=2$ and is a
minimum if, $V^{\prime\prime}_{eff}>0$. Using the expressions for the metric
functions with arbitrary $\sigma$ ($0<\sigma<1$) and $\kappa=0$ we obtain, for
a minimum, the requirement $\sigma>\frac{3+4m(m+1)}{5+6m(m+1)}$. Note that for
$m=0$ one can have a minimum as long as $\sigma>0.6$. For $m=1$,
$\sigma<0.647$, the potential remains a single barrier with a maximum at the
throat, whereas for $\sigma>0.647$ we have a minimum at the throat and a
double barrier. This generic behaviour, i.e. the existence of a critical
$\sigma$, holds true for all $m\geq 0$. Different values of $m$ will have
different $\sigma$ at which the transition occurs from single to double
barrier potential. Similar analysis holds for general $(\sigma,\kappa)$ values
too, but since the analytic expressions for the throat radius are cumbersome,
we have verified the behaviour numerically (not stated here) for each
parameter set. For black hole cases, the maximum will occur at some point away
from the horizon and can be obtained by finding the critical points of
$V_{eff}(r)$. Note that the condition in (29) is valid for arbitrary $f(r)$,
$f_{1}(r)$ and may be used to know (without writing the scalar wave equation)
if a geometry will exhibit a single or a double barrier effective potential
for scalar wave propagation.

Table 1 summarizes the properties of the spacetimes belonging to each
parameter set.

| Case I | Case II | Case III | Case IV | Case V | Case VI
---|---|---|---|---|---|---
Type | BH | WH | WH | WH | Regular BH | WH
Violation of EC | ✗ | ✓ | ✓ | ✓ | ✗ | ✓
Singularity | ✓ | ✗ | ✗ | ✗ | ✗ | ✗
Single barrier potential | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Double barrier potential | ✗ | ✗ | ✓ | ✗ | ✗ | ✓
Table 1:  Summary of various properties of spacetimes for different parameter
ranges of $(\sigma,\kappa)$.

##  V Stability analysis: quasi-normal modes and echoes

After obtaining the form of the effective potential, we calculate the
corresponding scalar quasi-normal modes associated with each case. The
calculation of the QNM frequencies can be executed using multiple techniques
available in the literature. One of the most convenient analytical techniques
for calculating QNMs is the WKB method initially discussed in Schutz and Will
(1985); Iyer and Will (1987). Later, a number of articles extended the
technique to higher orders, using Pade corrections, to achieve better
convergence Konoplya (2003); Konoplya et al. (2019); Matyjasek and Opala
(2017). In our spacetimes here, the WKB method is effective in obtaining
analytical results for potentials with a single barrier. Hence, QNMs cannot
quite be obtained using the WKB method for some specific wormhole geometries
where the effective perturbation potential is a double barrier.

To handle the double barrier cases and to verify the results obtained from the
WKB method for single barrier potentials, we implement numerical techniques
for calculating QNMs Franzin et al. (2022); Konoplya and Zhidenko (2011). We
will make use of the direct integration method and Prony extraction scheme to
obtain the QNM frequencies numerically. Let us go through the essence of the
two methods one-by-one. Direct integration or the shooting method was proposed
by Chandrasekhar and Detweiler Chandrasekhar (1985) where the differential
equation eq.(27), written in terms of the radial coordinate $r$, is integrated
numerically. For black holes, purely outgoing boundary conditions are
implemented at spatial infinity and at the horizon. The integration is then
done from the horizon to a finite intermediate point where the solution and
its derivative is matched with the solution obtained at infinity. The matching
condition gives discrete values of $\omega$ which are the QNMs. Numerical
errors may occur in the method, as the integration has to be done a small
distance away from horizon to ensure smooth behaviour of all functions. Also,
spatial infinity is taken at a finite value. These errors due to numerical
limitations can be minimised by choosing suitable precision and accuracy for
the integration scheme. The robustness of the method should be tested by
slightly changing the matching point but keeping the result unchanged. On the
other hand, for wormhole geometries, ideally, the integration needs to be done
from $r_{*}\rightarrow-\infty$ to $\infty$. All wormholes, arising from our
general metric, are symmetric about their throat at $r_{*}=0$ and will have
even and odd classes of solutions for the perturbation equation. The
integration is thus performed on positive $r_{*}$, i.e., from $r=r_{0}$
($r_{*}=0$) to $\infty$ and the condition on the solution at the throat
decides whether it is symmetric or antisymmetric, on reflection (i.e.
$r_{*}\rightarrow-r_{*}$). For our case, this method is especially suitable,
as implementing this technique does not require the evaluation of the tortoise
coordinate. Additional numerical errors might be introduced due to $r_{*}$, as
it can only be obtained numerically for our metric in the form of an
interpolation function. Avoiding the use of $r_{*}$ in direct integration
removes one possible source of numerical error.
The Prony extraction scheme Konoplya and Zhidenko (2011); Berti et al. (2007b)
begins by first obtaining the time domain (TD) evolution of an initial
Gaussian pulse in the background of our spacetime. Instead of using eq.(27),
we keep the time dependence so that the equation becomes

