# Echoes from bounded universes

Renan B. Magalhães<EMAIL_ADDRESS>Programa de Pós-Graduação em
Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazil.
Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia
- CSIC. Universidad de Valencia, Burjassot-46100, Valencia, Spain.  Andreu
Masó-Ferrando<EMAIL_ADDRESS>Departamento de Física Teórica and IFIC,
Centro Mixto Universidad de Valencia - CSIC. Universidad de Valencia,
Burjassot-46100, Valencia, Spain.  Flavio Bombacigno<EMAIL_ADDRESS>Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia
- CSIC. Universidad de Valencia, Burjassot-46100, Valencia, Spain.  Gonzalo J.
Olmo<EMAIL_ADDRESS>Departamento de Física Teórica and IFIC, Centro Mixto
Universidad de Valencia - CSIC. Universidad de Valencia, Burjassot-46100,
Valencia, Spain.  Departamento de Física, Universidade Federal do Ceará, Caixa
Postal 6030, Campus do Pici, 60455-760 Fortaleza, Ceará, Brazil  Luís C. B.
Crispino<EMAIL_ADDRESS>Programa de Pós-Graduação em Física, Universidade
Federal do Pará, 66075-110, Belém, Pará, Brazil.  Departamento de Matemática
da Universidade de Aveiro and Centre for Research and Development in
Mathematics and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro,
Portugal.

###### Abstract

We construct a general class of modified Ellis wormholes, where one asymptotic
Minkowski region is replaced by a bounded 2-sphere core, characterized by
asymptotic finite areal radius. We pursue an in-depth analysis of the
resulting geometry, outlining that geodesic completeness is guaranteed also
when the radial function asymptotically shrinks to zero. Then, we study the
evolution of scalar perturbations, bringing out how these geometric
configurations can in principle affect the time-domain profiles of quasinormal
modes, pointing out the distinctive features with respect to other black holes
or wormholes geometries.

##  I Introduction

The advent of gravitational waves astronomy [1] and the technical progress
made in very large base-line interferometry [2] is enabling us to confront
predictions of gravitational theories in the strong field regime with
observational data. In this sense, significant theoretical efforts are being
devoted to the search and study of exotic compact objects that may offer a
reasonable alternative to the black hole paradigm of Einstein’s theory of
General Relativity (GR). Such objects could lead to effects not expected in GR
and, for this reason, their analysis could help identify observational
signatures of new physics. Among the most popular alternatives, one finds
hairy or regularized black holes [3, 4, 5, 6], boson stars [7, 8], gravastars
[9], wormholes [10], and black bounces [11]. In this work, we put the focus on
a new kind of exotic object closely related with wormholes but with proper
observational features that could allow us to tell them apart from the usual
wormholes found in the literature.

In the conventional approach to the resolution of the field equations of a
given gravitational system, one typically begins with a reasonable description
of the matter sources and then employs the Einstein field equations to derive
the corresponding spacetime geometry. When dealing with wormhole
configurations, however, the process is usually inverted. In fact, once the
desired spacetime geometry is selected, it is possible to use Einstein’s
equations in reverse to determine the matter distribution responsible for it.
Despite violating various energy conditions, wormholes still stand out because
of their theoretical implications, as they offer valuable insights into the
foundational aspects of gravity models and their possible non trivial
topological structure. Wormholes are not only relevant for interstellar travel
[12], which is a well-known and popular application recurrently used in the
literature, but could be also crucial to better understand the nature of
quantum entanglement [13]. For these reasons, it is important to explicitly
address the theoretical implications of their existence and the proper
observational signatures that could signal their presence in astrophysical
scenarios [14, 15].

The emission of gravitational waves by the coalescence of compact objects, for
instance, can clearly distinguish between black holes (objects with an event
horizon) and wormholes (no event horizon), as the latter are expected to emit
a series of echoes [16] which are not present in the case of objects with a
horizon, from which nothing can come out (though these echoes are not unique
to wormholes, see for instance [17]). Also, the distortion of light
trajectories by black holes and wormholes might be quite different, not only
because wormholes may allow light to come out from their internal region but
also because the number and structure of light rings might exhibit different
behaviors [18, 19]. These aspects can be used to identify a number of treats
that allow to characterize the various specimens present in the zoo of compact
objects.

In this sense, it should be noted that among the billions of compact objects
present in the universe, the observation of events apparently compatible with
our predictions does not rule out the existence of objects of a different
nature. Current detectors’ sensitivities are still insufficient to break
significant degeneracies in the data [20], which demands for new generations
of observatories able to provide a clearer picture of the nature of such
objects [21]. In the meantime, the existence and consistency of exotic compact
objects can be pursued theoretically, with the aim of identifying their proper
signatures and potential phenomenological implications.

It is important, therefore, to have a clear understanding of the proper
features that define and identify wormholes. From an astrophysical
perspective, one would expect that wormholes should be spherical or
axisymmetric (rotating). Accordingly, the presence of spherically symmetric
wormholes is typically identified in the literature by the existence of a two-
sphere of minimal non-zero area, which represents the wormhole throat [10,
22]. However, the observation of a minimal two-sphere, which is a local
characteristic, cannot be used to infer anything about the global properties
of the spacetime, such as its topology. Thus, in order to broaden our
understanding of their observational features, it is necessary to consider
scenarios in which typical local properties of wormholes are present but where
their proper global aspects are not. In particular, since in standard
wormholes the throat connects two unbounded (typically asymptotically flat)
regions, one may consider spacetimes where the throat connects an unbounded
region to a bounded one. Spacetimes with this characteristic have been
recently found as the end product of the collapse of boson stars in quadratic
Palatini gravity [23, 24]. In that scenario, the central region of the
collapsed object experiences a kind of inflationary stretching that leads to
the formation of an expanding bubble (or baby universe) behind the event
horizon. Inspired by that phenomenon, here we consider a static setting in
which an asymptotically Minkowskian region is attached to a bounded bubble via
a minimal two-sphere. The gravitational echoes generated by such bounded
universes and their geodesic structure will be scrutinized and compared with
those of more standard wormholes.

