# Geodesic structure of generalized Vaidya spacetime through the K-essence

Bivash Majumder<EMAIL_ADDRESS>Department of Mathematics, Prabhat
Kumar College, Contai, Purba Medinipur 721404, India  Maxim Khlopov
<EMAIL_ADDRESS>Institute of Physics, Southern Federal University, 194
Stachki, Rostov-on-Don 344090, Russian Federation $\&$
National Research Nuclear University, MEPHI, Moscow, Russian Federation $\&$
Virtual Institute of Astroparticle Physics 10, rue Garreau, 75018 Paris,
France  Saibal Ray<EMAIL_ADDRESS>Centre for Cosmology, Astrophysics
and Space Science (CCASS), GLA University, Mathura 281406, Uttar Pradesh,
India  Goutam Mannab<EMAIL_ADDRESS>
bCorresponding author  Department of Physics, Prabhat Kumar College, Contai,
Purba Medinipur 721404, West Bengal, India $\&$
Institute of Astronomy, Space and Earth Science (IASES), Kolkata 700054, West
Bengal, India

(Received: date / Accepted: date)

###### Abstract

This article investigates on the radial and non-radial geodesic structures of
the generalized K-essence Vaidya spacetime. Within the framework of K-essence
geometry, it is important to note that the metric does not possess conformal
equivalence to the conventional gravitational metric. This study employs a
non-canonical action of the Dirac-Born-Infeld kind. In this work, we
categorize the generalized K-essence Vaidya mass function into two distinct
forms. Both the forms of the mass functions have been extensively utilized to
analyze the radial and non-radial time-like or null geodesics in great details
inside the comoving plane. Indications of the existence of wormhole can be
noted during the extreme phases of spacetime, particularly in relation to
black holes and white holes, which resemble the Einstein-Rosen bridge. In
addition, we have also detected the distinctive indication of the quantum
tunneling phenomenon around the central singularity.

K-essence emergent gravity, Modified theories of gravity, Friedman equations,
Dark energy

###### pacs:

04.20.-q, 04.50.-h, 04.50.Kd

##  I Introduction

S. Chandrasekhar extensively analyzed the time-like and null-geodesic features
of the Schwarzschild spacetime in his book [1]. In addition, he has examined
the orbital configurations of both the confined and unconfined trajectories
using graphical representations. In addition, the authors in [2] have examined
the geodesic structures of the Schwarzschild anti-de Sitter spacetime. The
researchers assessed both radial and non-radial paths for time-like and null
geodesics. Additionally, they have demonstrated that the geodesic structures
of this black hole exhibit distinct forms of motion that are not permitted by
the Schwarzschild spacetime. The geometric framework of the Schwarzschild
spacetime is also examined in [3]. The Jacobi metric for time-like geodesics
in static spacetimes has been examined in the reference [4]. They have
demonstrated that the unrestricted movement of large particles in stationary
spacetimes is determined by the geodesics of a Riemannian metric that depends
on the particle’s energy. This metric is similar to Jacobi’s metric in
classical dynamics. When the mass of an object approaches zero, Jacobi’s
metric becomes identical to the Fermat or optical metric, which does not
depend on energy. In addition, they have provided a detailed account of the
characteristics of the Jacobi metric pertaining to the motion of heavy
particles beyond the event horizon of a Schwarzschild black hole. The authors
of [5] derived the Jacobi metric for different stationary metrics and
developed the Jacobi-Maupertuis metric for time-dependent metrics by using the
Eisenhart-Duval lift [6, 7]. The authors in [8] have documented the remarkable
characteristics of the time-like geodesic structure when dark energy is
present in an emergent gravity framework, specifically for the Barriola-
Vilenkin metric [9]. The K-essence emergent gravity metric is precisely
correlated with the Barriola-Vilenkin (BV) metric for the Schwarzschild
background, specifically for a certain form of K-essence scalar field [10].
The researchers have analyzed the various paths that time-like geodesics can
take in the presence of dark energy in the Barriola-Vilenkin spacetime [8],
which is equivalent to the Schwarzschild spacetime in terms of its fundamental
structure. However, the permissible ranges for the maximum and minimum
distances from the central object are significantly distinct. For a constant
dark energy density, the orbits, both bound and unbound, are graphed.

In 1951, P. C. Vaidya proposed the first relativistic line element that
properly represented the spacetime of a conceivable star [11]. It extended the
specific solution of Schwarzschild by depicting the emission of radiation for
a mass that is not in a static state. The Schwarzschild solution describes the
geometry of spacetime around a spherically symmetric, non-rotating, black
object with a constant mass. Therefore, it is clear that the model is
incapable of accurately depicting spacetime outside the confines of a star.
The solution proposed by Vaidya [11], known as the Vaidya spacetime or the
radiating Schwarzschild metric, was introduced as a possible explanatory
framework. The main distinction between the two metrics is that the Vaidya
metric adds a time-dependent mass parameter, whereas the Schwarzschild metric
uses a constant mass value. As a result, the spacetime in the Vaidya metric
evolves with time. The Vaidya metric is primarily used to investigate
gravitational collapse. The occurrence of gravitational collapse is widely
acknowledged in the disciplines of general relativity and astrophysics, as
demonstrated by the research conducted by Joshi et al. [12, 13, 14, 15, 16,
21, 17, 18]. It plays a vital role in understanding several astrophysical
aspects of our cosmos. The phenomenon of gravitational collapse provides
useful insights into several elements of astronomy, including the evolution of
structures, the features of stars, the genesis of black holes, and the
construction of white dwarfs or neutron stars, among other events.
Gravitational collapse refers to the phenomenon in which a star collapses as a
result of its mass. The outcome of this collapse might vary depending on the
exact beginning mass conditions, leading to distinct stages of collapse.
Papapetrou [19] was the first to demonstrate that the solution of a null dust
fluid with spherical symmetry in gravitational collapse can lead to the
creation of naked singularities. This statement presents a counterexample of
the cosmic censorship hypothesis (CCH) as proposed by Penrose [20]. The
authors in [14, 22] have provided a detailed account of the causal paths that
connect the singularities in the continuing Vaidya scenario. Furthermore, a
comprehensive classification of the non-spacelike geodesics that link the
naked singularity in the past is presented, offering a rather thorough
discussion of the restrictions involved. It is subsequently demonstrated to be
a robust curvature singularity in a more significant manner.

The Vaidya solution, as a generalization, encompasses all the established
solutions of Einstein’s field equations that include a mix of Type-I and Type-
II matter fields [23, 24, 25, 26, 27, 28]. The composition of this work is
attributed to Husain [29] and Wang $\&$ Wu [30]. The extension of the Vaidya
solution is sometimes referred to as the generalized Vaidya spacetime. The
work performed in [31] examines the gravitational collapse of the generalized
Vaidya spacetime within the framework of the cosmic censorship theory. They
demonstrated that the categories of generalized Vaidya mass functions had
emerged in the situation, suggesting the end of collapse with a locally
visible central singularity. The authors computed the magnitude of these
singularities. A comprehensive mathematical framework was created to examine
the requirements for the mass function for non-spacelike geodesics going
towards the future to end at the singularity in the past. Furthermore, they
demonstrated that, when considering a certain generalized Vaidya mass
function, the ultimate outcome of the collapse can be precisely defined as
either a black hole or a naked singularity. The work by Patil [32] examines
the phenomenon of gravitational collapse in higher dimensions within the
context of the charged-Vaidya spacetime. It has been demonstrated that
singularities occur in a charged null fluid in a higher dimension. These
singularities consistently lack any form of covering, hence contradicting the
strong CCH. This idea does not specifically pertain to weak cosmic censorship.
The Vaidya metric has received significant attention in scholarly research,
with several major contributions to our comprehension of this subject. The
authors of the study [33] examined the geometric properties of Vaidya’s
spacetime while considering a white hole that undergoes a decrease in mass.
They found that the white hole can either stabilize and transform into a black
hole within a limited or indefinite amount of time, or entirely evaporate. The
researchers have focused specifically on the scenario of total evaporation
over an indefinite period of time. They have successfully demonstrated the
presence of an asymptotic light-like singularity in the conformal curvature,
which connects both the past space-like singularity and the future time-like
infinity. Vertogradov [34] conducted a study on the structure of the
generalized Vaidya spacetime, specifically focusing on the case when the
matter field of type-II follows the equation of state $P=\rho$. The findings
of the study revealed the presence of an eternal naked singularity in this
spacetime, which meets all energy conditions. Once formed, the singularity
will remain perpetually uncovered by the apparent horizon. Nevertheless, the
formation of the apparent horizon leads to the emergence of a white hole.
Solanki et al. [35] have derived precise mathematical equations that describe
the changes in the photon sphere and the angular radius of the shadow in a
certain Vaidya spacetime. The mass function $m(v)$ has been seen as a function
of time that either increases or decreases linearly. The initial scenario can
function as a basic representation of a black hole that is accumulating
matter, whereas the subsequent scenario can be seen as an illustration of a
black hole that is emitting radiation, as theorized by Hawking.

In the realm of K-essence geometry, Manna et al. [36] were the first to
establish a link between K-essence geometry and Vaidya spacetime. They
achieved this by introducing a new definition of the generalized Vaidya mass
function, which directly depends on the kinetic energy of the K-essence scalar
field. Subsequently, Manna et al. [37] demonstrated that the K-essence
emergent gravity metric bears a strong resemblance to the recently found
generalized Vaidya metrics for the collapse of a null fluid. This similarity
arises from the presence of a k-essence emergent mass function. Notably,
Manna’s analysis exclusively considers the K-essence scalar field as a
function of either the advanced or the retarded time. The recently developed
K-essence model, known as the K-essence emergent Vaidya spacetime, has
successfully met all the necessary energy conditions. The presence of the
centrally exposed singularity and the intensity and stability of the
singularities in the K-essence emergent Vaidya metric yield intriguing results
in their research. The evaporation of the dynamical horizon with the Hawking
temperature in the K-essence Vaidya Schwarzschild spacetime was investigated
by Manna et al. in [38]. This study uses the dynamical horizon equation to
quantify the reduction in mass caused by Hawking radiation. Additionally, the
tunneling formalism, namely the Hamilton-Jacobi technique, is utilized to
compute the Hawking temperature. In addition, Sawayama’s revised explanation
of the dynamical horizon [39] is utilized to demonstrate that the results
obtained differ from the conventional Vaidya spacetime geometry. The authors
have established by analytical measures that the mass of the black hole,
denoted as $m(v,r)$, in the K-essence emergent Schwarzschild-Vaidya spacetime,
consistently decreases over time but does not fully evaporate.