$\displaystyle\frac{\partial^{2}\psi}{\partial
t^{2}}-\frac{\partial^{2}\psi}{\partial r_{*}^{2}}+V_{eff}(r_{*})\psi=0.$ (30)

The wave equation for $\psi(r_{*},t)$ is recast using light cone coordinates
$(u,v)$ such that $(du=dt-dr_{*},dv=dt+dr_{*})$. The appropriate
discretization scheme is discussed in Konoplya and Zhidenko (2011); Gundlach
et al. (1994). With proper initial conditions defined on the null grid, the
wave equation is integrated to obtain the time domain profile. A detailed
study of the ringdown profile indicates the presence of QNM frequencies while
the damped signal will establish the stability of the corresponding spacetime
under the given perturbation. The signal is fitted with a superposition of
damped sinusoidal frequencies, in the Prony extraction method and the most
dominant quasi-normal mode is thereby extracted.
Apart from proving stability, the time domain signal plays crucial role in
studying the behaviour of a wormhole geometry characterized especially with a
double barrier effective perturbation potential. The double peaks cause
multiple reflection of the incident wave producing a phenomenon known as
‘echoes’. After the initial prompt ringdown, the signal gets dominated by the
repetitive echoes of decreasing amplitude Cardoso et al. (2016a); Maggio et
al. (2020); Bueno et al. (2018). The quasinormal modes of the wormhole starts
to influence the ringdown only at very late times when the echoes have died
down. These wormholes thus possess long-lived dominant modes that come into
play at late times. Such a distinctive feature of the ringdown is only shown
by a double potential barrier. Through our work we will observe how the
ringdown pattern changes for different spacetimes. We also note the
differences in the approach to stability for black holes and various wormhole
geometries. The phenomenon of ‘echoes’ have been widely studied for various
classes of spacetimes Mark et al. (2017); Conklin et al. (2018); Dutta Roy et
al. (2020); Roy (2022); Yang et al. (2021); Fang et al. (2021); Dey et al.
(2020) and are prime indicators of any new physics close to the event horizon
of black holesAbedi et al. (2017); Cardoso et al. (2016b); Uchikata et al.
(2019); Nakano et al. (2017); Oshita et al. (2020a); Wang and Afshordi (2018);
Cardoso et al. (2019); Price and Khanna (2017); Oshita et al. (2020b). They
also play a significant role as a distinguishing feature of exotic compact
objects (ECO) in the context of black hole mimicker analysis Cardoso and Pani
(2019); Bronnikov and Konoplya (2020); Churilova and Stuchlik (2020).
We now move on to studying the ringdown profile and obtain the scalar QNMs for
each case separately.

###  V.1 Case I: $\sigma=1$, $\kappa=0$(Schwarzschild black hole)

For the Schwarzschild black hole, the time domain signal (see Fig.(8)) is
dominated by the quasi-normal modes of the black hole Chandrasekhar (1985);
Nollert and Schmidt (1992); Nollert (1993) which are also the modes excited
from the perturbation of the photon ring Cardoso et al. (2009).

Figure 8: Typical Schwarzschild black hole ringdown profile as observed at
$r_{*}=30$ with initial Gaussian profile $\psi=e^{-\frac{(u-10)^{2}}{100}}$
and grid spacing $h=0.5$.

###  V.2 Case II: $\sigma=0$, $\kappa=0$ (Schwarzschild wormhole)

The wormhole with these parameters possesses quasinormal modes that will be
function of angular momentum number $m$ and the mass parameter $M$. Taking the
eikonal limit (large $m$),
$\omega\approx\Big{(}\frac{m}{2}\sqrt{1-\frac{i}{\sqrt{2}m}}\Big{)}M^{-1}$ and
hence the imaginary component tends to $Im(\omega)\rightarrow
0.17677\,M^{-1}$, as visible in the plot of Fig.(9).

(a) Variation of Re($\omega$) (in $M^{-1}$) with $m$.

(b) Variation of Im($\omega$) (in $M^{-1}$) with $m$.

Figure 9: QNM variation for Case II wormhole with angular momentum $m$. The
Im($\omega$) decreases with increasing $m$ as observed in most wormhole
geometries.