One of the simplest wormhole configurations is represented by the well-known
Ellis wormhole (EWH) [25], which is made of two asymptotically flat regions
connected by a throat111In this work we do not focus on wormhole geometries
involving Schwarzschild-like deformation (see for instance [11]), which will
be the object of a forthcoming analysis [26].. In this work we introduce the
idea that one of the infinite asymptotic region, i.e. the Minkowski spacetime
of EWHs, could be possibly replaced by some inner spatial volume endowed with
finiteness in one or more directions. The (partial) compactness of this new
internal region is thus expected to induce specific observable effects on both
the shadow of the object and on its spectroscopic properties, where peculiar
absorption effects, resonances, and echoes may arise depending on the typical
size of the internal region and the different boundary conditions involved.

In this work we move a first step on in this direction by looking at minimal
deformations of the classic EWH [25] considering that one of the asymptotic
Minkowski regions is replaced by a bounded 2-sphere with asymptotically
constant areal radius. We expect that the finiteness of the radial function in
this internal region plays a relevant role in the propagation of
perturbations, because the effective potential perceived by scalar modes,
hereinafter $V_{\Phi}$, critically depends on the areal radius behavior. In
particular, we refer to the possibility that echoes could be generated,
bringing the question of how to distinguish, in a phenomenological sense, our
geometrical setting from analogous results in different scenarios [27, 28, 29,
30, 31, 32]. Moreover, among the possible geometric realizations studied, we
will consider the case in which the asymptotic areal radius vanishe, implying
that the geodesic completeness of the resulting spacetime is not a priori
guaranteed (see [33, 34] for examples of geodesically incomplete wormholes).
In order to clarify this delicate issue we analyze to some extent the geodesic
motion for radial (analytically) and non-radial (numerically) trajectories,
confirming that geodesic completeness is insensitive to the actual value of
the asymptotic areal radius, which turns out to affect only the presence or
not of inner region bounces for nonradial motion.

The paper is organized as follows. In sec. II we introduce the model and we
discuss the general geometric properties, by also presenting a detailed
analysis of the energy condition violation; in sec. III we perform an
exhaustive investigation of geodesic motion and photon orbit stability; in
sec. IV scalar perturbations are addressed and we show how the appearance and
the properties of echoes are affected by the geometry; in sec. V we report our
final comments.

Spacetime signature is chosen mostly plus, i.e.
$\eta_{\mu\nu}=\text{diag}(-1,1,1,1)$ and we adopted units in which $8\pi G=1$
and $c=1$.

##  II The model

One of the simplest (symmetric) traversable wormholes in the literature is
represented by the EWH, whose line element is given by [25]

$ds^{2}=-dt^{2}+dx^{2}+r(x)^{2}(d\theta^{2}+\sin^{2}\theta d\varphi^{2}),$ (1)

with radial coordinate $x\in(-\infty,\infty)$ and areal radius
$r(x)=\sqrt{x^{2}+a^{2}}$. The areal radius has a regular minimum at $x=0$,
which corresponds to the wormhole throat with radius $a$. As $|x|$ increases,
the areal radius monotonically grows and the line element (1) approaches
Minkowski spacetime as $x\to\pm\infty$.

Here we present a toy model of an asymmetric wormhole-like object,
characterized by a bounded 2-sphere radius inner region surrounded by an
asymptotically flat exterior, formally described by the line element (1), but
with the modified radial function

$r^{2}(x)=\left\\{\begin{array}[]{ll}x^{2}+a^{2},&x\geq 0,\\\
x^{2}+a^{2}-(x^{2}+a^{2}-R^{2})\tanh^{2}(c\,x^{2}),&x<0,\end{array}\right.$
(2)

where $a$ and $R$ are constants with dimension of length, while $c$ has
dimension of $\text{length}^{-2}$. One can check that (2) is endowed with a
throat-like structure in $x=0$, where $r(x)$ exhibits the regular minimum
$r^{2}(0)=a^{2}$, with $a$ taken to be the throat radius. In the asymptotic
internal region, as $x\to-\infty$, the radial function $r^{2}(x)\to R^{2}$,
that is, the 2-sphere radius is asymptotically bounded. In particular, when
$R^{2}=0$, the 2-sphere radius shrinks to zero exponentially. The role of the
parameter $c$ is to control how much the radial function $r^{2}(x)$ departs
from a parabola close to the throat, so that the asymptotic value $R$ is
reached faster as one increases the value of $c$. If $c=0$, the radial
function (2) reduces to the one of the EWH, that is, $r(x)^{2}=x^{2}+a^{2}$.

In this model, the outer universe ($x\geq 0$) is described by the same line
element as the EWH spacetime. The modified areal radius (2), however, sharply
modifies the structure of the inner universe ($x<0$). In Fig. 1 we exhibit the
behavior of the modified areal radius squared in both sides of the throat and
compare it with the standard parabolic behavior of EWH
($r(x)^{2}=x^{2}+a^{2}$). In the top panel we fixed the value of the parameter
$c$ and consider some values of the asymptotic 2-sphere radius $R$. In the
bottom panel we fixed the asymptotic radius and consider different values of
$c$. From Fig. 1 one notices that the areal radius may present a local maximum
in the inner region. One can check that maximum location, $x_{m}$, must
satisfy $M(x_{m})=0$ and $r_{xx}(x_{m})<0$, where

$M(x)\equiv 1-2c(x^{2}+a^{2}-R^{2})\tanh(c\,x^{2})$ (3)

and $r_{xx}$ stands for the second derivative of $r(x)$ (similarly $r_{x}$
denotes the first derivative of $r(x)$). Since $M(x)$ is continuous in the
interval $(-\infty,0)$, and $M(x)\to-\infty$ as $x\to-\infty$ and $M(x)\to 1$
as $x\to 0^{-}$, there is at least one point $x_{m}\in(-\infty,0)$ such that
$M(x_{m})=0$, and this point is a maximum whether $r_{xx}(x_{m})<0$.