The K-essence theory is a scalar field model that deviates from the canonical
form. In this theory, the dominant energy component of the field is its
kinetic energy, rather than its potential energy. This concept has been
extensively studied by several researchers [40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 50]. The distinctions between the K-essence theory employing a non-
canonical Lagrangian and the relativistic field theories utilizing a canonical
Lagrangian are found in the sophisticated dynamical solutions of the K-essence
equation of motion. These solutions not only spontaneously violate Lorentz
invariance but also alter the metric for the perturbations around them. The
disturbances propagate in the emergent or analogous curved spacetime,
characterized by a metric distinct from the gravitational metric. The non-
canonical Lagrangian may be expressed as $\mathcal{L}(X)=-V(\phi)F(X)$, where,
$X=\frac{1}{2}g_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi$, $\phi$ is the
K-essence scalar field, $V(\phi)$ is the potential term. An alternate form of
the Lagrangian, as described by Tian [51], may be represented as
$\mathcal{L}=[1+f(y)]X+[1+g(y)]V_{exp}$, where
$V_{exp}=V_{0}\leavevmode\nobreak\ exp(-\lambda\phi)$, $V_{0}$ and $\lambda$
are constants, $y=X/V_{exp}$, and $f(y)$ and $g(y)$ are arbitrary functions.
The functions $f(y)$ and $g(y)$ are unrestricted and can have any form.
Furthermore, it is important to mention that there exist examples of K-essence
theories that are not minimally linked, as mentioned in the Refs. [52, 53,
54]. Nevertheless, this article only addresses the minimally coupled K-essence
theory, as investigated by the Refs. [40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50]. In a general sense, the Lagrangian has the capacity to depend on any
functions of $\phi$ and $X$. The K-essence theory offers the benefit of
circumventing both the fine-tuning problem and the coincidence problem [55] of
the current universe. Additionally, it generates the negative pressure
required for the universe’s acceleration only through the kinetic energy of
the field. The kinetic term of the field dominates over the potential term.
The article [46] presents attractor solutions where the dynamics of the cosmos
are governed by the scalar field of the models. During the radiation-dominated
phase, the K-essence field mimicked the equation of state of radiation and had
a constant ratio to the radiation density. The K-essence field was unable to
replicate the dust-like equation of state (EoS) due to dynamical limits during
the time dominated by dust. However, it rapidly reduced its energy value by
many orders of magnitude and eventually reached a constant value.
Subsequently, over a period approximately equivalent to the current age of the
universe, the density of matter was diminished by the K-essence field, leading
to the commencement of cosmic acceleration. The equation of state (EoS) of the
K-essence theory ultimately converges to a value within the range of 0 to -1.
Although in theory, it has the potential to extend beyond $-1$. Another
intriguing aspect of the K-essence idea is its potential to generate a type of
dark energy where the speed of sound is consistently slower than that of
light. This feature may mitigate the cosmic microwave background (CMB)
disruptions on large angular scales [56, 57, 58]. In this specific situation,
Manna et al. [8, 10, 36, 37, 38, 62, 63, 64, 65, 66, 67] have developed a
fascinating emergent gravity metric referred to as $\bar{G}_{\mu\nu}$. This
metric possesses distinct attributes in contrast to the standard gravitational
metric $g_{\mu\nu}$ and is derived from the notions of the Dirac-Born-Infeld
(DBI) type action, as outlined in the works [70, 71, 72, 73]. Dirac et al.
proposed a non-canonical Lagrangian in order to eliminate the infinite self-
energy of the electron, as described in their work [73]. The specific reasons
and objectives for selecting the non-canonical theory, such as the K-essence
theory, may be found in the Refs.[68, 69]. The Planck collaborations’
findings, as shown in Refs. [59, 60, 61], have examined the empirical evidence
supporting the concept of K-essence with a DBI-type non-canonical Lagrangian,
along with other modified theories. Furthermore, it has been noted that the
K-essence theory may be applied in a model that combines dark energy and dark
matter [8, 10, 62, 63, 64, 65, 47], as well as from a purely gravitational
perspective [36, 37, 38, 66, 67].

This article is organized as follows: In section 2, we provided a concise
explanation of the K-essence geometry and its connection to the conventional
generalized Vaidya spacetime, which leads to the construction of a new
generalized K-essence Vaidya spacetime. Section 3 offers a comprehensive
analysis of the geodesic structures observed in the generalized K-essence
Vaidya spacetime. This analysis considers two forms of mass function while
ensuring that the condition on the kinetic energy of the K-essence scalar
field is maintained. This section also provides a detailed analysis of the
radial and non-radial geodesics used to examine the structure of time-like and
null geodesics in the given spacetime. This is achieved by solving the Euler-
Lagrange equations. The graphical and numerical analysis is also done in this
section. In section 4, we will wrap up both the conclusion and the discussion.

##  II Brief of the relation between K-essence with generalized Vaidya
spacetime

This section offers a short introduction to the geometry of K-essence and the
generalized Vaidya spacetime. Initially, we present a brief summary of the
geometric aspects related to the K-essence, as extensively explored in many
scholarly references [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. The action
performed by this geometry is

$\displaystyle S_{k}[\phi,g_{\mu\nu}]=\int
d^{4}x{\sqrt{-}g}\mathcal{L}(X,\phi)$ (1)

where the expression $X=\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$
represents the canonical kinetic term, whereas $\mathcal{L}(X,\phi)$ denotes
the non-canonical Lagrangian. In this scenario, the conventional gravitational
metric $g_{\mu\nu}$ has formed a minimum coupling with the K-essence scalar
field ($\phi$).

The energy-momentum tensor that corresponds solely to the K-essence scalar
field is:

$\displaystyle T_{\mu\nu}\equiv\frac{-2}{\sqrt{-g}}\frac{\delta S_{k}}{\delta
g^{\mu\nu}}=-2\frac{\partial\mathcal{L}}{\partial
g^{\mu\nu}}+g_{\mu\nu}\mathcal{L}$
$\displaystyle=-\mathcal{L}_{X}\nabla_{\mu}\phi\nabla_{\nu}\phi+g_{\mu\nu}\mathcal{L},$
(2)

where $\mathcal{L}_{\mathrm{X}}=\frac{d\mathcal{L}}{dX},\leavevmode\nobreak\
\mathcal{L}_{\mathrm{X}X}=\frac{d^{2}\mathcal{L}}{dX^{2}},\leavevmode\nobreak\
\mathcal{L}_{\mathrm{\phi}}=\frac{d\mathcal{L}}{d\phi}$ and $\nabla_{\mu}$ is
the covariant derivative defined with respect to the gravitational metric
$g_{\mu\nu}$.

The equation of motion (EOM) for the K-essence scalar field is

$\displaystyle-\frac{1}{\sqrt{-g}}\frac{\delta
S_{k}}{\delta\phi}=\tilde{G}^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi+2X\mathcal{L}_{X\phi}-\mathcal{L}_{\phi}=0,$
(3)

where

$\displaystyle\tilde{G}^{\mu\nu}\equiv\frac{c_{s}}{\mathcal{L}_{X}^{2}}[\mathcal{L}_{X}g^{\mu\nu}+\mathcal{L}_{XX}\nabla^{\mu}\phi\nabla^{\nu}\phi],$
(4)

with $1+\frac{2X\mathcal{L}_{XX}}{\mathcal{L}_{X}}>0$ and
$c_{s}^{2}(X,\phi)\equiv{(1+2X\frac{\mathcal{L}_{XX}}{\mathcal{L}_{X}})^{-1}}$.

Following [8, 10, 36, 37, 62], the inverse metric can be written as

$\displaystyle\bar{G}_{\mu\nu}=g_{\mu\nu}-\frac{\mathcal{L}_{XX}}{\mathcal{L}_{X}+2X\mathcal{L}_{XX}}\nabla_{\mu}\phi\nabla_{\nu}\phi.$
(5)

The Eqs. (4), (5) have physical relevance when $\mathcal{L}_{X}$ is nonzero,
assuming a positive definite $c_{s}^{2}$. Eq. (5) states that the emergent
metric, represented as $\bar{G}_{\mu\nu}$, differs in its conformal properties
from the metric $g_{\mu\nu}$ when considering non-trivial configurations of
the scalar field $\phi$. Like canonical scalar fields, the variable $\phi$
exhibits diverse local causal structural properties. It also differs from
those that are defined using $g_{\mu\nu}$. The EOM, as stated in Eq. (3), is
valid even when taking into account the implicit relationship between $L$ and
$\phi$. Then the EOM Eq. (3) is:

$\displaystyle\frac{1}{\sqrt{-g}}\frac{\delta
S_{k}}{\delta\phi}=\bar{G}^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=0.$ (6)

This study addresses the Dirac-Born-Infeld (DBI) type non-canonical
Lagrangian, which is represented as $\mathcal{L}(X,\phi)\equiv\mathcal{L}(X)$
[8, 10, 36, 70, 71, 72, 73, 68, 69]:

$\displaystyle\mathcal{L}(X)=1-\sqrt{1-2X}.$ (7)

The K-essence paradigm posits that the prevalence of kinetic energy over
potential energy results in the exclusion of the potential term in the
Lagrangian equation (7) [70, 68, 69]. The squared speed of sound, represented
as $c_{s}^{2}$, is determined by the expression $(1-2X)$. Therefore, the Eq.
(5) for the effective emergent metric is expressed as:

$\displaystyle\bar{G}_{\mu\nu}=g_{\mu\nu}-\nabla_{\mu}\phi\nabla_{\nu}\phi=g_{\mu\nu}-\partial_{\mu}\phi\partial_{\nu}\phi,$
(8)

since $\phi$ is a scalar.

The Christoffel symbol corresponding to the emergent gravity metric given by
Eq. (8) can be written as [8, 10, 68, 69]:

$\displaystyle\bar{\Gamma}^{\alpha}_{\mu\nu}$
$\displaystyle=\Gamma^{\alpha}_{\mu\nu}-\frac{1}{2(1-2X)}\Big{[}\delta^{\alpha}_{\mu}\partial_{\nu}+\delta^{\alpha}_{\nu}\partial_{\mu}\Big{]}X\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (9)

where $\Gamma^{\alpha}_{\mu\nu}$ is the usual Christoffel symbol associated
with the gravitational metric $g_{\mu\nu}$.

Hence, the geodesic equation governing the K-essence geometry may be expressed
as:

$\displaystyle\frac{d^{2}x^{\alpha}}{d\lambda^{2}}+\bar{\Gamma}^{\alpha}_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}=0,$
(10)

where $\lambda$ is an affine parameter.

The covariant derivative $D_{\mu}$ [41, 68, 69] linked with the emergent
metric $\bar{G}_{\mu\nu}$ $(D_{\alpha}\bar{G}^{\alpha\beta}=0)$ gives

$\displaystyle
D_{\mu}A_{\nu}=\partial_{\mu}A_{\nu}-\bar{\Gamma}^{\lambda}_{\mu\nu}A_{\lambda},$
(11)

and the inverse emergent metric is $\bar{G}^{\mu\nu}$ such as
$\bar{G}_{\mu\lambda}\bar{G}^{\lambda\nu}=\delta^{\nu}_{\mu}$.

Therefore, considering the extensive behavior that defines the dynamics of
K-essence and general relativity [43, 68, 69], the Emergent Einstein’s
Equation (EEE) may be formulated as:

$\displaystyle\mathcal{\bar{G}}_{\mu\nu}=\bar{R}_{\mu\nu}-\frac{1}{2}\bar{G}_{\mu\nu}\bar{R}=\kappa\bar{T}_{\mu\nu},$
(12)

where $\kappa=8\pi G$ is constant, $\bar{R}_{\mu\nu}$ is Ricci tensor and
$\bar{R}\leavevmode\nobreak\ (=\bar{R}_{\mu\nu}\bar{G}^{\mu\nu})$ is the Ricci
scalar. Moreover, the energy-momentum tensor $\bar{T}_{\mu\nu}$ is linked to
this emergent spacetime.