###  V.3  Case III: $0<\sigma<1,\kappa=0$ (Damour-Solodukhin wormhole)

The Damour-Solodukhin wormhole deserves attention since both single and double
potential barriers occur, depending on the value of $\sigma$. When
$\sigma=0.9$, the two potential peaks are well separated and the profile is
dominated by the wormhole QNM just after the initial ringdown has ended (see
Fig.(10)). For even higher $\sigma$ values, say $\sigma=0.9999$, the potential
peak separation widens further and leads to significant change in the ringdown
behaviour. The initial ringdown for $\sigma=0.9999$ wormhole occurs at
$t\approx 50-90$ and resembles that of the Schwarzschild black hole but is
soon followed by well-defined echoes. Extracting the frequency, using Prony
method, from $t\approx 50-90$ gives a QNM of the order $0.292267-i0.122944$
$M^{-1}$ which is very close to value fof the Schwarzschild black hole
dominant mode. The echoes start appearing after the prompt ringdown from
$t\approx 120$. The echo signal is dominated by modes of the order
$0.285289-i0.03223$ $M^{-1}$ for $t\approx 130-180$ and $0.2297-i0.00403$
$M^{-1}$ for $t\approx 350-400$. Expectedly, modes with long damping period
dominate the signal at late times. These modes are very close to higher
overtone QNMs of the wormhole as also noted in Bueno et al. (2018). The
fundamental QNM of the wormhole is $0.09205-i10^{-6}$ $M^{-1}$ obtained using
DI. The small magnitude of Im($\omega$) of the fundamental mode indicates that
at late times the signal is dominated entirely by the fundamental wormhole
QNM. Such behavior has been observed in other black hole/ wormhole transition
spacetimes as wellCardoso et al. (2016a); Churilova and Stuchlik (2020); Yang
et al. (2021).

(a) $\sigma=0.9$

(b) $\sigma=0.9999$

Figure 10: TD profile for Damour-Solodukhin wormhole with $\sigma$ values close to 1 observed at $r_{*}=30$ with initial Gaussian profile $\psi=e^{-\frac{(u-10)^{2}}{100}}$, grid spacing $h=0.5$ and $m=1$. m |  $\omega$ M (DI)
---|---
1 | 0.32013 -i 0.012934
2 | 0.484687 -i 0.003451
3 | 0.6478119 -i 0.000712
4 | 0.8088245 -i 0.0001199
5 | 0.968591 -i 0.000018068
Table 2:  Fundamental $\omega_{QNM}$ values for Case III wormhole with
$\sigma=0.9$. DI is used as WKB cannot be applied due to the double barrier
nature of the potential. The Im($\omega$) decreases rapidly with increasing
$m$, indicating the large $m$ modes to be long-lived and dominating at late
times.

Note the time domain profile for wormholes with well-separated double barrier
potentials needs to be computed for long time if one intends to extract QNM
frequencies of the wormhole from the signal. This process is computationally
exhaustive as high precision is required for computing the QNM frequencies
with such low damping rate. The DI method thus becomes more suitable for these
wormholes. To verify the authenticity of the DI result, we have checked its
convergence over wide range of matching points. The QNM values for
$\sigma=0.9$ is shown in Table 2 that indicate the low damping rate of the
modes.

$\sigma$ |  $\omega$ M (WKB) |  $\omega$ M (DI)
---|---|---
0.01 | 0.751719 -i 0.177449 | 0.7516804 -i 0.1774696
0.1 | 0.718926 -i 0.165179 | 0.718891 -i 0.165198
0.2 | 0.680852 -i 0.150609 | 0.6808197 -i 0.1506254
0.3 | 0.640844 -i 0.1348974 | 0.64081047 -i 0.134899
0.4 | 0.598844 -i 0.11791406 | 0.598592 -i 0.11782059
0.5 | 0.554175 -i 0.0986858 | 0.5538142 -i 0.0992029
0.6 | 0.509473 -i 0.0676018 | 0.505933 -i 0.078929
0.7 | - | 0.453844 -i 0.0571522
0.8 | - | 0.394867 -i 0.0344919
0.9 | - | 0.3201304 -i 0.0129345
0.95 | - | 0.2655024 -i 0.0043492
0.99 | - | 0.182275 -i 0.0003563
Table 3:  Fundamental $\omega_{QNM}$ for different Damour-Solodukhin wormholes
(Case III) with $m=1$. The double barrier nature of potential becomes
prominent as $\sigma\rightarrow 1$ due to which WKB method cannot be applied.

The QNMs will also depend on parameter $\sigma$, for any particular $m$ value,
which are shown in Table 3. To visualize the dependence analytically, we have
obtained the fitting functions for $\omega_{r}$ and $\omega_{i}$ as a function
of $\sigma$ using the NonlinearModelFit feature of Mathematica and are denoted
within the plots of Fig.(11).

(a) Fitting $Re(\omega)$

(b) Fitting $Im(\omega)$

Figure 11: Analytical fitting function mentioned in the inset boxes in the
figures indicate the dependence of the Re($\omega$) and Im($\omega$) (in
$M^{-1}$) on $\sigma$ for Case III wormholes and is shown by the red curve for
$m=1$.