Figure 1: Modified areal radius (2). Top panel: $r(x)^{2}$ for fixed value of
$c$ and different asymptotic 2-sphere radius $R$. Bottom panel: $r(x)^{2}$ for
fixed $R$ and some choices of $c$. We normalized the plots with the throat
radius $a$.

###  II.1 Embedding diagrams

The modified areal radius (2), therefore, significantly changes the structure
of the modified EWH internal region. That can be properly visualized by
drawing the embedding diagrams of the model for different asymptotic 2-sphere
radius. To do so, we consider at a given time coordinate $t$ the equatorial
plane $\theta=\pi/2$, where the metric (1) reduces to
$ds^{2}=dx^{2}+r(x)^{2}d\phi^{2}$. Now, one may embed this hypersurface in the
three-dimensional Euclidean space (written in cylindrical coordinates)
$ds^{2}=dz^{2}+d\rho^{2}+\rho^{2}d\phi^{2}$. By identifying the polar radius
with the radial function, i.e. $\rho=r(x)$, one obtains the differential
equation

$z_{x}^{2}=1-r_{x}^{2},$ (4)

which can be solved for $z(x)$. Together with (2), they can be plotted in the
plane $r-z$ to build the embedding diagram as the revolution surface of the
curve $\gamma(x)=(r(x),z(x))$ about axis $r=0$. In Fig. 2 we show the
embedding diagrams of four modified EWHs with bounded 2-sphere radius in the
inner region. In particular, in the top-left panel, we exhibit the embedding
diagram of a wormhole-like object whose 2-sphere radius shrinks to zero in the
asymptotic region, creating a sort of “bubble” below the throat in the
embedding diagram. Additionally, we also show three configurations of finite
$R$, with $R<a$, $R=a$ and $R>a$.

Figure 2: Embedding diagrams of finite 2-sphere radius EWHs with fixed value
of $c$. In the top-left panel we exhibit a wormhole-like object in which the
2-sphere radius shrinks to zero, creating a sort of “bubble” in the internal
region. The other panels correspond to asymptotic 2-sphere radius $R$ smaller
(top-right), equal (bottom-left) and bigger (bottom-right) than the wormhole
throat $a$ (represented by the red line).

###  II.2 Curvature, energy density and pressures

To have a better physical view of the solutions modeled by the line element
(1) with areal radius (2), one can compute the curvature invariants related
with it, which read

$\displaystyle
g_{\mu\nu}R^{\mu\nu}=\dfrac{2\left(1-r_{x}^{2}-2rr_{xx}\right)}{r^{2}},$ (5)
$\displaystyle
R_{\mu\nu}R^{\mu\nu}=\dfrac{2\left(\left(r_{x}^{2}-1\right)^{2}+2r\left(r_{x}^{2}-1\right)r_{xx}+3r^{2}r_{xx}^{2}\right)}{r^{4}},$
(6) $\displaystyle
R_{\alpha\beta\gamma\lambda}R^{\alpha\beta\gamma\lambda}=\frac{4\left(1-2r_{x}^{2}+r_{x}^{4}+2r^{2}r_{xx}^{2}\right)}{r^{4}}.$
(7)

Since the spacetime is asymptotically flat in the exterior region all the
curvature scalars vanish very far from the throat. In the internal region,
instead, curvature invariants may be bounded or unbounded depending on the
asymptotic 2-sphere radius $R$. As the radial coordinate approaches
$x\to-\infty$, one finds that $g_{\mu\nu}R^{\mu\nu}\to 2/R^{2}$,
$R_{\mu\nu}R^{\mu\nu}\to 2/R^{4}$ and
$R_{\alpha\beta\gamma\lambda}R^{\alpha\beta\gamma\lambda}\to 4/R^{4}$,
corresponding to the curvature invariants of a 2-sphere of radius $R$. If
$R=0$ the curvature scalars are unbounded in the internal region, as they
diverge as $x\to-\infty$. However, such a behaviour does not represent
actually any pathology in the spacetime, since all the geodesics can be
extended forever in this geometry, and no spacetime singularity is present
(see discussion in Sec. III).

It is a well established fact that standard EWHs can be sustained in GR only
in the presence of exotic matter fields, resulting in the explicit violation
of the different energy conditions [12, 35]. Here, we demonstrate that our
modified EWH model, considered in the context of GR, does not evade such a
restriction, and exotic matter sources are still required222Here we do not
discuss into detail modified theories of gravity, where traversable wormhole
configurations can emerge in the absence of exotic matter sources [36, 37, 38,
39, 40, 41, 42, 43, 44]. For this purpose, it is convenient to introduce the
orthonormal frame

$\mathbf{e}_{\hat{t}}=\partial_{t},\;\mathbf{e}_{\hat{x}}=\partial_{x},\;\mathbf{e}_{\hat{\theta}}=\frac{\partial_{\theta}}{r(x)},\;\mathbf{e}_{\hat{\phi}}=\frac{\partial_{\phi}}{r(x)\sin\theta},$
(8)

that satisfy
$g_{\mu\nu}e^{\mu}_{\hat{a}}e^{\nu}_{\hat{b}}=\eta_{\hat{a}\hat{b}}$, where
$\eta_{\hat{a}\hat{b}}\equiv\text{diag}(-1,1,1,1)$ is the Minkowski metric. In
such a frame the Einstein tensor takes the form
$G_{\hat{a}\hat{b}}=G_{\mu\nu}e^{\mu}_{\hat{a}}e^{\nu}_{\hat{b}}$, whose
components are:

$\displaystyle G_{\hat{t}\hat{t}}=\dfrac{1-r_{x}^{2}-2r_{xx}}{r^{2}},$ (9)
$\displaystyle G_{\hat{x}\hat{x}}=\dfrac{r_{x}^{2}-1}{r^{2}},$ (10)
$\displaystyle
G_{\hat{\theta}\hat{\theta}}=G_{\hat{\phi}\hat{\phi}}=\dfrac{r_{xx}}{r}.$ (11)

By assuming for the source of the wormhole a fluid description, in the
orthonormal basis we can write the energy momentum tensor in the form
$T_{\hat{a}\hat{b}}=\text{diag}(\rho(x),-\tau(x),p(x),p(x))$, where its
components have a well-known physical interpretation in terms of the energy
density $\rho(x)$, the radial tension $\tau(x)$ and the lateral pressure
$p(x)$. Now, requiring that our model is a solution of GR, i.e.
$G_{\hat{a}\hat{b}}=8\pi T_{\hat{a}\hat{b}}$, we obtain the
expressions333Henceforth we will omit the dependence on $x$ in the components
of the energy momentum tensor.