Now, we will provide a concise overview of the K-essence emergent generalized
Vaidya spacetime. In the cited work [37], the author has introduced the
concept of K-essence emergent generalized Vaidya spacetime. This framework
considers the background gravitational metric to be the typical generalized
Vaidya metric [29, 30], while also satisfying the necessary energy
requirements. The line element for the emergent generalized Vaidya metric in
K-essence theory is as follows:

$\displaystyle dS^{2}$
$\displaystyle=-\Big{[}1-\frac{2m(t,r)}{r}-\phi_{t}^{2}\Big{]}dt^{2}+2dtdr+r^{2}d\Omega^{2}$
(13)
$\displaystyle=-\Big{[}1-\frac{2\mathcal{M}(t,r)}{r}\Big{]}dt^{2}+2dtdr+r^{2}d\Omega^{2}$

with $d\Omega^{2}=d\theta^{2}+d\Phi^{2}$. They have defined the K-essence
emergent Vaidya mass function

$\displaystyle\mathcal{M}(t,r)=m(t,r)+\frac{r}{2}\phi_{t}^{2}$ (14)

where $m(t,r)$ is the usual generalized Vaidya mass function and
$\phi_{t}^{2}$ ($\phi_{t}=\frac{\partial\phi}{\partial t}$) is the non-zero
kinetic energy of the K-essence scalar field. The above-mentioned mass
function pertains to the gravitational energy associated with the K-essence
emergent gravity within a specified radius $r$. Here, we substitute the
Edington advanced time coordinate with the conventional time coordinate,
without any loss of generality, denoted as $v\rightarrow t$. In this study
[37], the author has examined the effective K-essence emergent metric, as
denoted by Eq. (8). Additionally, the author has calculated all the components
of the EEE (Eq. (12)) and the necessary energy conditions. It is important to
mention that the assumption about $\phi$ contradicts local Lorentz invariance
since, in general, spherical symmetry only requires $\phi(x)=\phi(t,r)$. The
inclusion of the assumption of the independence of $\phi$, denoted as
$\phi(t,r)=\phi(t)$, suggests that beyond this specific frame selection, a
spherically symmetric $\phi$ is indeed a function of both $t$ and $r$. The
K-essence theory permits the occurrence of Lorentz violation due to the fact
that the dynamic solutions of the K-essence equation of motion spontaneously
break Lorentz invariance and alter the metric for the perturbations around
these solutions.

Furthermore, the authors in [36] have successfully established a connection
between the geometry of K-essence and the Vaidya spacetime. The researchers
have developed a model of the Vaidya spacetime with generalized K-essence,
which takes into account any spherically symmetric static black hole as the
underlying spacetime. The line element of the new geometry is (using Eq. (8):

$\displaystyle dS^{2}$
$\displaystyle=-\Big{[}f(r)-\phi_{t}^{2}\Big{]}dt^{2}+2dtdr+r^{2}d\Omega^{2}$
(15)
$\displaystyle=-\Big{(}1-\frac{2\mathcal{M}(t,r)}{r}\Big{)}dt^{2}+2dtdr+r^{2}d\Omega^{2}$

gives the mass function

$\displaystyle\mathcal{M}(t,r)=\frac{1}{2}r\Big{[}1+\phi_{t}^{2}-f(r)\Big{]}.$
(16)

In this article [36], the authors also have calculated all the components of
EEE and required energy conditions. If we consider $f(r)=(1-2M/r)$, i.e., the
background physical spacetime is Schwarzschild spacetime, the mass function
may be expressed as:

$\displaystyle\mathcal{M}(t,r)=M+\frac{r}{2}\phi_{t}^{2}.$ (17)

Again, if we select the function $f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}$,
$Q$ represents the charge of the Reissner-Nordstrom (RN) black hole in the
physical spacetime. In this case, the related mass function [36] is modified
as

$\displaystyle\mathcal{M}(t,r)=M-\frac{Q^{2}}{2r}+\frac{r}{2}\phi_{t}^{2}.$
(18)

It is important to mention that the values of $\phi_{t}^{2}$ must be between 0
and 1. Otherwise, the metric (13) and (15) cannot be specified properly, and
the presence of a dynamical horizon is also questionable [36, 37]. In order to
maintain the energy conditions, it is evident that the $\phi_{t}^{2}$ must be
a monotonically increasing function of $t$, with the condition
$\phi_{t}^{2}<1$. The admissible configurations for the K-essence scalar field
in the generalized Vaidya solution, in order to have a dynamical horizon, are
subject to a highly restrictive constraint. It is important to note that the
metrics mentioned above represent dynamical horizons rather than isolated or
event horizons, as explained extensively in Ref. [36, 37].

It is also noted that the time dependence in the given mass functions (Eqs.
(17), (18)) arises from the kinetic energy of the K-essence scalar field.
However, in the mass function Eq. (14), the time dependence comes from both
the usual generalized Vaidya mass and the K-essence scalar fields. Thus,
considering the above situations of the K-essence generalized Vaidya
spacetime, we may conclude that the K-essence Vaidya mass function adheres to
the general form specified in Eq.(14). Therefore, we may conclude that the
background metric can be chosen from any standard gravitational metric, with
the only alteration being the replacement of their masses with a background
mass, which likewise satisfies the EEE equation.

##  III Geodesics for the generalized K-essence Vaidya spacetime

This section focuses on analyzing the geodesic structure of the generalized
K-essence Vaidya spacetime. In this context, we define our investigative
metric as Eq. (13), where the K-essence emergent Vaidya mass function is
represented by Eq. (14). For the metric (13), we can write the Lagrangian as
[1, 8, 34, 35]

$2\mathcal{L}=-\Big{(}1-\frac{2\mathcal{M}(t,r)}{r}\Big{)}\dot{t}^{2}+2\dot{t}\dot{r}+r^{2}\dot{\theta}^{2}+r^{2}\sin^{2}{\theta}\leavevmode\nobreak\
\dot{\Phi}^{2}.$ (19)

where $\dot{t}=\frac{dt}{d\tau},\leavevmode\nobreak\
\dot{r}=\frac{dr}{d\tau},\leavevmode\nobreak\
\dot{\theta}=\frac{d\theta}{d\tau},\leavevmode\nobreak\
\dot{\Phi}=\frac{d\Phi}{d\tau}$, $\tau$ is to be identified with the proper
time. Now, using the Euler-Lagrange equation we have

$\displaystyle\frac{d}{d\tau}\Big{(}\frac{\partial\mathcal{L}}{\partial\dot{t}}\Big{)}=\frac{\mathcal{M}_{t}}{r}\dot{t}^{2};$
(20) $\displaystyle
2r\dot{\theta}+r^{2}\ddot{\theta}=r^{2}\sin{\theta}\cos{\theta}\leavevmode\nobreak\
\dot{\Phi}^{2}$ (21) $\displaystyle and\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle
r^{2}\sin^{2}{\theta}\leavevmode\nobreak\ \dot{\Phi}=Constant$ (22)
$\displaystyle with\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\frac{\partial\mathcal{L}}{\partial\dot{t}}=-\Big{(}1-\frac{2\mathcal{M}}{r}\Big{)}\dot{t}+\dot{r}$
(23)

where $\mathcal{M}_{t}=\frac{\partial\mathcal{M}(t,r)}{\partial t}$ and we
write $\mathcal{M}(t,r)$ as $\mathcal{M}$.

Because our object and metric are spherically symmetric, we can simplify
everything by examining just motion on the equatorial plane
$\theta=\frac{\pi}{2}$ and therefore, $\dot{\theta}=0$. For the above choice
of equatorial plane, the Eq. (22) becomes

$\displaystyle r^{2}\dot{\Phi}=Constant=L(say)$ (24)

Thus, by employing Eq. (24) on the equatorial plane, we may write from Eq.
(19) that the Lagrangian is

$\displaystyle
2\mathcal{L}=-\Big{(}1-\frac{2\mathcal{M}(t,r)}{r}\Big{)}\dot{t}^{2}+2\dot{t}\dot{r}+\frac{L^{2}}{r^{2}}.$
(25)

Due to the inclusion of the generalized K-essence Vaidya mass function
($\mathcal{M}(t,r)$) in the Lagrangian formulation provided above, further
analysis is not possible as it can have varying values based on the
gravitational mass. Within this particular situation, we have the option to
select the mass function. For our subsequent analysis, we have selected two
distinct mass functions. Moreover, it is mentioned that the K-essence Vaidya
mass function (14) depends on $\phi_{t}^{2}$, which has values between $0$ and
$1$. Therefore, we can select $\phi_{t}^{2}$ as an explicit function of time
in order to keep the values of $\phi_{t}^{2}$ throughout the article as [36]

$\displaystyle\phi_{t}^{2}=e^{-t/t_{0}}$ (26)

where $t_{0}$ is a positive constant.

###  III.1 Case-I: $\mathcal{M}(t,r)=M+\frac{r}{2}\phi_{t}^{2}$

In this portion, we will look at the K-essence Vaidaya mass function as

$\displaystyle\mathcal{M}(t,r)=M+\frac{r}{2}e^{-t/t_{0}}$ (27)

where $M$ is the Schwarzschild black hole’s mass. In this scenario, the time
dependence of the mass parameter is derived from the K-essence scalar field
via Eq. (26), which was previously discussed in the preceding section. Given
that the mass parameter in Eq. (25) is directly influenced by time through
equations (14) and (26), we may infer from Refs. [34, 35] that the energy $E$
can be expressed as a function of time $t$:

$\displaystyle\frac{\partial\mathcal{L}}{\partial\dot{t}}=E(t).$ (28)

Now using Eqs. (27) and (28) in Eq. (20), we have

$\displaystyle\dot{E}(t)=-\frac{e^{-\frac{t}{t_{0}}}}{2t_{0}}\leavevmode\nobreak\
\dot{t}^{2}$ (29)

The solution of the aforementioned equation, as denoted by Eq. (29), is highly
intricate and cannot be solved directly. To determine the expression for
$E(t)$ in the above equation, we converted our measurement to a comoving
plane, where $\frac{dt}{d\tau}=1$. This conversion is consistently maintained
throughout the article. Thus, on the comoving plane ($d\tau\equiv dt$), the
expression for $E(t)$ is

$\displaystyle
E(t)=\frac{1}{2}e^{-\frac{t}{t_{0}}}\equiv\frac{1}{2}\phi_{t}^{2}.$ (30)

Thus, we can say that in our model, specifically when we select a specific
form (26) for the kinetic energy of the K-essence scalar field while
satisfying the imposed conditions, we have found a direct relationship between
the energy of the system we have chosen and the kinetic energy of the
K-essence scalar field in the comoving plane. So, the K-essence Vaidya mass
function (27) can be written as

$\displaystyle\mathcal{M}(t.r)=M+E(t)r.$ (31)

It is important to point out that the Vaidya metric defines the gravitational
field surrounding a massive object, often a dying star, that emits radiation
in the form of null dust. The notion of the Vaidya spacetime is expanded in
the generalized version to encompass a wide range of scenarios, accommodating
different forms of matter and radiation. The spacetime is dynamic and
undergoes evolution as matter compresses, with the metric describing the
changing curvature [29, 30]. Within the framework of the generalized Vaidya
spacetime, employing comoving observers that satisfy the condition
“$dt=d\tau$” simplifies the mathematical representation of the spacetime. It
enables us to utilize a temporal reference that tracks the movement of matter
as it undergoes gravitational collapse to become a black hole or emits
radiation as a star. Using a time parameter that evolves with the behavior of
matter is a practical approach for investigating gravitational collapse or
radiating stars. It improves the intuitiveness and physical significance of
describing the collapse process. When examining geodesic structures in the
generalized Vaidya spacetime from the perspective of a comoving observer, our
focus lies on the trajectories that objects or particles take when they deal
with the changing spacetime caused by the reducing matter. These geodesics
illustrate the paths that things follow as they move through spacetime, which
is influenced by changes in curvature caused by the dynamics of matter.
Comprehending these geodesic structures is essential for analyzing the
dynamics of particles, photons, and observers in spacetime. It facilitates
forecasting the movement and interaction of objects inside the gravitational
field generated by collapsing matter and is a crucial component in the
analysis of the physics and astrophysical phenomena occurring in these
spacetimes.