Machine precision (precision upto 16 decimal places) has been used for
obtaining the fitting functions in Mathematica. Fig.(11) shows the best-fit
model where the analytical red curve matches well with the data points shown
in blue. The goodness of the fit is measured by calculating the $\chi^{2}$
value associated with the fit. The above fitting function for $\omega_{r}$
(shown in inset of Fig.(11)) gives a ‘p value’ as 0.87 calculated using
Mathematica which gives the probability of obtaining the observed results from
the fit as $93\%$. The probability value is taken from the chi-square
distribution table corresponding to 4 degrees of freedom chi . This indicates
that the model considered is suitable and can be accepted to represent the
real parts of QNM frequencies. Similarly, for $\omega_{i}$ the probability is
about $94\%$ ensuring the goodness of the fitting function. We can now use the
fitting function to estimate the frequency and damping time corresponding to
different wormhole parameters. In physical units, throat radius is $r_{0}=2M$
and the oscillation frequency becomes

$\nu=\frac{c^{3}}{GM}\frac{f(\sigma)}{2\pi}Hz=\frac{32390.9}{M}f(\sigma)\,Hz$
(31)

with the mass parameter $M$ expressed in units of solar mass $M_{\odot}$ and
$G=6.67\times 10^{-11}$ $m^{3}kg^{-1}s^{-2}$ , $c=3\times 10^{8}ms^{-1}$. The
mass parameter $M$ denotes the corresponding ADM mass of the wormhole Shaikh
(2018). Let us consider wormhole with $\sigma=0.01$ and we take the detector
sensitivity range from 10Hz- 10 kHz Buikema et al. (2020). To obtain the
throat radius ($r_{0}=2M$) in length units we multiply a factor $G/c^{2}$. The
throat radius then becomes $7.16472\times 10^{3}$ km and mass of wormhole
$2430.25M_{\odot}$ for frequency of 10Hz. Similarly, if we take $\nu=10kHz$,
then throat radius and mass is $7.164$ km and $2.43025M_{\odot}$ respectively.
Comparing with the masses of the astrophysical objects mentioned in the
LIGO/VIRGO data catalogue, we deduce the mass of wormhole corresponding to 10
Hz frequency to be very large and although possible, is not of interest at the
moment. So a realistic wormhole, if they exist in nature, may emit
scalar/gravitational waves of a very high frequency of the order of $\sim$kHz.
The analysis can be carried out for other $\sigma$ values as well as for
gravitational perturbations, but the results possibly remain qualitatively the
same.
On the other hand, the damping time is given by the imaginary part of $\omega$

$\tau=\frac{2\pi GM}{c^{3}g(\sigma)}sec$ (32)

If we consider $\sigma=0.01$ and take the mass of the wormhole as
$2.43025M_{\odot}$ obtained earlier, then damping time is of the order
0.000422 sec. Similar fitting functions can also be obtained for other mode
values where the values of the constants $a,b,c,d$ will change accordingly.

###  V.4 Case IV: $\sigma=0,\kappa\neq 0$ (Hayward wormhole)

The quasi-normal modes of the Hayward wormhole are calculated using the WKB
method improved with Pade approximation along with the standard numerical
routines as the potential is a single barrier for all parameter values. We
begin by exploring the dependence of the QNM values on the angular momentum
number $m$ for different $\kappa$ wormholes, where some interesting non-
trivial behavior is observed.

Figure 12: Variation of Re($\omega$)(in $M^{-1}$) with $m$ for different
$\kappa$ Hayward wormholes.

Figure 13: Variation of imaginary component of QNM frequency (in units of
$M^{-1}$) for different $\kappa$ Hayward wormholes with $m$.

Fig.(12) shows the real $\omega$ to be increasing with increasing $m$ which
holds true for all values of $\kappa$. On the other hand, imaginary $\omega$
decreases with increasing $m$ until $\kappa$ is small (see Fig.(13)). As we
consider higher values of $\kappa$, the $\omega_{i}$ starts to increase with
$m$ where the intermediate values of $\kappa$ show the transitional behaviour.
Such behaviour of the imaginary part of $\omega$ with respect to $m$ has been
observed in other wormhole geometries as well Kim et al. (2018); Roy (2022).
Next, we study the dependence of the QNMs on parameter $\kappa$ and calculate
the corresponding fitting functions as discussed in the previous section. For
$\omega_{r}$, $m=1$ we obtain the fitting function using the NonlinearModelFit
in Mathematica. Such fitting model has a probability of $94\%$ from chi-square
estimation and hence is taken as a good fit. The behaviour will hold true for
all $m$ values but with different values of constants in the fitting function
(see Fig.(14)). To get an estimate on the mass and throat radius of the
wormholes for different $\kappa$, we consider

$\nu=\frac{c^{3}f(\kappa)}{2\pi GM}$ (33)

where $f(\kappa)$ is the fitting function and $M$ denotes the ADM mass in
solar mass units. The throat radius $x_{0}$ needs to be multiplied by a factor
of $GM/c^{2}$ to get the radius in length unit. For $\kappa=0.01$ and $\nu=10$
Hz, $M=2427.76M_{\odot}$ and throat radius is $7.15\times 10^{3}$ km.
Similarly, for 10 kHz frequency we get the mass of the order of $2.4M_{\odot}$
and throat radius 7.15 km. The analysis can be repeated for other $\kappa$
values as well.
For the imaginary component of QNM, we derive the fitting function where the
values of the constant change for different $m$. Although the dependence on
$m$ changes for different $\kappa$, we find the fitting function mentioned
within the plots of Fig.(15) to work well for all $m$ values. For both $m=4$
and $6$, chi-square fitting gives a probability of about $96\%$. This happens
because the value of $\omega_{i}$ decreases with increasing $\kappa$
irrespective of $m$. The fitting function matches well with the QNM values
evaluated for two different $m$, $m=4$ and $6$, as shown in Fig.(15).