$\displaystyle\rho=\dfrac{1-r_{x}^{2}-2rr_{xx}}{8\pi
r^{2}},\,\;\tau=\dfrac{1-r_{x}^{2}}{8\pi r^{2}},\,\;p=\dfrac{r_{xx}}{8\pi r},$
(12)

with $\rho=-2p+\tau$. In GR, energy conditions may be posed in terms of the
energy momentum tensor components in the above orthonormal system [45, 46].
They are the null energy condition (NEC)

$\rho-\tau\geq 0\text{ and }\rho+p\geq 0,$ (13)

the weak energy condition (WEC)

$\rho\geq 0,\,\,\rho-\tau\geq 0\text{ and }\rho+p\geq 0,$ (14)

the strong energy condition (SEC)

$\rho-\tau+2p\geq 0,\,\,\rho-\tau\geq 0\text{ and }\rho+p\geq 0,$ (15)

and the dominant energy condition (DEC)

$\rho\geq 0,\,\,\tau\in[-\rho,\rho]\text{ and }p\in[-\rho,\rho].$ (16)

As one expects, at the throat all four energy conditions are automatically
violated since $r_{x}(0)=0$ and $r_{xx}(0)>0$, i.e.

$\displaystyle\rho|_{x=0}=-\dfrac{2ar_{xx}(0)+1}{8\pi a^{2}}<0,$ (17)
$\displaystyle(\rho-\tau)|_{x=0}=-2p|_{x=0}=-\dfrac{2ar_{xx}(0)}{8\pi
a^{2}}<0.$ (18)

Since in the inner region the areal radius goes smoothly to a finite value,
its first and second derivatives vanishes asymptotically. Therefore, inside
the throat, when $r\to R>0$, the lateral pressure vanishes in the very inner
region of the object, whereas the density and radial tension are finite and
positive in this limit. The finiteness of these quantities were expected since
the curvature scalars are bounded in those models. On the other hand, when the
areal radius shrinks until it vanishes, all the components of
$T_{\hat{a}\hat{b}}$ blow up in the very inner region of the object. This is
due to the fact that $r$ goes to zero more rapidly than its derivatives,
therefore even the lateral pressure diverges in this limit.

##  III Geodesic analysis

In this section we examine into some details the geodesic trajectories of free
point-like particles in their motion over the two regions of the modified EWH.
The geodesic equation is obtained from the Lagrangian ${\cal
L}=\dot{s}^{2}/2=k/2$, where the overdot denotes a derivative with respect to
an affine parameter $\lambda$ and $k$ is the normalization of the four-
velocity ($k=-1,0$ for massive particles and light-rays, respectively). Due to
the symmetries of the Lagrangian, two quantities are conserved along the
geodesics, namely $E$ and $\ell$, respectively related with the time
translation symmetry (the Lagrangian is independent of $t$) and with the
rotational symmetry (the Lagrangian is independent of $\varphi$). Therefore,
in the equatorial plane the geodesic equation reads

$\dot{x}^{2}=E^{2}-\left(\dfrac{L^{2}}{r(x)^{2}}-k\right),$ (19)

and by conducting a thorough analysis of (19) together with the radial
function (2), we can unveil the underlying geodesic structure of the models we
propose.

Let us first consider radial geodesics ($L=0$) moving toward the asymptotic
internal region, which, regardless of the asymptotic internal 2-sphere radius
$R$, are given in this case by

$\displaystyle\dot{x}^{2}$ $\displaystyle=E^{2}+k,$ (20)
$\displaystyle\dot{r}^{2}$ $\displaystyle=r_{x}^{2}(E^{2}+k).$ (21)

Upon integration of (20), one obtains the trajectory for outgoing particles
that cross the throat into the inner region
$x(\lambda)=-\sqrt{(E^{2}+k)}\lambda+x_{0}$, where $x_{0}$ is an integration
constant. Therefore, one notices that radial geodesics can extend
indefinitely, regardless of the asymptotic 2-sphere radius. This is
particularly relevant when then inner 2-sphere radius shrinks to zero, and the
curvature scalars, energy density and pressures diverge in the asymptotic
limit $x\to-\infty$. Since it takes an infinite affine time $\lambda$ to reach
the asymptotic infinity, the region with ill-defined quantities is actually
inaccessible for massive or massless particles in radial motion. From Eq.
(21), one notices that the areal velocity $\dot{r}$ of particles in radial
motion goes to zero when $\lambda\to\infty$ (particles going to
$x\to-\infty$).

The analysis for non-radial geodesics is more involved, since we cannot obtain
an analytical expression for the geodesics. However, some approximations and
numerical analysis lead to some interesting conclusions. First, let us
consider that the asymptotic radius $R$ is finite and non-vanishing. When it
does happen, as one approaches the asymptotic internal region, $\dot{x}^{2}$
is approximately

$\dot{x}^{2}\approx E^{2}-\left(\dfrac{L^{2}}{R^{2}}-k\right),$ (22)

which can also be integrated leading to the conclusion that even non-radial
geodesic are complete for both massive and massless particles moving in the
internal region of the modified EWH with $R\neq 0$.

When $R=0$ an interesting feature happens. The effective potential
$V_{\text{eff}}=1/r^{2}(x)$ grows without bound in the internal region,
therefore any particle with non-zero angular momentum must suffer a bounce in
the internal region being reflected to the outer universe. The only particle
capable of propagating indefinitely within this object is one with zero
angular momentum, whether it is massive or massless, exhibiting purely radial
motion. Hence, even in the vanishing $R$ case, all geodesics are complete.