Using Eqs. (28), (31) in Eq. (23), we have

$\displaystyle\frac{dr}{dt}+E(t)=1-\frac{2M}{r}$ (32)

On the other hand, using Eq. (31) in Eq. (25), we get

$\displaystyle\frac{dr}{dt}+E(t)=2\mathcal{L}-\frac{L^{2}}{r^{2}}$ (33)

By substituting Eqs. (24) and (30) into Eq. (33), we obtain the angular
relation as follows:

$\displaystyle r+L\Phi=2\mathcal{L}t+t_{0}E(t)$ (34)

taking integration constant to be zero.

For non-radial geodesic, using Eqs. (32) and (33) we have

$\displaystyle\Big{(}\frac{dr}{dt}+E(t)\Big{)}^{2}+\Big{(}D^{2}-2\Big{)}\Big{(}\frac{dr}{dt}+E(t)\Big{)}+\Big{(}1-2D^{2}\mathcal{L}\Big{)}=0$
(35)

where $D=\frac{2M}{L}$ and $L\neq 0$. Solving the above Eq. (35), we obtain

$\displaystyle\frac{dr}{dt}+E(t)=\frac{1}{2}\Big{[}(2-D^{2})\pm
D\sqrt{D^{2}-4(1-2\mathcal{L})}\Big{]}$ (36)

where we tate only positive solutions for our study. By employing Eq. (30) and
performing integration on the aforementioned Eq. (36), we obtain

$\displaystyle r=t_{0}E(t)+\frac{t}{2}\Big{[}(2-D^{2})\pm
D\sqrt{D^{2}-4(1-2\mathcal{L})}\Big{]}+c$ (37)

where $c$ is an integration constant.

####  III.1.1 Time-like Geodesics for Case-I

In order to analyze the structure of the time-like geodesics in the specified
spacetime (13) with the mass function (27), we impose the condition
$2\mathcal{L}\equiv\bar{G}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-1$. In this
particular circumstance, Eq. (33) is transformed as

$\displaystyle\frac{dr}{dt}+E(t)=-1-\frac{L^{2}}{r^{2}}.$ (38)

First, we will analyze the radial geodesics with $L=0$, and then we will go on
to the non-radial geodesics with $L\neq 0$. These geodesics are studied from
the perspective of time-like geodesics of the generalized K-essence Vaidya
metric (13), considering the mass function (27), inside a comoving system.

In order to track the radial geodesics ($\dot{\Phi}=0$), we consider the
motion of a particle with no angular momentum ($L=0$) that starts its journey
from a state of rest at a distance of $r=r_{a}$ and time $t=t_{a}$, so that
the rate of change of its radial position with respect to time,
$\frac{dr}{dt}$, is zero. Thus, by referring to Eq. (32), we obtain

$\displaystyle r_{a}=\frac{2M}{1-E(t_{a})}.$ (39)

Using Eq. (30) in Eq. (38), we get

$\displaystyle r=-t+t_{0}E(t)+c_{1}$ (40)

where $c_{1}$ is an integration constant. Using the aforementioned two Eqs.
(39) and (40) in conjunction with the previously mentioned radial geodesics
criteria for a particle, we get the expression for constant $c_{1}$ as

$\displaystyle c_{1}=\frac{2M}{1-E(t_{a})}+t_{a}-t_{0}E(t_{a})$ (41)

Hence from Eq. (40), we obtain

$\displaystyle r=-t+t_{0}E(t)+\frac{2M}{1-E(t_{a})}+t_{a}-t_{0}E(t_{a}).$ (42)

Now, we have the capability to compute the duration it takes for a particle to
reach the singularity ($r=0$) at a specific time $t=t_{s}$ along the radial
geodesic in our given spacetime which is

$\displaystyle-
t_{s}+t_{0}E(t_{s})+\frac{2M}{1-E(t_{a})}+t_{a}-t_{0}E(t_{a})=0$
$\displaystyle\implies
t_{s}=t_{0}\ln{\Big{[}\frac{1}{2W_{0}\Big{(}\frac{1}{2}e^{\frac{p}{t_{0}}}\Big{)}}\Big{]}}$
(43)

where $p=\big{(}-\frac{2M}{1-E(t_{a})}-t_{a}+t_{0}E(t_{a})\big{)}$ and
$W_{0}(e^{\frac{p}{t_{0}}})$ is the Lambert W function [74] provided that
$e^{\frac{p}{t_{0}}}\geq 0$. Additionally, $W_{0}(q)$ is the solution of the
equation $xe^{x}=q$ when $q$ is a non-negative real number. It is also noted
that the Lambert $W_{0}$ function is a single-valued function. It is important
to note that Eq. (35) can be used to investigate radial geodesics. Because
$D\to\infty$ when $L\to 0$ i.e., $\frac{1}{D}\to 0$ when $L\to 0$. So that the
Eq. (35) transformed to

$\displaystyle\frac{dr}{dt}+E(t)=2\mathcal{L}$ (44)

so that for radial time-like geodesics if we substitute $2\mathcal{L}=-1$ then
it the exactly same with Eq. (38) for $L=0$.

At the moment, we are tracking the non-radial scenario using the time-like
geodesic framework. In this study, we use the Eqs. (35) and (36).

Case-A: First, look at $D=2\sqrt{2}$ for the real root of the Eq. (36), and
then we use Eq. (30) to find

$\displaystyle r-t_{0}E(t)+3t-c_{2}=0$ (45)

where $c_{2}$ is an integration constant, and from Eq. (34) we get

$\displaystyle\Phi=\frac{1}{L}\Big{(}2t-c_{2}\Big{)}.$ (46)

Let a particle starts its journey from $r=r_{a}$ at time $t=t_{a}$, then from
(45) we obtain

$\displaystyle c_{2}=r_{a}-t_{0}E(t_{a})+3t_{a}$ (47)

By observing that $\frac{t_{a}}{t_{0}}\geq 0$ and $0\leq
E(t_{a})<\frac{1}{2}$, we obtain the condition as long as all the values are
finite is

$\displaystyle c_{2}\leq r_{a}+3t_{a}<\frac{t_{0}}{2}+c_{2}.$ (48)

So that the Eqs. (45) and (46) transformed to

$\displaystyle r-t_{0}E(t)+3t-r_{a}+t_{0}E(t_{a})-3t_{a}=0$ (49)
$\displaystyle\Phi=\frac{1}{L}\Big{(}2t-r_{a}+t_{0}E(t_{a})-3t_{a}\Big{)}.$
(50)

If we assume that the particle is moving towards the singularity
$(r\rightarrow 0)$ when time $t\rightarrow t_{s}$, then according to Eq. (49),
we obtain

$\displaystyle
t_{s}=t_{0}\ln{\Big{[}\frac{1}{6W_{0}\Big{(}\frac{1}{6}e^{\frac{p_{1}}{3t_{0}}}\Big{)}}\Big{]}}$
(51)

where $p_{1}=-r_{a}+t_{0}E(t_{a})-3t_{a}$ and $e^{\frac{p_{1}}{3t_{0}}}\geq
0$.

Hence, Eqs. (45) and (46) demonstrate the possibility of tracing several non-
radial time-like geodesics for varying values of $c_{2}$ under the condition
$D=2\sqrt{2}$. As a result, a particle starts its journey from $r=r_{a}$ when
$\Phi=\Phi_{a}=\frac{1}{L}\big{(}-r_{a}+t_{0}E(t_{a})-t_{a}\big{)}$, $t=t_{a}$
and will approach to singularity $r\rightarrow 0$ when
$\Phi=\Phi_{s}=\frac{1}{L}\big{(}2t_{s}-r_{a}+t_{0}E(t_{a})-3t_{a}\big{)}$,
$t=t_{s}=t_{s}=t_{0}\ln{\big{[}\frac{1}{6W_{0}\big{(}\frac{1}{6}e^{\frac{p_{1}}{3t_{0}}}\big{)}}\big{]}}$
and it will leave the singularity when
$t>t_{s}=t_{0}\ln{\big{[}\frac{1}{6W_{0}\big{(}\frac{1}{6}e^{\frac{p_{1}}{3t_{0}}}\big{)}}\big{]}}$.
Given that $c_{2}\leq r_{a}+3t_{a}<\frac{t_{0}}{2}+c_{2}$, it follows that the
values of $r_{a}$ and $t_{a}$ are finite for all finite values of $c_{2}$ and
$t_{0}$. Also, since
$t_{s}=t_{0}\ln{\big{[}\frac{1}{6W_{0}\big{(}\frac{1}{6}e^{\frac{p_{1}}{3t_{0}}}\big{)}}\big{]}}$,
the value of $t_{s}$ is finite. By using Eqs. (45) and (46), we have plotted
two non-radial time-like geodesics for $L=1$ and $t_{0}=0.1$, each
corresponding to distinct values of $c_{2}=0.1$ and $c_{2}=15$. These
geodesics are depicted in Figs. (1) and (2), respectively. To plot the
geodesics in these diagrams, we have converted the coordinate system from
polar coordinates $(r,\Phi)$ to cartesian coordinates $(x,y)$. It should be
noted that there exists central singularity for both the conventional
generalized Vaidya spacetime [29, 30] and the generalized K-essence Vaidya
spacetime [36, 37], meaning that both $r\rightarrow 0$ and $t\rightarrow 0$.
However, a singularity is typically defined as only $r\rightarrow 0$. In this
case, we may see that the particle’s track will allow a future observer to
watch the particle reach $r\rightarrow 0$ at a specific time and escape the
singularity. These phenomena are discussed in the conclusion section.

Figure 1: Time-like geodesics for $D=2\sqrt{2}$, $L=1$, $t_{0}=0.1$,
$c_{2}=0.1$ Figure 2: Time-like geodesics for $D=2\sqrt{2}$, $L=1$,
$t_{0}=0.1$, $c_{2}=15$

Case-B: To calculate the real solutions of Eq. (35) for the purpose of
analyzing non-radial time-like geodesics, we assume that $D>2\sqrt{2}$. So
that the Eq. (35) can be written as

$\displaystyle\Big{(}\frac{dr}{dt}+E(t)-A\Big{)}\Big{(}\frac{dr}{dt}+E(t)-B\Big{)}=0$
$\displaystyle\implies$ $\displaystyle\Big{(}r-t_{0}E(t)-At-
c_{3}\Big{)}\Big{(}r-t_{0}E(t)-Bt-c_{3}\Big{)}=0$ (52)

where

$\displaystyle
A=1-\frac{D^{2}}{2}+\frac{D}{2}\sqrt{D^{2}-8}\leavevmode\nobreak\ ,$
$\displaystyle\leavevmode\nobreak\
B=1-\frac{D^{2}}{2}-\frac{D}{2}\sqrt{D^{2}-8}$ (53)

and $c_{3}$ is an integration constant.

The Eq. (52) demonstrates that there are two non-radial time-like geodesics
that may be followed for any finite value of $c_{3}$.