Figure 14: Analytical fitting function mentioned in the inset boxes in the
plot indicates the dependence of the Re($\omega$) (in $M^{-1}$ units) on
$\kappa$ for Hayward wormholes and is shown as the red curve with $m=1$. It
matches well with the QNM values shown in blue.

(a) Fitting $Im(\omega)$ for $m=4$

(b) Fitting $Im(\omega)$ for $m=6$

Figure 15: Analytical fitting functions mentioned in the inset boxes in the
plots indicate the dependence of the Im($\omega$) (in $M^{-1}$ units) on
$\kappa$ for Hayward wormholes and is shown as the red curve with $m=4$ and
$m=6$.

###  V.5 Case V: $\sigma=1,\kappa\neq 0$ (Regular Hayward black hole)

This case corresponds to the Hayward black hole which is regular at $r=0$.
Once again we aim to study the dependence of the QNM frequencies on the
parameter $\kappa$ as $\sigma$ is fixed to one. First, we consider the real
part of $\omega$ and proceed by considering the fitting function plotted in
Fig.(16) determined by NonLinearModelFit in Mathematica. Estimating the mass
and horizon radius for say $\kappa=0.01$, we find $M=948M_{\odot}$ and radius
$2.79\times 10^{3}$ km corresponding to 10 Hz frequency, while for higher 10
kHz we get mass and radius as $0.948M_{\odot}$ and 2.79 km respectively. In
essence, the black hole with $\kappa=0.01$ is much lighter when compared to a
Case IV wormhole with same $\kappa$, for a particular frequency.
Similarly, for the imaginary part we obtain a fitting function that matches
well with the calculated QNM values shown in Fig.(16). From the chi-square
distribution we have also ensured the goodness of the above fitting functions
–they turn out to be around $95\%$ and $93\%$ respectively.

(a) Fitting $Re(\omega)$

(b) Fitting $Im(\omega)$

Figure 16: Analytical fitting functions mentioned in the inset boxes in the
plots indicate the dependence of the Re($\omega$) and Im($\omega$) (in
$M^{-1}$) on $\kappa$, for Hayward regular black hole and is shown by the red
curve with $m=1$.

###  V.6 Case VI: $0<\sigma<1$, $0<\kappa\leq\frac{4}{3\sqrt{3}}$ (Hayward-
Damour-Solodukhin wormhole)

We finally arrive at the most general parametrized wormhole metric which
supports effective potentials with both single and double peaks depending on
the value $\sigma$. Hence, the numerical routines mentioned earlier are better
suited for calculating the QNMs. Nonetheless, for single barriers, the WKB
method acts as a verification mechanism for the numerical results. Distinct
echo patterns are also visible for wormholes that have double potential
barriers as shown in Fig.(17) for a particular wormhole example belonging to
this case. Similar to Case III wormholes, the initial prompt damping is
dominated by photon ring modes with the fundamental wormhole QNMs coming into
play once the echoes have died down at very late times.

Figure 17: Time domain profile showing distinct echoes as observed at
$r_{*}=30$ for $\sigma=0.9999$, $\kappa=0.5$ Case VI wormhole. The grid
spacing $h=0.5$ and a Gaussian initial profile is used.

The study of the QNM dependence on parameters is more intricate for these
wormholes as the metric is controlled by both $\sigma$ and $\kappa$. To
observe the QNM behaviour, we will keep one of the parameters fixed at a time
and study the variation of the QNMs with the other. First, we fix $\sigma$ and
choose different fitting functions to explore the dependence of QNMs on
$\kappa$ when calculated in units of $M^{-1}$.

(a) Fitting $Re(\omega)$ for $\sigma=0.1$

(b) Fitting $Im(\omega)$ for $\sigma=0.1$

(c) Fitting $Re(\omega)$ for $\sigma=0.5$

(d) Fitting $Im(\omega)$ for $\sigma=0.5$

Figure 18: Analytical fitting functions mentioned in the inset boxes in the
figures indicate the dependence of the Re($\omega$) and Im($\omega$) (in
$M^{-1}$) on $\kappa$ for Case VI wormholes and is shown by the red curve with
$m=1$ and fixed $\sigma$.

(a) Fitting $Re(\omega)$ for $\sigma=0.88$

(b) Fitting $Im(\omega)$ for $\sigma=0.88$

(c) Fitting $Re(\omega)$ for $\sigma=0.9$

(d) Fitting $Im(\omega)$ for $\sigma=0.9$

Figure 19: The analytical fitting functions mentioned in the figures indicate
the dependence of the Re($\omega$) and Im($\omega$) (in $M^{-1}$) on $\kappa$
for Case VI wormholes and is shown by the red curve with $m=1$ and fixed
$\sigma$.