###  III.1 Photon orbits and photon spheres

To better understand the geodesic structure of the modified EWH, one may study
the orbits in these geometries. For simplicity let us consider null geodesics
($k=0$), and rewrite Eq. (19) as

$\dfrac{1}{r^{4}}\left(\dfrac{dx}{d\varphi}\right)^{2}=\dfrac{1}{b^{2}}-V_{\text{eff}}(x),$
(23)

where $b=L/E$ is the impact parameter that a photon has in the exterior
region. Since $x=0$ is a regular minimum of $r^{2}(x)$, it is also a local
maximum of the effective potential. Therefore, the throat radius $a$ plays the
role of a critical impact parameter of light-rays, since photons impinging
from infinity with impact parameter greater than the throat radius, $b>a$, do
not enter in the inner universe and are scattered back to infinity; while
photons with impact parameters smaller than the throat radius, $b<a$, do cross
the throat and enter in the inner universe; whereas, photons with impact
parameter equal to the throat radius, $b=a$, stay trapped in an unstable
orbit. The location of this unstable orbit is at the local maximum of the
effective potential, that is, $x=0$ or $r=a$. This orbit is called photon
sphere, and since it is at an unstable point it is called unstable photon
sphere. If $r(x)$ has a local maximum, the effective potential also exhibits a
local minimum in the inner universe. This local minimum corresponds to a
stable photon sphere, whose existence may support long-lived modes (radiation
may be trapped by these compact object). In Sec. IV we perform a discussion of
trapped scalar modes.

The behavior of the effective potential inside the throat deeply departs from
the standard profile of the EWH. For a non-vanishing asymptotic 2-sphere
radius, the potential goes to a barrier of magnitude $1/R^{2}$ as
$x\to-\infty$, while for $R=0$, the effective potential exponentially grows
inside the throat. Any photon able to enter in the inner universe ($b<a$),
propagates in a bounded 2-sphere universe, and the behavior of these curves
depends on the asymptotic radius $R$. For $R\geq a$, any photon crossing the
throat must propagate toward the asymptotic infinity, since the asymptotic
value of the effective potential $1/R^{2}\leq 1/a^{2}$. However for $R<a$,
after crossing the throat, depending on the photons’ impact parameter, photons
may suffer a bounce in the inner universe at $x_{b}$ and be scattered back to
the outer universe, since $1/R^{2}>1/a^{2}$. For non-radial geodesics ($L\neq
0$), the bounce happens if given an impact parameter $b$, there is an
$x=x_{b}$ such that $V_{\text{eff}}(x_{b})=1/b^{2}$ and
$V_{\text{eff}}>1/b^{2}$ for $x<x_{b}$. At $x_{b}$ therefore a bounce happens
and the particle is scattered to the outer universe. In Fig. 3 we show the
effective potential of the configurations depicted in Fig. 2, together with
the inverse of the impact parameter squared of some photons able to cross the
throat ($1/b^{2}>1/a^{2}$). The point $x=x_{b}$ where the vertical lines have
the same value of the effective potential is the bounce location, where a
photon is reflected in the inner universe and scattered back to the outer one.

Figure 3: Effective potential of four modified EWHs with fixed $c$ and four
choices of $R$, namely $R/a=0,0.5,1$ and 5. We also plot the inverse of the
impact parameter squared of some photons that enter in the inner region of the
modified EWH.

In Fig. 4 we show how the four configurations shown in Fig. 3 scatter light
rays with the same impact parameter in the outer universe. In the top-left
panel, the geodesics propagate in a modified EWH with vanishing asymptotic
2-sphere radius. As previously discussed, any non-radial geodesic moving
through the inner region must suffer a bounce. This behavior is shown in the
top-left panel of Fig. 4. When $R<a$, depending on the impact parameter the
geodesic can be scattered to the outer universe or propagate to the asymptotic
internal region. We exhibit this behavior in the top-right panel of Fig. 4. We
can see that, geodesics with impact parameter $b\leq R$ are scattered back to
the outer universe, and any other geodesic crossing the throat must propagate
to the asymptotic internal region with bounded 2-sphere. The bottom-left and
bottom-right panels, respectively, exhibit geodesics in $R=a$ and $R>a$
modified EWHs. In these configurations any photon crossing the throat to the
inner universe, propagates toward it to the asymptotic internal region with
bounded 2-sphere.

Figure 4: Null geodesics in the modified EWH. Solid lines correspond to light
rays propagating in the outer universe, while dashed lines represent geodesics
in the inner universe. We are considering photons with the same values of
impact parameter in the outer universe, and showing how these photons are
scattered or absorbed depending on the modified EWH configuration. The circle
with radius 1 corresponds to the throat of the wormhole; the outermost circle
corresponds to the local maximum of the areal radius $r(x)$ in the inner
universe. The other circles are the asymptotic 2-sphere radius of each
configuration.

##  IV Scalar field perturbations

In order to extract valuable information about the geometries investigated in
this study, we turn our attention to the analysis of scalar perturbations on
the background metric $g_{\mu\nu}$. By examining the evolution of these
perturbations, we can discern the distinctive effects that these geometries
imprint on the time-domain profiles, making them distinguishable from other
black hole and wormhole configurations.