For

$\displaystyle r-t_{0}E(t)-At-c_{3}=0$ (54)

then from Eq. (34),

$\displaystyle\Phi=\frac{1}{L}\Big{[}-t-At-c_{3}\Big{]}$ (55)

and for

$\displaystyle r-t_{0}E(t)-Bt-c_{3}=0$ (56)

we have from (34),

$\displaystyle\Phi=\frac{1}{L}\Big{[}-t-Bt-c_{3}\Big{]}.$ (57)

Let a particle starts its journey from $r=r_{a}$ in the path (54) at time
$t=t_{a}$, the from the equation (54)

$\displaystyle c_{3}=r_{a}-t_{0}E(t_{a})-At_{a}$ (58)

Clearly, since $\frac{t_{a}}{t_{0}}\geq 0$ and $0\leq E(t_{a})<\frac{1}{2}$ so
that

$\displaystyle c_{3}\leq r_{a}-At_{a}<\frac{t_{0}}{2}+c_{3}$ (59)

then the Eqs. (54) and (55) transformed to

$\displaystyle r-t_{0}E(t)-At-r_{a}+t_{0}E(t_{a})+At_{a}=0$ (60)
$\displaystyle\Phi=\frac{1}{L}\Big{[}-t-At-r_{a}+t_{0}E(t_{a})+At_{a}\Big{]}$
(61)

And if we consider that the particle approaches to $r\rightarrow 0$ when
$t\rightarrow t_{s}$ on the path (54), then from (60) we have

$\displaystyle
t_{s}=t_{0}\ln{\Big{[}\frac{1}{-2AW_{0}\Big{(}-\frac{1}{2A}e^{\frac{p_{2}}{At_{0}}}\Big{)}}\Big{]}}$
(62)

where $p_{2}=r_{a}-t_{0}E(t_{a})-At_{a}$ and
$\frac{1}{A}e^{\frac{p_{2}}{At_{0}}}<0$.

Similarly, as before, let us consider a particle that begins its trajectory at
$r=r_{a}$ along the path (56) at time $t=t_{a}$ and eventually approaches
singularity at $t=t_{s}$. By analysing equations (56) and (57), we may derive
the following results:

$\displaystyle r-t_{0}E(t)-Bt-r_{a}+t_{0}E(t_{a})+Bt_{a}=0$ (63)
$\displaystyle\Phi=\frac{1}{L}\Big{[}-t-Bt-r_{a}+t_{0}E(t_{a})+Bt_{a}\Big{]}$
(64) $\displaystyle c_{3}=r_{a}-t_{0}E(t_{a})-Bt_{a}$ (65) $\displaystyle
c_{3}\leq r_{a}-Bt_{a}<\frac{t_{0}}{2}+c_{3}$ (66) $\displaystyle
t_{s}=t_{0}\ln{\Big{[}\frac{1}{-2BW_{0}\Big{(}-\frac{1}{2B}e^{\frac{p_{3}}{Bt_{0}}}\Big{)}}\Big{]}}$
(67)

where $p_{3}=r_{a}-t_{0}E(t_{a})-Bt_{a}$ and
$\frac{1}{B}e^{\frac{p_{3}}{Bt_{0}}}\leq 0$

Therefore, Eqs. (54), (55), (56), and (57) demonstrate the possibility of
tracking several non-radial time-like geodesics for distinct values of $c$
given that $D>2\sqrt{2}$. Hence, for the path (54), a particle starts its
journey from $r=r_{a}$ when
$\Phi=\Phi_{a}=\frac{1}{L}\big{[}-t_{a}-r_{a}+t_{0}E(t_{a})\big{]}$, $t=t_{a}$
and approaching to $r\rightarrow 0$ when
$\Phi=\Phi_{s}=\frac{1}{L}\big{[}-t_{s}-At_{s}-r_{a}+t_{0}E(t_{a})+At_{a}\big{]}$,
$t=t_{s}=t_{0}\ln{\big{[}\frac{1}{-2AW_{0}\big{(}-\frac{1}{2A}e^{\frac{p_{2}}{At_{0}}}\big{)}}\big{]}}$
and it will leave the singularity when
$t>t_{s}=t_{0}\ln{\big{[}\frac{1}{-2AW_{0}\big{(}-\frac{1}{2A}e^{\frac{p_{2}}{At_{0}}}\big{)}}\big{]}}$.
Also, for the path (56), a particle starts its journey from $r=r_{a}$ when
$\Phi=\Phi_{a}=\frac{1}{L}\big{[}-t_{a}-r_{a}+t_{0}E(t_{a})\big{]}$, $t=t_{a}$
and will arrive at singularity $r=0$ when
$\Phi=\Phi_{s}=\frac{1}{L}\big{[}-t_{s}-Bt_{s}-r_{a}+t_{0}E(t_{a})+Bt_{a}\big{]}$,
$t=t_{s}=t_{0}\ln{\big{[}\frac{1}{-2BW_{0}\big{(}-\frac{1}{2B}e^{\frac{p_{3}}{Bt_{0}}}\big{)}}\big{]}}$
and it will leave the singularity when
$t>t_{s}=t_{0}\ln{\big{[}\frac{1}{-2BW_{0}\big{(}-\frac{1}{2B}e^{\frac{p_{3}}{Bt_{0}}}\big{)}}\big{]}}$.
Since $c_{3}\leq r_{a}-At_{a}<\frac{t_{0}}{2}+c_{3}$ and $c_{3}\leq
r_{a}-Bt_{a}<\frac{t_{0}}{2}+c_{3}$, then for any finite values of $c_{3}$,
$t_{0}$, $A$ and $B$, the values of $r_{a}$ and $t_{a}$ are also finite. Using
the Eqs. (54), (55) and (56), (57), for $D=2.9$, $L=1$ and $t_{0}=0.1$, we
have traced two non-radial time-like for $c_{3}=0.1$ in Figure-(3) and also
for $c_{3}=10$ in Figure-(4).

Figure 3: Time-like geodesics for $D=2.9$, $L=1$, $t_{0}=0.1$, $c_{3}=0.1$:
Blue Colour line for $r-t_{0}E(t)-At-c_{3}=0$ and Red colour line for
$r-t_{0}E(t)-Bt-c_{3}=0$ Figure 4: Time-like geodesics for $D=2.9$, $L=1$,
$t_{0}=0.1$, $c_{3}=10$: Blue Colour line for $r-t_{0}E(t)-At-c_{3}=0$ and Red
colour line for $r-t_{0}E(t)-Bt-c_{3}=0$

####  III.1.2 Null Geodesics for Case-I

In this subsection, we want to elucidate the null geodesic behavior of the
generalized K-essence Vaidya spacetime, as described by Eq. (13) and the mass
function given by Eq. (27). We assume that $2\mathcal{L}=0$. For this study,
the Eqs. (33) and (34) reduces to

$\displaystyle\frac{dr}{dt}+E(t)=-\frac{L^{2}}{r^{2}}$ (68) $\displaystyle
r+L\Phi=t_{0}E(t).$ (69)

In order to track the radial geodesics, we assume that a particle with zero
angular momentum ($L=0$) begins its motion from a state of rest at a distance
of $r=r_{a}$ and time $t=t_{a}$, such that the rate of change of $r$ with
respect to $t$ is zero ($\frac{dr}{dt}=0$). Thus, based on Eq. 32, the value
of $r_{a}$ may be determined as $\frac{2M}{1-E(t_{a})}$. Furthermore,
according to Eq. (68),

$\displaystyle r=t_{0}E(t)+c_{4}$ (70)

where $c_{4}$ is an integration constant. At $t=t_{a}$, we obtain

$\displaystyle c_{4}=\frac{2M}{1-E(t_{a})}-t_{0}E(t_{a})$ (71)

so that

$\displaystyle r=t_{0}E(t)+\frac{2M}{1-E(t_{a})}-t_{0}E(t_{a}).$ (72)

Assuming that the particle reaches the singularity ($r=0$) at a specific time
$t=t_{s}$, we have determined

$\displaystyle
t_{s}=t_{0}\ln{\Big{[}\frac{1-E(t_{a})}{2E(t_{a})\Big{(}1-E(t_{a})\Big{)}-(4M/t_{0})}\Big{]}}.$
(73)

Now, we will examine the characteristics of non-radial ($L\neq 0$) geodesics
inside the framework of null geodesics in the given spacetime (13), which is
governed by the particular mass function (27). The requirement for the
existence of real roots of Eq. (35) under the assumption that $2\mathcal{L}=0$
is that $D\geq 2$. Now we discuss the following two cases:

Case-A: When $D=2$, we have from Eq. (36)

$\displaystyle r-t_{0}E(t)+t-c_{5}=0$ (74)

and from Eq. (34) we get

$\displaystyle\Phi=\frac{1}{L}\Big{(}t-c_{5}\Big{)}$ (75)

Let a particle starts its journey from $r=r_{a}$ at time $t=t_{a}$, the from
(74)

$\displaystyle c_{5}=r_{a}-t_{0}E(t_{a})+t_{a}$ (76)

It is evident that the inequality $\frac{t_{a}}{t_{0}}\geq 0$ is true, and we
also have the condition $0\leq E(t_{a})<\frac{1}{2}$ so that

$\displaystyle c_{5}\leq r_{a}+t_{a}<\frac{t_{0}}{2}+c_{5}$ (77)

then the Eqs. (74) and (75) transformed to

$\displaystyle r-t_{0}E(t)+t-r_{a}+t_{0}E(t_{a})-t_{a}=0$ (78)
$\displaystyle\Phi=\frac{1}{L}\Big{(}t-r_{a}+t_{0}E(t_{a})-t_{a}\Big{)}$ (79)

And if we consider that the particle reaches the singularity $(r=0)$ at time
$t=t_{s}$, then from (78 ) we have

$\displaystyle
t_{s}=t_{0}\ln{\Big{[}\frac{1}{2W_{0}\Big{(}\frac{1}{2}e^{\frac{p_{4}}{t_{0}}}\Big{)}}\Big{]}}$
(80)

where $p_{4}=-r_{a}+t_{0}E(t_{a})-t_{a}$ and $e^{\frac{p_{4}}{t_{0}}}\geq 0$.

Therefore, Eqs. (74) and (75) demonstrate that we may track several non-radial
null geodesics for varying values of $c_{5}$ in the case of $D=2$. Hence, a
particle starts its trajectory at a distance of $r=r_{a}$ when
$\Phi=\Phi_{a}=\frac{1}{L}\big{[}-r_{a}+t_{0}E(t_{a})\big{]}$, $t=t_{a}$ and
will arrive at singularity $r=0$ when
$\Phi=\Phi_{s}=\frac{1}{L}\big{(}t_{s}-r_{a}+t_{0}E(t_{a})-t_{a}\big{)}$,
$t=t_{s}=t_{0}\ln{\big{[}\frac{1}{2W_{0}\big{(}\frac{1}{2}e^{\frac{p_{4}}{t_{0}}}\big{)}}\big{]}}$
and it will leave the singularity when
$t>t_{s}=t_{0}\ln{\big{[}\frac{1}{2W_{0}\big{(}\frac{1}{2}e^{\frac{p_{4}}{t_{0}}}\big{)}}\big{]}}$.
Given that $c_{5}\leq r_{a}+t_{a}<\frac{t_{0}}{2}+c_{5}$, it follows that both
$r_{a}$ and $t_{a}$ are finite for any finite values of $c_{5}$ and $t_{0}$.
Also, since
$t_{s}=t_{0}\ln{\big{[}\frac{1}{2W_{0}\big{(}\frac{1}{2}e^{\frac{p_{4}}{t_{0}}}\big{)}}\big{]}}$,
the value of $t_{s}$ is finite. By using Eqs. (74) and (75), we have plotted
two non-radial time-like geodesics for $L=1$ and $t_{0}=0.1$, with two
distinct values of $c_{5}$, namely $c_{5}=1$ and $c_{5}=10$. These geodesics
are depicted in Figure-(5) and Figure-(6) correspondingly.