The Fig.(18) and Fig.(19) show the fitting function as well as the variation
of real and imaginary parts of QNM with parameter $\kappa$ for fixed $\sigma$
values. It is observed that while $\omega_{i}$ always decreases with
increasing $\kappa$, there is no such general trend for $\omega_{r}$. For
small values of $\sigma$, $\omega_{r}$ increases with increasing $\kappa$. But
as we approach values of $\sigma$ close to unity, $\omega_{r}$ varies slowly
with the change occurring in third decimal place of two consecutive
frequencies. Initially $\omega_{r}$ increases followed by a decrease in its
value with $\kappa$. These behaviour are reflected in the fitting functions as
well. Using the fitting function we estimate the mass and throat radius of the
wormhole for different $\sigma$ values as shown in Tab.(4).

$\sigma$ | $\kappa$ | M $(M_{\odot})$ | Throat Radius (km)
---|---|---|---
0.1 | 0.01 | 2.32 | 6.84
| 0.7 | 2.75 | 6.63
0.5 | 0.01 | 1.79 | 5.28
| 0.7 | 2.004 | 4.82
0.88 | 0.01 | 1.092 | 3.22
| 0.7 | 1.087 | 2.61
0.9 | 0.01 | 1.037 | 3.05
| 0.7 | 1.017 | 2.44
Table 4: Numerical estimates on mass and throat radius of Case VI wormholes
for different sets of ($\sigma,\kappa$) values corresponding to $10$ kHz
frequency obtained using the fitting functions mentioned earlier.

A similar analysis can be performed by fixing $\kappa$ and observing the
variation of QNM with $\sigma$. We consider $\kappa=0.1$ and $0.7$ and notice
the general behavior of QNMs to be similar in both cases with variation of
$\sigma$. As we see in Fig.(20), with increasing $\sigma$ both the real and
imaginary parts of QNM decreases.

(a) Fitting $Re(\omega)$ for $\kappa=0.1$

(b) Fitting $Im(\omega)$ for $\kappa=0.1$

(c) Fitting $Re(\omega)$ for $\kappa=0.7$

(d) Fitting $Im(\omega)$ for $\kappa=0.7$

Figure 20: Analytical fitting functions mentioned in the inset boxes in the
figures indicate the dependence of the Re($\omega$) and Im($\omega$) (in
$M^{-1}$) on $\sigma$ for Case VI wormholes and is shown by the red curve with
$m=1$ and fixed $\kappa$.

The fitting function mentioned above for $\omega_{r}$ corresponding to
$\kappa=0.7$ and $\sigma=0.5$ gives the mass of the wormhole as
$2.008M_{\odot}$ and throat radius $4.832$ km. Note that these estimates match
with the values given in Tab.4 for the same wormhole. This verifies the
effectiveness of the fitting functions obtained in two different scenarios for
the same wormhole geometries.

##  VI Distinguishing between different types of spacetimes

In the previous section we have thoroughly explored the stability and
corresponding quasi-normal modes of each spacetime arising in our parametrized
family. We have also observed the dependence of the frequency and damping time
on the metric parameters for each geometry and calculated the corresponding
fitting functions. In some cases, values of the parameters and their change
correspond to switch-over from one geometry to another. An interesting feature
of our results is that we are able to study those spacetimes where the
parameters lie very close to (above or below) those switch-over values. For
example, taking values of $(\sigma,\kappa)$ as (0,0),(0,0.01),(0.01,0) and
(0.01,0.01) will result in different wormhole geometries but with very
identical ringdown behavior and QNMs. In essence, they can be thought as
mimickers of each-other. Once a gravitational wave signal is detected, the
signal is then probed to obtain information about the source. Quasi-normal
modes, through parameter estimation, help in narrowing down the properties of
the source. If two distinct astrophysical objects emit similar gravitational
wave signal with QNM values almost identical, then the process of identifying
the source becomes tricky. We explore this scenario assuming a scalar wave
perturbing the wormhole. We compare and contrast the scalar ringdown signal
and the QNMs for parameter values that are around the above-mentioned switch-
over values . Table 5 shows the fundamental QNMs calculated using direct
integration for four different wormhole spacetimes that result in near
identical modes. In order to distinguish them, very high precision is required
as the change occurs after second decimal place. Similarly, in Fig.(21) we
compare the scalar ringdown behaviour of wormholes belonging to Case II and
Case III with $\sigma=0.01$ which appear to be very identical.
In contrast, when we compare the scalar QNMs for wormholes which are close to
black hole parameter values, we observe a distinct difference (see Table 6).
While the real component of the QNM is very similar for both geometries, the
imaginary part of the QNM for the wormhole is very small. Also, if we compare
wormholes belonging to Case VI with the Hayward regular black hole scenario,
then we have $\sigma$ close to unity. This leads to distinct double potential
barriers giving rise to echoes that dominate the ringdown signal of the
wormhole. Hence, the wormhole and black hole geometries can be easily
distinguished from the ringdown profile and also through their respective QNM
values.
The above discussion once again highlights the significance of our metric.
Apart from shifting between different wormhole and black hole spacetimes by
choosing specific parameter values to study their ringdown behaviour, the form
of the metric makes it suitable to compare between different spacetimes. We
explored the possibility of using scalar QNMs as a tool to distinguish various
geometries. We found different wormholes to be very identical in their
behaviour under scalar perturbations and thus will need very high precision to
distinguish them only through such scalar quasi-normal modes.