For a massless scalar field $\Psi$ localized within the background metric
$g_{\mu\nu}$, the dynamics of the field are governed by the Klein-Gordon
equation

$\Box\Psi(t,x,\theta,\varphi)=0.$ (24)

Due to the spherical symmetry of the problem, we can decompose the field in
the following way

$\Psi(x^{\mu})=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\frac{\Phi(x,t)}{r(x)}Y_{\ell
m}(\theta,\varphi),$ (25)

where $Y_{\ell m}(\theta,\varphi)$ denotes spherical harmonics of degree
$\ell$ and order $m$. By substituting (25) in (24), one obtains that the
radial function $\Phi(x,t)$ must satisfy

$\displaystyle\left(\dfrac{d^{2}}{dt^{2}}-\dfrac{d^{2}}{dx^{2}}+V_{\Phi}\right)\Phi=0,$
(26)

where the effective potential $V_{\Phi}$ is given by

$V_{\Phi}=\dfrac{\ell(\ell+1)}{r(x)^{2}}+\dfrac{r_{xx}}{r(x)}.$ (27)

Now, in order to integrate the wave equation (26) we follow the procedure
described in [47]. This involves introducing light-cone coordinates,
specifically the advanced time coordinate denoted as $v\equiv t+x$ and the
retarded time coordinate denoted as $u\equiv t-x$. Thus, the wave equation can
be expressed as

$\left(4\frac{d^{2}}{dudv}+V_{\Phi}\right)\Phi=0.$ (28)

The integration of this differential equation is done numerically on a null
grid which leads to the following expression for the discretized scalar field
evolution

$\Phi_{N}=\Phi_{E}+\Phi_{W}-\Phi_{S}-\frac{h^{2}}{8}V_{\Phi}(S)(\Phi_{W}+\Phi_{E})+O(h^{4}),$
(29)

where $h$ is the stepsize between two neighboring grid points and subscripts
indicate the point in the grid where the function is evaluated. Explicitly,
$S=(u,v)$, $W=(u+h,v)$, $E=(u,v+h)$ and $N=(u+h,v+h)$, as can be seen more
clearly in Fig 5.

Figure 5: Representation of the numerical grid used for the integration of
(28). The evaluation points of (29) are represented with $S$, $W$, $E$ and
$N$. The stepsize of the grid can be visualized by the distance between to
consecutive points in the same axis $h=u_{E}-u_{S}=v_{N}-v_{E}$.

The initial distribution for the scalar perturbation is set on the null
surfaces $u=0$ and $v=0$. Then, the grid is computed line by line using the
mechanism described in (29), with a stepsize $h=0.1$ and grid values ranging
from $u_{\text{min}}=0$ to $u_{\text{max}}=1000$ and $v_{\text{min}}=0$ to
$v_{\text{max}}=1000$. As initial conditions for the scalar perturbation we
use a Gaussian distribution on the $u=0$ surface, together with a constant
profile on the $v=0$ surface, i.e.:

$\Phi(0,v)=Ae^{-(v-v_{c})^{2}/(2\sigma^{2})},$ (30)

with height $A=1$, width $\sigma^{2}=4.5$ and centered at $v_{c}=20$.

Figure 6: The radial profiles of the effective potential (27) are presented
for three $\ell$-modes and $R/a=0$. The top panel displays the case for
$\ell=0$, the central panel shows $\ell=1$, and the bottom panel shows
$\ell=2$. The solid line represents the radial profile of the effective
potential for the standard EWH, while dashed lines correspond to three
different configurations of modified EWH with varying $c$ parameters.  Figure
7: The radial profiles of the effective potential (27) are presented for four
configurations with different $R$ parameter and $\ell=1$.

The effective potential is then calculated by applying (27) and using the
radial function $r(x)$ as presented in (2). Radial profiles of the effective
potential with $R/a=0$ are depicted in Fig. 6, where, in order to optimize the
grid we designed, the throat of the wormhole has been conveniently shifted to
$x=-20$. As one can see, the effective potential exhibits a peak like in
standard EWHs, which is associated to the throat of the wormhole, and with
their maximum value increasing in proportion to $\ell$. Conversely, as $x$
tends to $-\infty$, the potential does not drop to zero, but it shows a rapid
and smooth growth, remaining finite for all finite radial values. It is
remarkable that the effective potential exhibits a significantly slower growth
as $x$ approaches $-\infty$ for the fundamental $\ell$-mode as compared to
higher $\ell$-modes. Moreover, with increasing values of $\ell$, it approaches
infinity at a faster rate, although the most prominent discrepancy in growth
rate is observed between $\ell=0$ and $\ell=1$. Such a behavior, combined with
the first peak, gives rise to a well, which is expected to lead to echoes in
the time-domain spectrum (see discussion in the next section). For the
fundamental $\ell$-mode, the effective potential assumes negative values
within the well near the peak associated with the throat of the wormhole. The
radial extent over which negative effective potential occurs diminishes as
$\ell$ increases. Finally, it is worth mentioning that the parameter $c$
exerts an influence on the effective potential, causing the well formed
between the throat peak and the asymptotic boundary to narrow as its value
increases.

For the non-vanishing $R$ case, one can show that the asymptotic behavior of
$V_{\Phi}$ reads

$\displaystyle\lim_{x\to+\infty}V_{\Phi}=0,$ (31)
$\displaystyle\lim_{x\to-\infty}V_{\Phi}=\left\\{\begin{array}[]{ll}0,&\ell=0,\\\
\dfrac{\ell(\ell+1)}{R^{2}},&\ell\neq 0.\end{array}\right.$ (34)

It is worth nothing that in the asymptotic internal region, whenever $\ell\neq
0$ and $R\neq 0$, the effective potential goes to a threshold value. The
massless scalar field propagating in this region would behave equivalently to
a scalar field with effective mass in a Minkowski background, namely

$\left(\dfrac{d^{2}}{dt^{2}}-\dfrac{d^{2}}{dx^{2}}+\mu_{e}^{2}\right)\Phi=0,$
(35)

where $\mu_{e}=\sqrt{\ell(\ell+1)}/R$. This kind of effective mass in scalar
field dynamics typically arises in non-asymptotically flat spacetimes, such as
when scalar waves propagate around a black hole immersed in a magnetic field
[48, 49]. For the case $\ell=0$, the effective potential vanishes
asymptotically in both sides of the modified EWH, therefore no effective mass
term appears.

In Fig. 7, we depict the radial profiles of the effective potential for
$\ell=1$ and various values of the $R$ parameter. As observed in the geodesic
analysis, the presence of a non-zero $R$ leads to the asymptotic finiteness of
the potential as $x\to-\infty$. The finite value towards which the potential
tends is inversely proportional to the magnitude of $R$, with the potential
becoming approximately two orders of magnitude smaller than the throat peak
for $R/a=5$.