Figure 5: Null geodesics for $D=2$, $L=1$, $t_{0}=0.1$, $c_{5}=1$ Figure 6:
Null geodesics for $D=2$, $L=1$, $t_{0}=0.1$, $c_{5}=10$

Case-B: When $D>2$, we have from Eq. (35)

$\displaystyle\Big{(}r-t_{0}E(t)-At-c_{6}\Big{)}\Big{(}r-t_{0}E(t)-Bt-
c_{6}\Big{)}=0,$ (81)

where

$\displaystyle A=1-\frac{D^{2}}{2}+\frac{D}{2}\sqrt{D^{2}-4},$ $\displaystyle
B=1-\frac{D^{2}}{2}-\frac{D}{2}\sqrt{D^{2}-4}$ (82)

and $c_{6}$ is an integration constant. The Eq. (81) demonstrates the
existence of two non-radial null geodesics that may be followed for any finite
value of $c_{6}$. The Eq. (81) is of the same form as (52) in the context of
time-like geodesics, but with different constants. Therefore, based on this
closeness, we may infer that equations (75) and (81) as

$\displaystyle r-t_{0}E(t)-At-c_{6}=0$ (83)
$\displaystyle\Phi=\frac{1}{L}\Big{(}-At-c_{6}\Big{)}.$ (84)

Also,

$\displaystyle r-t_{0}E(t)-Bt-c_{6}=0$ (85)
$\displaystyle\Phi=\frac{1}{L}\Big{(}-Bt-c_{6}\Big{)}.$ (86)

Let a particle starts its journey from $r=r_{a}$ in the path (83) or (85) at
time $t=t_{a}$, then from the Eq. (83), we have

$\displaystyle c_{6}=r_{a}-t_{0}E(t_{a})-At_{a}$ (87) and $\displaystyle
c_{6}\leq r_{a}-At_{a}<\frac{t_{0}}{2}+c_{6}$ (88)

also then from the equation (85), we have

$\displaystyle c_{6}=r_{a}-t_{0}E(t_{a})-Bt_{a}$ (89) and $\displaystyle
c_{6}\leq r_{a}-Bt_{a}<\frac{t_{0}}{2}+c_{6}$ (90)

Therefore we have

$\displaystyle r-t_{0}E(t)-At-r_{a}+t_{0}E(t_{a})+At_{a}=0$ (91)
$\displaystyle\Phi=\frac{1}{L}\Big{[}-At-r_{a}+t_{0}E(t_{a})+At_{a}\Big{]}$
(92)

and also

$\displaystyle r-t_{0}E(t)-Bt-r_{a}+t_{0}E(t_{a})+Bt_{a}=0$ (93)
$\displaystyle\Phi=\frac{1}{L}\Big{[}-Bt-r_{a}+t_{0}E(t_{a})+Bt_{a}\Big{]}$
(94)

If we consider that the particle approaches the singularity $(r\rightarrow 0)$
at time $t=t_{s}$ along the path (83) or (85), then we may deduce from
equation (91)

$\displaystyle
t_{s}=t_{0}\ln{\Big{[}\frac{1}{-2AW_{0}\Big{(}-\frac{1}{2A}e^{\frac{p_{5}}{At_{0}}}\Big{)}}\Big{]}}$
(95)

and from (93)

$\displaystyle
t_{s}=t_{0}\ln{\Big{[}\frac{1}{-2BW_{0}\Big{(}-\frac{1}{2B}e^{\frac{p_{6}}{Bt_{0}}}\Big{)}}\Big{]}}$
(96)

where $p_{5}=r_{a}-t_{0}E(t_{a})-At_{a}$, $p_{6}=r_{a}-t_{0}E(t_{a})-Bt_{a}$,
$\frac{1}{A}e^{\frac{p_{5}}{At_{0}}}\leq 0$ and
$\frac{1}{B}e^{\frac{p_{6}}{Bt_{0}}}\leq 0$.

As the outcome of the Eqs. (83), (84), and (85), (86), we may trace distinct
non-radial null geodesics for different values of $c_{6}$ when $D>2$. As a
result, given the path (83), a particle begins its journey from $r=r_{a}$ when
$\Phi=\Phi_{a}=\frac{1}{L}\big{[}-r_{a}+t_{0}E(t_{a})\big{]}$, $t=t_{a}$ and
will approaches singularity $r\rightarrow 0$ when
$\Phi=\Phi_{s}=\frac{1}{L}\big{[}-At_{s}-r_{a}+t_{0}E(t_{a})+At_{a}\big{]}$,
$t=t_{s}=t_{0}\ln{\big{[}\frac{1}{-2AW_{0}\big{(}-\frac{1}{2A}e^{\frac{p_{5}}{At_{0}}}\big{)}}\big{]}}$
and it will leave from the singularity when
$t>t_{s}=t_{0}\ln{\big{[}\frac{1}{-2AW_{0}\big{(}-\frac{1}{2A}e^{\frac{p_{5}}{At_{0}}}\big{)}}\big{]}}$.
Also, for the path (85), a particle starts its journey from $r=r_{a}$ when
$\Phi=\Phi_{a}=\frac{1}{L}\big{[}-r_{a}+t_{0}E(t_{a})\big{]}$, $t=t_{a}$ and
will approaches singularity $r\rightarrow 0$ when
$\Phi=\Phi_{s}=\frac{1}{L}\big{[}-Bt_{s}-r_{a}+t_{0}E(t_{a})+Bt_{a}\big{]}$,
$t=t_{s}=t_{0}\ln{\big{[}\frac{1}{-2BW_{0}\big{(}-\frac{1}{2B}e^{\frac{p_{6}}{Bt_{0}}}\big{)}}\big{]}}$
and it will leave the singularity when
$t>t_{s}=t_{0}\ln{\big{[}\frac{1}{-2BW_{0}\big{(}-\frac{1}{2B}e^{\frac{p_{6}}{Bt_{0}}}\big{)}}\big{]}}$.
Since $c_{6}\leq r_{a}-At_{a}<\frac{t_{0}}{2}+c_{6}$ and $c_{6}\leq
r_{a}-Bt_{a}<\frac{t_{0}}{2}+c_{6}$, then for any finite values of $c_{6}$,
$t_{0}$, $A$ and $B$, the values of $r_{a}$ and $t_{a}$ are also finite. By
substituting the values $D=2.1$, $L=1$, and $t_{0}=0.1$ into equations (83),
(84), (85), and (86), we have obtained two non-radial nulls for $c_{6}=0.1$ in
Figure-(7) and also for $c_{6}=10$ in Figure-(8).

Figure 7: Null geodesics for $D=2.1$, $L=1$, $t_{0}=0.1$, $c_{6}=0.1$: Blue
Colour line for $r-t_{0}E(t)-At-c_{6}=0$ and Red colour line for
$r-t_{0}E(t)-Bt-c_{6}=0$ Figure 8: Null geodesics for $D=2.1$, $L=1$,
$t_{0}=0.1$, $c_{6}=10$: Blue Colour line for $r-t_{0}E(t)-At-c_{6}=0$ and Red
colour line for $r-t_{0}E(t)-Bt-c_{6}=0$

###  III.2 Case-II: $\mathcal{M}(t,r)=\mu t+\frac{r}{2}\phi_{t}^{2}$

In this part, we consider the generalized K-essence Vaidya mass function (14)
with the assumption (26) as

$\displaystyle\mathcal{M}(t,r)=\mu t+\frac{r}{2}e^{-\frac{t}{t_{0}}}$ (97)

where we take the usual generalized Vaidya mass function $m(t,r)\equiv
m(t)=\mu t$ i.e., is a linear function of $t$ [34, 35, 91] and $\mu$ is a
positive constant.

From the above mass function (97), we have

$\displaystyle\frac{\mathcal{\bar{M}}_{t}}{r}=-\frac{e^{-\frac{t}{t_{0}}}}{2t_{0}}$
(98)

where we redefine the first derivative of the given mass function with respect
to time as $\mathcal{\bar{M}}_{t}\equiv\mathcal{M}_{t}-\mu$.

Following the same procedure as for Case-I, and using Eqs. (20-25) and (28),
and again assuming the comoving plane ($d\tau\equiv dt$), we obtain

$\displaystyle E(t)=\frac{1}{2}e^{-\frac{t}{t_{0}}}$ (99) and
$\displaystyle\mathcal{M}(t.r)=\mu t+E(t)r.$ (100)

Again, continuing the similar procedure as Case-I, we found the following
relations:

$\displaystyle\frac{dr}{dt}+E(t)=2\mathcal{L}-\frac{L^{2}}{r^{2}}$ (101)
$\displaystyle\frac{dr}{dt}+E(t)=1-\frac{2\mu t}{r}$ (102) $\displaystyle
r+L\Phi=2\mathcal{L}t+t_{0}E(t)$ (103)

####  III.2.1 Time-like Geodesics for Case-II

In this portion, we analyze the behavior of time-like geodesics in the
generalized K-essence Vaidya spacetime, as described by Eq. (13) with the mass
function given by Eq. (97). For the sake of this investigation, we assume
$2\mathcal{L}=-1$.

Following from the previous analysis for Case-I, we will now focus on tracking
the radial geodesics. Specifically, we will examine a particle with zero
angular momentum ($L=0$) that begins its motion from a state of rest at a
distance $r=r_{a}$ and time $t=t_{a}$,

$\displaystyle r_{a}=\frac{2\mu t_{a}}{1-E(t_{a})}$ (104)

and using Eq. (38), we obtain

$\displaystyle r=-t+t_{0}E(t)+c_{7}$ (105) $\displaystyle c_{7}=\frac{2\mu
t_{a}}{1-E(t_{a})}+t_{a}-t_{0}E(t_{a})$ (106) $\displaystyle
r=-t+t_{0}E(t)+\frac{2\mu t_{a}}{1-E(t_{a})}+t_{a}-t_{0}E(t_{a})$ (107)

Similarly, if the particle approaches the singularity ($r\rightarrow 0$) at
time $t=t_{s}$, the same type of consideration applies, we get

$\displaystyle
t_{s}=t_{0}\ln{\Big{[}\frac{1}{2W_{0}\Big{(}\frac{1}{2}e^{\frac{p_{7}}{t_{0}}}\Big{)}}\Big{]}}$
(108)

provided $e^{\frac{p_{7}}{t_{0}}}\geq 0$ where $p_{7}=-\frac{2\mu
t_{a}}{1-E(t_{a})}-t_{a}+t_{0}E(t_{a})$.