Figure 21: The time domain profiles for Case II wormhole (red) and Case III Damour-Solodukhin wormhole (blue) with $\sigma=0.01$. The signal is observed at $r_{*}=5$, grid size $h=0.5$ and initial Gaussian signal of the form $e^{\frac{-(u-10)^{2}}{100}}$. With $\sigma$ values being close to 0, the two wormhole geometries are difficult to distinguish as evident from their TD profiles and QNM values. m | Case II | Case III $(\sigma=0.01)$ | Case IV ($\kappa=0.01$) | Case VI ($\sigma=\kappa=0.01)$
---|---|---|---|---
1 | 0.755245 -i 0.1787874 | 0.7516804 -i 0.177469 | 0.7552616 -i 0.178785 | 0.751697 -i 0.17746636
2 | 1.253101 -i 0.177527 | 1.246957 -i 0.1762021 | 1.2531305 -i 0.1775244 | 1.246987 -i 0.1761995
3 | 1.752240 -i 0.1771437 | 1.7435064 -i 0.175854 | 1.752232 -i 0.1771796 | 1.743549 -i 0.175852
Table 5:  $\omega_{QNM}$ values in units of $M^{-1}$ for different modes calculated using DI. m | Case V ($\kappa=0.5$) | Case VI $(\sigma=0.9,\kappa=0.5)$
---|---|---
1 | 0.2992639 -i 0.0928689 | 0.319806 -i 0.0089489
2 | 0.4936223 -i 0.09224176 | 0.4895156 -i 0.0020109
3 | 0.689167 -i 0.09208116 | 0.6578297 -i 0.00034295
Table 6:  Comparing $\omega_{QNM}$ ($M^{-1}$) values for Hayward regular black
hole and Case VI wormhole calculated using DI.

##  VII Conclusions

Black hole solutions in GR have a singularity hiding behind the event horizon.
One possible route towards avoiding the singularity in black hole spacetimes
is to search for regular spacetimes with or without event horizon(s). Such
spacetimes, devoid of any singularity, were first introduced by Bardeen and
have been named as ‘regular’ black holes. Later work by Hayward, Neves and Saa
provided newer examples of such regular black holes. In our work here, we
begin with the Hayward spacetime and generalize the metric by introducing two
different mass parameters $M_{1}$ and $M$ which appear in the $g_{tt}$ and
$g_{rr}$ components of the metric, respectively. The parametrized metric thus
obtained not only gives regular black holes but also represents various
wormhole geometries as well as the singular Schwarzschild black hole, for
specific choices of parameters. We have analysed in detail each of the six
distinct spacetimes (included in our general metric) for different values of
the dimensionless parameters $(\sigma,\kappa)$. The allowed parameter space
extends from $\sigma\in[0,1]$ and $\kappa\in[0,\frac{4}{3\sqrt{3}}]$.

As mentioned in the Introduction, we acknowledge the problems associated with
non-singular spacetimes namely; energy-condition violating matter threading
wormholes and the mass-inflation instability associated with the inner horizon
of regular black holes. However, instead of being deterred by such issues,
through our work, we have tried to shed light on the stability question of
these geometries under scalar perturbations, by calculating the scalar QNMs.
The analysis of this ringdown gives information about the viability of these
models as astrophysical entities. Our proposed generalized Hayward metric is
especially suitable for this purpose as the behaviour of multiple types of
regular spacetimes can be visualised simultaneously in a single framework.