As the 2-sphere approaches its asymptotic value, the effective potential
exhibits a sort of effective centrifugal barrier for $R=0$, playing the role
of an effective mirror for the scalar field perturbation. For $R\neq 0$ in
this same limit, one can see that the effective potential tends to a plateau
which height is proportional to $1/R^{2}$.

###  IV.1 Discussion of the results

Figure 8: Time-domain profile of the absolute value of the scalar field
perturbation for the $\ell=1$ mode. The top panels depict identical geometries
associated with the EWH. In the left column, three configurations with varying
values of the parameter $c$ and $R/a=0$ are displayed. In the right column,
three configurations with different values of the parameter $c$ are shown, but
with $R/a=5$. The disparity between the EWH and the other cases are the so-
called echoes.

After solving the discretized wave equation, we extract the scalar field
values at the observation point $x_{\text{obs}}=0$ using the coordinate
transformation $x=(u-v)/2$ and $t=(u+v)/2$. The numerical integration results
are displayed in Fig. 8. In the top panels, the time-domain evolution of the
scalar field for an EWH is shown as a reference. Even though our model allows
arbitrary large values of $c$, here we are focusing on small deviations of
EWHs in the throat scale, therefore restricting our analysis to $a^{2}c\leq
0.01$. This makes with the potential peak at the throat to be almost the same
as EWH (see Figs. 6 and 7). As the initial wave packet impinges on the
modified EWH, it encounters the throat peak first, causing a portion of the
wave to return to the observation point and exhibiting a characteristic
ringdown. Due to the similarities between the throat peaks of modified and
standard EWHs, the prompt contribution and initial ringdown signal in the time
profile of scalar perturbations are expected to be basically the same in both
scenarios.

The portion of the scalar wave packet that is transmitted by the throat peak
passes through the well and encounters the potential barrier extending to
$x\to-\infty$. It then reflects back towards the observation point. However,
in order to reach $x_{\text{obs}}$, it must interact with the throat peak once
again. A portion of the incident wave is reflected, repeating the same
process, while another portion is transmitted. The transmitted portion, after
being detected at the observation point, evolves towards $x\to\infty$ and does
not pass through the observation point again. Each time a wave is reflected
and passes through the observation point, it is recorded as a peak in the
time-domain profile of $|\Phi_{l}|$. As expected based on the description of
the effective potential of modified EWH, (unstable) photon sphere modes from
the scalar perturbation exist and ring in the same way as in standard EWH.
However, in this case, there is a stable photon sphere in the inner universe,
causing the bounded 2-sphere region of the wormhole to act as a cavity,
trapping photon sphere modes. This results in a series of echoes in the scalar
perturbation time profile, which are illustrated in Fig. 8. The
characteristics of these echoes vary depending on the parameters used to
construct the spacetime, since the width and height of the cavity in the
effective potential deeply depend on $c$ and $R$.

Let us now discuss the $R=0$ spacetimes, corresponding to the left column of
Fig. 8. It can be observed that there is a relationship between the value of
the parameter $c$ and the frequency of the echoes. This arises from the fact
that $c$ influences the width of the effective potential well. As the well
becomes narrower with increasing $c$, it can be observed that the time
interval between two consecutive echoes decreases (or, equivalently the
frequency increases), as the waves have to travel a shorter distance.

Another notable feature that can be observed is the gradual decay of peak
amplitudes. Each time the scalar wave packet interacts with the throat peak,
it is divided into transmitted and reflected parts, resulting in weaker
successive echoes. This effect contrasts with the case of the EWH, where the
signal exhibits a rapid decay compared to the modified Ellis scenario. The
presence of two asymptotic infinities and the absence of a potential well in
the EWH prevent the emergence of echoes, thereby contributing to the faster
signal decay. When the potential well is narrower, the superposition of
different echoes becomes more noticeable, leading to deformations in their
waveforms. This behavior is particularly evident in the late-time regime, as
shown by the bottom plot in the left column of Fig. 8.

Similar features are noticeable also in the right column of Fig. 8,
corresponding to the $R/a=5$ configuration. However, the echoes’ amplitudes
are notably smaller compared to the previous scenario, making it difficult to
differentiate distinct echoes due to their reduced amplitude.

The absence of an unbounded growing effective potential results in wave
packets interacting with a finite barrier, leading once again to the division
of the package into a reflected part and a transmitted part that propagates
towards $x\to-\infty$ without bouncing back to the outer universe. The height
of the barrier diminishes as $R$ increases, making it evident that the
reflected part of the wave also is smaller as one considers bigger values of
$R$, as shown in Fig. 9. Notably, there is a distinct transition regime
depending on the asymptotic value to which the effective potential tends.
Specifically, the asymptotic value aligns with the height of the throat peak
for $R/a\approx 0.8$ for the considered configurations. When the barrier
exceeds the throat peak height ($R/a<0.8$), the echoes are easily
distinguishable. However, when the barrier is smaller ($R/a>0.8$), a higher
proportion of the wave is lost, leading to a reduction in the amplitude of the
echoes.

For late-times, we observe a diminishment in the damping of the echoes of the
scalar wave. At this point, so-called quasi-resonances are present—arbitrarily
long-lived QNMs [50]. These modes were discussed in the QNM analysis of
massive fields in asymptotically flat spacetimes [50, 51], and also appears in
non-asymptotically flat scenarios where the wave equation acquires a sort of
effective mass [49, 52]. Our model pertains to the latter case.

Remarkably, in the asymptotic limit $R\rightarrow\infty$, the effective
potential $V_{\Phi}$ recovers the usual behaviour of the one of EWH in the
inner universe, that is, $V_{\Phi}\to 0$ as $x\to-\infty$. Therefore, the
time-domain profile of modified EWH with large asymptotic 2-sphere radius,
tends towards the expected profile of a standard EWH.

Finally, it is noteworthy that while the results presented here are for the
$\ell=1$ mode, the qualitative characteristics are present in the other
$\ell$-modes as well.