When studying non-radial time-like geodesics, we obtain the same expression as
Eq. (35), but with a different value for the parameter $D$ ($D=\frac{2\mu
t}{L}$). In order to trace the geodesics, it is necessary to satisfy the given
requirement

$\displaystyle D\geq 2\sqrt{2}\implies t\geq\frac{\sqrt{2}L}{\mu}.$ (109)

In this non-radial study, the scenario where $D=2\sqrt{2}$ cannot be taken
into consideration. This is because if $D=2\sqrt{2}$, then
$t=\frac{\sqrt{2}L}{\mu}$, resulting in a constant value for time. As a
result, the spacetime is not like the generalized K-essence Vaidya type. So,
in this non-radial time-like geodesic investigation, we only look into
$D>2\sqrt{2}$, ensuring that the spacetime is time-dependent, which is an
essential characteristic of both the normal Vaidya spacetime and the
generalized K-essence Vaidya spacetime. As before, we solve the Eq. (35) for
$D>2\sqrt{2}$ where $D=\frac{2\mu t}{L}$, we obtain

$\displaystyle\Big{(}r-t_{0}E(t)-\bar{A}(t)-c_{8}\Big{)}\Big{(}r-t_{0}E(t)-\bar{B}(t)-c_{8}\Big{)}=0$
(110)

where

$\displaystyle\bar{A}(t)=t-\frac{2\mu^{2}t^{3}}{3L^{2}}+\frac{2\mu^{2}}{3L^{2}}\Big{(}t^{2}-\frac{2L^{2}}{\mu^{2}}\Big{)}^{\frac{3}{2}}$
(111)
$\displaystyle\bar{B}(t)=t-\frac{2\mu^{2}t^{3}}{3L^{2}}-\frac{2\mu^{2}}{3L^{2}}\Big{(}t^{2}-\frac{2L^{2}}{\mu^{2}}\Big{)}^{\frac{3}{2}}.$
(112)

Clearly, Eq. (110) shows that there are two non-radial time-like geodesics
that can be traced for any finite value of $c_{8}$. Therefore, we have two
non-radial time-like geodesics, either,

$\displaystyle r-t_{0}E(t)-\bar{A}(t)-c_{8}=0$ (113)
$\displaystyle\Phi=\frac{1}{L}\Big{[}-t-\bar{A}(t)-c_{8}\Big{]},$ (114)

or,

$\displaystyle r-t_{0}E(t)-\bar{B}(t)-c_{8}=0$ (115)
$\displaystyle\Phi=\frac{1}{L}\Big{[}-t-\bar{B}(t)-c_{8}\Big{]}$ (116)

Let a particle starts its journey from $r=r_{a}$ in the path (113) at time
$t=t_{a}>\frac{\sqrt{2}L}{\mu}$, we have

$\displaystyle c_{8}=r_{a}-t_{0}E(t_{a})-\bar{A}(t_{a})$ (117)

Since $\frac{t_{a}}{t_{0}}\geq 0$ and $0\leq E(t_{a})<\frac{1}{2}$

$\displaystyle c_{8}\leq r_{a}-\bar{A}(t_{a})<\frac{t_{0}}{2}+c_{8}$ (118)

then the Eqs. (113) and (114) transformed to

$\displaystyle r-t_{0}E(t)-\bar{A}(t)-r_{a}+t_{0}E(t_{a})+\bar{A}(t_{a})=0$
(119)
$\displaystyle\Phi=\frac{1}{L}\Big{[}-t-\bar{A}(t)-r_{a}+t_{0}E(t_{a})+\bar{A}(t_{a})\Big{]}$
(120)

then at $t=t_{a}$

$\displaystyle\Phi=\Phi_{a}=\frac{1}{L}\Big{[}-t_{a}-r_{a}+t_{0}E(t_{a})\Big{]}.$
(121)

If we assume that the particle reaches the singularity $(r\rightarrow 0)$ at
time $t=t_{s}$ along the path (113), then we can deduce from Eq. (119) the
following:

$\displaystyle t_{0}E(t_{s})+\bar{A}(t_{s})+c_{8}=0$ $\displaystyle
E(t_{s})-\frac{2\mu^{2}t^{3}_{s}}{3t_{0}L^{2}}+\frac{c_{8}}{t_{0}}=-\frac{t_{s}}{t_{0}}-\frac{2\mu^{2}}{3t_{0}L^{2}}\Big{(}t^{2}_{s}-\frac{2L^{2}}{\mu^{2}}\Big{)}^{\frac{3}{2}}$
(122)

In this scenario, the precise expression for $t_{s}$ cannot be determined due
to the presence of a distinct mass function (97). However, an expression for
the transcendental equation (112) is available, allowing for numerical and
graphical analysis to get the finite time for any given values of $\mu$ and
$L$. Now let
$f(t_{s})=E(t_{s})-\frac{2\mu^{2}t^{3}_{s}}{3t_{0}L^{2}}+\frac{c}{t_{0}}$ and
$g(t_{s})=-\frac{t_{s}}{t_{0}}-\frac{2\mu^{2}}{3t_{0}L^{2}}\big{(}t^{2}_{s}-\frac{2L^{2}}{\mu^{2}}\big{)}^{\frac{3}{2}}$.
Then the only possibility for a finite value $t_{s}$ if and only if
$f(t_{s})=g(t_{s})$.
To start with let us consider $\mu=0.05$ and $L=1$ then $t_{a}>28.2843$. Let
$t_{a}=28.3$ Then from (111), $\bar{A}(t_{a})\approx-9.4739$. Now if we
consider $t_{0}=0.1$ then for $c_{8}=0.1$, $r_{a}\approx-9.3739$
$\Phi_{a}\approx-18.9261$ and also for $c_{8}=30$, $r_{a}\approx 20.5261$ and
$\Phi_{a}\approx-48.8261$ by using (117) and (118). Figure-(9) shows that the
for $c_{8}=0.1$, the Eq. (122) has no solution for $t_{s}$ since the curves
$y=f(t_{s})$ and $y=g(t_{s})$ do not intersect each other. So for $c_{8}=0.1$,
the particle which starts from $r=r_{a}$ at time $t=t_{a}>28.2843$ but it will
never reach the singularity as shown in Figure-(11) (blue line) using the Eqs.
(113) and (114). However, for $c_{8}=30$, from Figure-(10) shows that it has a
solution for $t_{s}$ since the the curves $y=f(t_{s})$ and $y=g(t_{s})$
intersect each other. For $c_{8}=30$, the particle initially located at
$r=r_{a}$ at time $t=t_{a}>28.2843$ will eventually reach the singularity
($r=0$) at time $t=t_{s}$, as depicted by the red line in Figure-(11). The
particle will then depart from the singularity ($r=0$), which is seen
graphically by Eqs. (113) and (114).

Figure 9: $\mu=0.05$, $L=1$, $t_{0}=0.1$ and $c_{8}=0.1$: Blue Colour line for
$f(t_{s})$ and Red colour line for $g(t_{s})$ Figure 10: $\mu=0.05$, $L=1$,
$t_{0}=0.1$ and $c_{8}=30$: Blue Colour line for $f(t_{s})$ and Red colour
line for $g(t_{s})$ Figure 11: Time-like geodesics for
$r-t_{0}E(t)-\bar{A}-c_{8}=0$: When $\mu=0.05$, $L=1$, $t_{0}=0.1$: Blue
Colour line for $c_{8}=0.1$ and Red colour line for $c_{8}=30$

Again, if we consider a particle starts its journey from $r=r_{b}$ in the path
(115) at time $t=t_{b}>\frac{\sqrt{2}L}{\mu}$, we have

$\displaystyle c_{8}=r_{b}-t_{0}E(t_{b})-\bar{B}(t_{b})$ (123)

In this path, the condition is $\frac{t_{b}}{t_{0}}\geq 0$ and $0\leq
E(t_{b})<\frac{1}{2}$

$\displaystyle c_{8}\leq r_{b}-\bar{B}(t_{b})<\frac{t_{0}}{2}+c_{8}$ (124)

then the Eqs. (115) and (116) transformed to

$\displaystyle r-t_{0}E(t)-\bar{B}(t)-r_{b}+t_{0}E(t_{b})+\bar{B}(t_{b})=0$
(125)
$\displaystyle\Phi=\frac{1}{L}\Big{[}-t-\bar{B}(t)-r_{b}+t_{0}E(t_{b})+\bar{B}(t_{b})\Big{]}$
(126)

then at $t=t_{b}$

$\displaystyle\Phi=\Phi_{b}=\frac{1}{L}\Big{[}-t_{b}-r_{b}+t_{0}E(t_{b})\Big{]}.$
(127)

Once more, assuming it is possible, let us contemplate the scenario in which
the particle moves towards the singularity $(r\rightarrow 0)$ at time
$t=t_{s}$ along the trajectory (115). Consequently, based on equation (125),
we may establish the following relation:

$\displaystyle
E(t_{s})-\frac{2\mu^{2}t^{3}_{s}}{3t_{0}L^{2}}+\frac{c_{8}}{t_{0}}=-\frac{t_{s}}{t_{0}}+\frac{2\mu^{2}}{3t_{0}L^{2}}\Big{(}t^{2}_{s}-\frac{2L^{2}}{\mu^{2}}\Big{)}^{\frac{3}{2}}$
(128)

Once again, this equation is transcendental, meaning it can be analyzed
numerically and in pictures for any finite values of $\mu$ and $L$. Now let
$f(t_{s})=E(t_{s})-\frac{2\mu^{2}t^{3}_{s}}{3t_{0}L^{2}}+\frac{c_{8}}{t_{0}}$
and
$g(t_{s})=-\frac{t_{s}}{t_{0}}+\frac{2\mu^{2}}{3t_{0}L^{2}}\big{(}t^{2}_{s}-\frac{2L^{2}}{\mu^{2}}\big{)}^{\frac{3}{2}}$.
Then the only possibility for a finite value $t_{s}$ if and only if
$f(t_{s})=g(t_{s})$.
To start with, let us consider $\mu=0.05$ and $L=1$ then $t_{b}>28.2843$. Let
$t_{b}=28.3$ Then, from (112), $\bar{B}(t_{b})\approx-9.4767$. Now if we
consider $t_{0}=0.1$ then for $c_{8}=0.1$, $r_{b}\approx-9.3767$
$\Phi_{b}\approx-18.9233$ and also for $c_{8}=30$, $r_{b}\approx 20.5233$ and
$\Phi_{b}\approx-48.8233$ by using (123) and (127). Figure-(12) demonstrates
that when $c_{8}=0.1$, the Eq. (128) does not have a solution for $t_{s}$
because the curves $y=f(t_{s})$ and $y=g(t_{s})$ do not cross. So for
$c_{8}=0.1$, the particle which starts from $r=r_{b}$ at time
$t=t_{b}>28.2843$ but it will never reach the singularity as shown in
Figure-(14) (blue line) using the Eqs. (115) and (116). However, Figure-(13)
shows that when $c_{8}=30$, Eq. (128) has a solution for $t_{s}$ since the
curves $y=f(t_{s})$ and $y=g(t_{s})$ intersect. For a value of $c_{8}$ equal
to 30, the particle begins at $r=r_{b}$ at a time $t=t_{b}$ greater than
28.2843. It will eventually reach the singularity at $r=0$ at time $t=t_{s}$,
as depicted by the red line in Figure-(14). The particle will then depart from
the location $r=0$, which can be determined from Figure-(14) using Eqs. (115)
and (116).