In Section III, we discussed the nature of the spacetimes as determined by the
behaviour of the corresponding curvature scalars, energy conditions and also
through the embedding diagrams for the geometries. The black hole solutions
(both regular and singular) occur only when $\sigma=1$ while rest of the
parameter space gives rise to distinct wormhole spacetimes. For
$\sigma=\kappa=0$ we get the Schwarzschild wormhole, $0<\sigma<1,\kappa=0$
gives the well-known Damour-Solodukhin wormhole and the remaining parameter
ranges lead to various unique wormhole solutions that have not been explored
yet. The later half of our work deals with the stability analysis of these
spacetimes under massless scalar wave propagation. The stability is deduced by
obtaining the time domain profile and calculating the corresponding scalar
quasi-normal modes of the spacetimes. We have found all the geometries to
harbour modes with negative imaginary component indicating damping of the
signal over time. Some wormholes also show double barrier potential that
result in repetitive ‘echo’ patterns in the signal. The dependence of the
quasinormal modes on the metric parameters $(\sigma,\kappa)$ has also been
explored by obtaining fitting functions for each spacetime and is summarized
in Tab.7. The dependence on parameters is rather intriguing for the Hayward
wormhole belonging to Case IV. The imaginary component of $\omega$ is found to
initially decrease with $m$ for small $\kappa$ but increasing with $m$ for
larger $\kappa$ values. The shift in behaviour occurs around $\kappa\approx
0.4$. Again for the Case VI wormholes, the dependence on the parameters is
worth noting. Now the real part of $\omega$ is seen to increase with $\kappa$
for small $\sigma$. But as $\sigma$ approaches unity, $Re(\omega)$ starts to
decrease with increasing $\kappa$. Such interesting dependences of the QNMs on
the metric parameters are also reflected through the fitting functions
mentioned within our work.

Spacetime | Re($\omega$) | Im($\omega$) | $\sigma$ | $\kappa$
---|---|---|---|---
Schwarzschild BH | - | - | 1 | 0
Schwarzschild WH | - | - | 0 | 0
DS WH | $\downarrow$ | $\downarrow$ | Increasing | 0
Hayward WH | $\uparrow$ | $\downarrow$ | 0 | Increasing
Hayward BH | $\uparrow$ | $\downarrow$ | 1 | Increasing
Case VI WH | $\uparrow$ | $\downarrow$ | Fixed ($\sigma<<1$) | Increasing
| $\downarrow$ | $\downarrow$ | Fixed ($\sigma\rightarrow 1$) | Increasing
| $\downarrow$ | $\downarrow$ | Increasing | Fixed
Table 7: Table shows the dependence of the QNMs on the parameters of the
metric for different spacetimes. $\uparrow$ and $\downarrow$ indicates
respectively increasing and decreasing behaviour of the corresponding quantity
with the change in parameter. For Case VI wormholes, since both
($\sigma,\kappa$) change we have kept one fixed and have studied the
dependence on the other. The inferences are for fixed $m$ value.

Some spacetimes lying close to each other in the parameter space are found to
have near identical ringdown behaviour and quasinormal modes which makes their
possible detection even more involved. Through specific examples we have shown
the how a similar ringdown profile arise for wormholes even if they belong to
different parameter ranges. Thus, in any real detection scenario, segregating
such wormholes would demand high precision observations as the modes differ
only after second decimal place. On the contrary, the wormholes close to black
holes in the parameter space behave quite distinctly. Such wormholes possess
well-separated double potential peaks that lead to echo patterns in the
ringdown signal. The wormholes also have a very long damping time indicating
differences, in comparison to a black hole.
In conclusion, our parametrized metric provides an elegant way of probing
multiple spacetimes with diverse characterisitics, within a single framework.
The metric also facilitates comparison among the different spacetimes,
generated from our parametrized metric, about their behaviour under scalar
wave propagation. Our work can be extended in many possible directions. The
wormhole spacetimes obtained can be studied for their geodesics and their
stability can be explored under axial and polar gravitational perturbations.
Future studies along different directions (perturbations or other) will
hopefully throw more light on how such wormholes or black holes, as introduced
here, may serve as possible candidates of black hole mimickers.

## Acknowledgements

PDR thanks Indian Institute of Technology, Kharagpur, India for support and
for allowing her to use available facilities there.

##  Appendix A Computation of tortoise coordinate for black holes and
wormholes

The tortoise coordinate defined in eq.(16) is integrated numerically for any
general values of parameter $(\sigma,\kappa)$ using Mathematica. Depending on
the nature of the spacetime, the initial conditions are specified for solving
the differential equation of the tortoise coordinate i.e. eq.(16). For
wormholes we set $r_{*}=0\hskip 7.22743pt(r=r_{0}+\epsilon)$ where
$\epsilon<<1$. This ensures smooth numerical integration as the location of
the throat is a root of the function $f(r)$ making eq.(16) diverge exactly at
the throat. The accuracy and precision of the integration is chosen such that
changing them slightly does not affect our results providing robustness to the
calculated $r_{*}$. The tortoise coordinate for black holes rapidly goes to
$-\infty$ as $r\rightarrow$ horizon. So the condition for integrating the
tortoise coordinate is specified at some positive value of $r$ where the
behavior of $r_{*}$ is not drastic. The authenticity of the computation has
been verified by comparing results with Schwarzschild black hole where the
tortoise coordinate is exactly known. The following figures (see Figs.(22(a))
and (22(b))) show the time domain profiles obtained for scalar potential with
$m=1$ of the Schwarzschild black hole. We find the two integration schemes
giving identical profiles validating the numerical integration approach for
tortoise coordinate for general spacetimes.

(a) TD using exact form of tortoise coordinate

(b) TD using tortoise coordinate obtained by numerical interpolation

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