Figure 9: Time-domain profile of the absolute value of the scalar field
perturbation for the $\ell=1$ mode. Seven configurations with the following
values $R/a=\left\\{0,0.5,0.9,1,10,20,40\right\\}$ are plotted, along with the
EWH as a reference.

##  V Final remarks

We have introduced a new class of modified EWHs, where one side of the global
spacetime geometry, usually consisting in an asymptotic Minkowski region, is
supplanted by a bounded 2-sphere patch, where the areal radius is finite and
asymptotically constant. The resulting asymmetric wormhole is still endowed
with a local minimum in the areal radius, i.e. a throat-like structure, where
a transition between the two regions takes place. This was achieved by means
of a modified radial function, which smoothly interpolates from the standard
Ellis case to the bounded 2-sphere core on the other side.

In the internal region of the modified EWH, the areal radius exhibits a local
maximum and eventually collapses to its asymptotic value $R$. Such a peculiar
behavior is ultimately responsible for a rich geodesic structure. We performed
an extensive analysis on causal geodesics, finding that regardless of the
model parameters, the modified EWH is geodesically complete, even in the
vanishing asymptotic 2-sphere scenario, $R=0$. This is indeed relevant,
because in the vanishing 2-sphere radius case, the curvature scalars, energy
density and pressures are unbounded as $x\to-\infty$, but these divergences
remain unreachable for causal geodesics. In the $R\neq 0$ scenario, the
curvature scalars are well-defined everywhere and go to the ones of a 2-sphere
with radius $R$ in the asymptotic internal region.

We also investigated photon orbits in this model by numerically solving the
geodesic equation. When the asymptotic value of the 2-spheres radius is lower
than the throat radius, $R<a$, there are light rays that depending on their
impact parameter suffer a bounce in the inner universe and are scattered back
to the outer universe. As $R$ diminishes, and the inner region becomes more
bounded, the photon’s impact parameter necessary for the bounce to happen is
smaller (see Fig. 3). The extreme scenario is then for $R=0$, where all
geodesics entering the inner universe are scattered back to the outer one, and
the only null geodesic that can propagate forever in the inner region is the
one of a photon in purely radial motion. Lastly, for $R\geq a$, any photon
that crosses the throat propagates towards the asymptotic infinity with
bounded 2-sphere radius.

The analysis of the effective potential shows that an unstable photon sphere
is present at the throat, similarly to the standard EWH. Moreover, the inner
universe has an additional structure, a stable photon sphere, that is located
in the local minimum of the effective potential. The existence of the stable
photon sphere allows long-lived modes to be trapped, giving rise to the
appearance of echoes in the ringdown profile, usually absent in standard EWHs.
In order to probe this aspect, we performed an analysis of massless scalar
excitations in the modified EWH background. After being transmitted to the
inner universe the scalar perturbation interacts with an effective centrifugal
barrier for $R=0$ or a step potential for $R\neq 0$. Therefore, part of the
modes are trapped in a potential well and observed as a series of echoes in
the time profile of the scalar perturbation. In particular, we noticed that
the role of the parameter $c$ is to modify the time delay between two
successive echoes. When $c$ increases, the cavity in the effective potential
becomes narrower and the echoes’ time delay diminishes.

The role of the asymptotically bounded 2-sphere radius $R$ is more subtle. For
the $R=0$ case, the scalar perturbations get trapped between the unbounded
barrier and the photon sphere at the throat. By increasing $R$, this behavior
persists, but since the unbounded barrier is replaced by a step potential
barrier, part of the modes can be transmitted through the asymptotic internal
region.

We notice that by replacing the asymptotically Minkowski spacetime with a
bounded 2-sphere core, the scalar field acquires an effective mass in the
asymptotic internal region for non-vanishing values of the multipole number
$\ell$ and non-zero asymptotic 2-sphere radius $R$. A similar effect is
observed with scalar waves propagating around magnetized black holes [49].
This leads to the presence of quasi-resonances in the late-time profile, which
are arbitrarily long-lived QNMs. Whether $R=0$ and $\ell\neq 0$, the effective
mass grows without bound acting as an effective mirror for the scalar wave.
Consequently, the scalar perturbation cannot reach the asymptotic internal
region, and the modes trapped near the stable photon sphere at some late time
should tunnel to the outer universe.

Our results indicate that the observational features of wormholes are
crucially dependent on their global characteristics, even though they can
share a very similar throat-like structure. The study of perturbations can be
used to extract valuable information about the compactness of the inner
universe, and this could be useful in future spectroscopy experiments trying
to identify new compact objects. We remark that the geometrical structure of
the model described in the present work can be enriched further either by
considering Schwarzschild-like scenarios, or by restricting the domain of the
bounded 2-sphere patch, i.e. truncating the range of the possible values
spanned by the coordinate $x$ and imposing specific boundary conditions. These
configurations are expected to generate a great variety of phenomenological
signatures due to the appearance of additional potential barriers or closed
universe effects in geodesic motion. The study of such aspects is currently
underway and will be the subject of forthcoming works.

###### Acknowledgements.

The authors would like to acknowledge Fundação Amazônia de Amparo a Estudos e
Pesquisas (FAPESPA), Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES) – Finance Code 001, from Brazil, for partial financial
support. This work is supported by the Spanish Grant PID2020-116567GB- C21
funded by MCIN/AEI/10.13039/501100011033, and the project PROMETEO/2020/079
(Generalitat Valenciana). This research has further been supported by the
European Union’s Horizon 2020 research and innovation (RISE) programme
H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740 and by the European Horizon
Europe staff exchange (SE) programme HORIZON-MSCA-2021-SE-01 Grant No.
NewFunFiCO-101086251. LC thanks University of Aveiro, in Portugal, for the
kind hospitality during the completion of this work. AMF is supported by the
Spanish Ministerio de Cienciea y Innovación with the PhD fellowship
PRE2018-083802. The work of FB is supported by the postdoctoral grant
CIAPOS/2021/169.

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