Figure 12: $\mu=0.05$, $L=1$, $t_{0}=0.1$ and $c_{8}=0.1$: Blue Colour line
for $f(t_{s})$ and Red colour line for $g(t_{s})$ Figure 13: $\mu=0.05$,
$L=1$, $t_{0}=0.1$ and $c_{8}=30$: Blue Colour line for $f(t_{s})$ and Red
colour line for $g(t_{s})$ Figure 14: Time-like geodesics for
$r-t_{0}E(t)-\bar{B}-c_{8}=0$: When $\mu=0.05$, $L=1$, $t_{0}=0.1$: Blue
Colour line for $c_{8}=0.1$ and Red colour line for $c_{8}=30$

####  III.2.2 Null Geodesics for Case-II

Within this part, we have studied the characteristics of both radial ($L=0$)
and non-radial ($L\neq 0$) null geodesics in the generalized K-essence Vaidya
spacetime (13) with the specific mass function (97).

By tracing the radial null geodesic using the same reasoning as previously, we
obtain the identical value of $r_{a}$ as stated in Eq.(104). Continuing in a
similar manner as before, we have:

$\displaystyle r=t_{0}E(t)+\frac{2\mu t_{a}}{1-E(t_{a})}-t_{0}E(t_{a})$ (129)

Again, we able to calculate the time $t=t_{s}$ for a particle approaches to
singularity $r\rightarrow 0$, we obtain

$\displaystyle
t_{s}=t_{0}\ln{\Big{[}\frac{1-E(t_{a})}{2E(t_{a})[1-E(t_{a})]-\frac{4\mu
t_{a}}{t_{0}}}\Big{]}}.$ (130)

To trace the characteristics of non-radial ($L\neq 0$) null geodesics for the
given spacetime (13) through the Eq. (35) we consider $D>2\implies
t\geq\frac{L}{\mu}$ ($D=\frac{2\mu t}{L}$). Similarly, we exclude the
situation when $D=2$ due to the aforementioned rationale for non-radial time-
like geodesics. Therefore, when $D>2$, the Eq. (35) can be written as

$\displaystyle\Big{(}r-t_{0}E(t)-\bar{A}(t)-c_{9}\Big{)}\Big{(}r-t_{0}E(t)-\bar{B}(t)-c_{9}\Big{)}=0$
(131)

where

$\displaystyle\bar{A}(t)=t-\frac{2\mu^{2}t^{3}}{3L^{2}}+\frac{2\mu^{2}}{3L^{2}}\Big{(}t^{2}-\frac{L^{2}}{\mu^{2}}\Big{)}^{\frac{3}{2}}$
$\displaystyle\implies\bar{A}(t)=\frac{2\mu^{2}}{3L^{2}}\mathcal{O}\Big{(}\frac{1}{t}\Big{)}$
(132) and
$\displaystyle\bar{B}(t)=t-\frac{2\mu^{2}t^{3}}{3L^{2}}-\frac{2\mu^{2}}{3L^{2}}\Big{(}t^{2}-\frac{L^{2}}{\mu^{2}}\Big{)}^{\frac{3}{2}}$
$\displaystyle\implies\bar{B}(t)=2t-\frac{4\mu^{2}t^{3}}{3L^{2}}-\frac{2\mu^{2}}{3L^{2}}\mathcal{O}\Big{(}\frac{1}{t}\Big{)}$
(133)

The equation (131) indicates the existence of two non-radial null geodesics
that may be followed for any finite value of $c_{9}$. Proceeding as before, we
have

$\displaystyle
r=t_{0}E(t)+\frac{2\mu^{2}}{3L^{2}}\mathcal{O}\Big{(}\frac{1}{t}\Big{)}+c_{9}$
(134) with
$\displaystyle\Phi=\frac{1}{L}\Big{[}-\frac{2\mu^{2}}{3L^{2}}\mathcal{O}\Big{(}\frac{1}{t}\Big{)}-c_{9}\Big{]}$
(135) and $\displaystyle
r=t_{0}E(t)+2t-\frac{4\mu^{2}t^{3}}{3L^{2}}-\frac{2\mu^{2}}{3L^{2}}\mathcal{O}\Big{(}\frac{1}{t}\Big{)}+c_{9}$
(136) with
$\displaystyle\Phi=\frac{1}{L}\Big{[}-2t+\frac{4\mu^{2}t^{3}}{3L^{2}}+\frac{2\mu^{2}}{3L^{2}}\mathcal{O}\Big{(}\frac{1}{t}\Big{)}-c\Big{]}$
(137)

Again, we consider a particle starts its journey from $r=r_{a}$ in the path
(134) at time $t=t_{a}>\frac{L}{\mu}$, we get

$\displaystyle c_{9}=r_{a}-t_{0}E(t_{a})-\bar{A}(t_{a})$ (138)

For this scenario, the condition imposed on the particle trajectory is as
follows:

$\displaystyle c_{9}\leq r_{a}-\bar{A}(t_{a})<\frac{t_{0}}{2}+c_{9}$ (139)

since $\frac{t_{a}}{t_{0}}\geq 0$ and $0\leq E(t_{a})<\frac{1}{2}$. Hence, the
Eqs. (134) and (135) becomes

$\displaystyle r-t_{0}E(t)-\bar{A}(t)-r_{a}+t_{0}E(t_{a})+\bar{A}(t_{a})=0$
(140)
$\displaystyle\Phi=\frac{1}{L}\Big{[}-\bar{A}(t)-r_{a}+t_{0}E(t_{a})+\bar{A}(t_{a})\Big{]}$
(141)

then at $t=t_{a}$

$\displaystyle\Phi=\Phi_{a}=\frac{1}{L}\Big{[}-r_{a}+t_{0}E(t_{a})\Big{]}$
(142)

If it is achievable, let us assume that the particle travels along the path
(134) and reaches the singularity $(r=0)$ at time $t=t_{s}$. Then, from (140)
we get

$\displaystyle
E(t_{s})-\frac{2\mu^{2}t^{3}_{s}}{3t_{0}L^{2}}+\frac{c}{t_{0}}=-\frac{t_{s}}{t_{0}}-\frac{2\mu^{2}}{3t_{0}L^{2}}\Big{(}t^{2}_{s}-\frac{L^{2}}{\mu^{2}}\Big{)}^{\frac{3}{2}}$
(143)

Again, it is also a transcendental equation and it can be analyzed numerically
and graphically for any finite value of $\mu$ and $L$. As before, let
$f(t_{s})=E(t_{s})-\frac{2\mu^{2}t^{3}_{s}}{3t_{0}L^{2}}+\frac{c_{9}}{t_{0}}$
and
$g(t_{s})=-\frac{t_{s}}{t_{0}}-\frac{2\mu^{2}}{3t_{0}L^{2}}\big{(}t^{2}_{s}-\frac{L^{2}}{\mu^{2}}\big{)}^{\frac{3}{2}}$.
Then the only possibility for a finite value $t_{s}$ if and only if
$f(t_{s})=g(t_{s})$.
To start with let us consider $\mu=0.01$ and $L=3$ then $t_{a}>300$. Let
$t_{a}=301$ Then from (132), $\bar{A}(t_{a})\approx 99.1025$. Now if we
consider $t_{0}=0.1$ then for $c_{9}=0.1$, $r_{a}\approx 99.2025$
$\Phi_{a}\approx-33.0675$ and also for $c_{9}=20$, $r_{a}\approx 119.1025$ and
$\Phi_{a}\approx-39.7008$ by using (138) and (140). Figure-(15) and
Figure-(16) shows that the for $c_{9}=0.1$ and $c_{9}=20$, the Eqs. (143) do
not have solution for finite $t_{s}$ since the the curves $y=f(t_{s})$ and
$y=g(t_{s})$ does not intersect each other. From the Eqs. (134) and (135), we
get
$r=t_{0}E(t)+\frac{2\mu^{2}}{3L^{2}}\mathcal{O}\big{(}\frac{1}{t}\big{)}+c_{9}$
and
$\Phi=\frac{1}{L}\big{[}-\frac{2\mu^{2}}{3L^{2}}\mathcal{O}\big{(}\frac{1}{t}\big{)}-c_{9}\big{]}$
so that for large $t$, we have $r\to c_{9}$ and $\Phi\to-\frac{c_{9}}{L}$. So
there is no way to trace any null geodesics for the non-zero value of $c_{9}$
like $c=0.1$ and $c=20$ as shown in Figure-(17) using the equations (140) and
(141). But if we take $c_{9}=0$, a particle starts its journey from
$r_{a}\approx 99.1025$ and $\Phi_{a}\approx-33.0342$, it finally plunges to
the singularity as shown in Figure-(18).

Figure 15: $\mu=0.01$, $L=3$, $t_{0}=0.1$ and $c_{9}=0.1$: Blue Colour line
for $f(t_{s})$ and Red colour line for $g(t_{s})$ Figure 16: $\mu=0.01$,
$L=3$, $t_{0}=0.1$ and $c_{9}=20$: Blue Colour line for $f(t_{s})$ and Red
colour line for $g(t_{s})$ Figure 17: Null geodesics for
$r-t_{0}E(t)-\bar{A}-c_{9}=0$: When $\mu=0.01$, $L=3$, $t_{0}=0.1$: Blue
Colour line for $c_{9}=0.1$ and Red colour line for $c_{9}=20$ Figure 18: Null
geodesics for $r-t_{0}E(t)-\bar{A}-c_{9}=0$: When $\mu=0.01$, $L=3$,
$t_{0}=0.1$: Blue Colour line for $c_{9}=0$

Similar to the previous example, if we look at a particle that begins its
journey from $r=r_{b}$ in the path described in Eq. (136) at time
$t=t_{b}>\frac{L}{\mu}$, we obtain the following relations with the same type
of meaning from the Eqs. (136), (137), and (133):

$\displaystyle c_{9}=r_{b}-t_{0}E(t_{b})-\bar{B}(t_{b})$ (144) $\displaystyle
c_{9}\leq r_{b}-\bar{B}(t_{b})<\frac{t_{0}}{2}+c_{9}$ (145) $\displaystyle
r-t_{0}E(t)-\bar{B}(t)-r_{b}+t_{0}E(t_{b})+\bar{B}(t_{b})=0$ (146)
$\displaystyle\Phi=\frac{1}{L}\Big{[}-\bar{B}(t)-r_{b}+t_{0}E(t_{b})+\bar{B}(t_{b})\Big{]}$
(147)
$\displaystyle\Phi=\Phi_{b}=\frac{1}{L}\Big{[}-r_{b}+t_{0}E(t_{b})\Big{]}$
(148) $\displaystyle
E(t_{s})-\frac{2\mu^{2}t^{3}_{s}}{3t_{0}L^{2}}+\frac{c_{9}}{t_{0}}=-\frac{t_{s}}{t_{0}}+\frac{2\mu^{2}}{3t_{0}L^{2}}\Big{(}t^{2}_{s}-\frac{L^{2}}{\mu^{2}}\Big{)}^{\frac{3}{2}}$
(149)

Again, as before, Eq. (149) is a transcendental equation, so it can be
analyzed numerically and graphically for any finite value of $\mu$ and $L$.
Now let
$f(t_{s})=E(t_{s})-\frac{2\mu^{2}t^{3}_{s}}{3t_{0}L^{2}}+\frac{c_{9}}{t_{0}}$
and
$g(t_{s})=-\frac{t_{s}}{t_{0}}+\frac{2\mu^{2}}{3t_{0}L^{2}}\big{(}t^{2}_{s}-\frac{L^{2}}{\mu^{2}}\big{)}^{\frac{3}{2}}$.
Then the only possibility for a finite value $t_{s}$ if and only if
$f(t_{s})=g(t_{s})$.
To start with let us consider $\mu=0.05$ and $L=1$ then $t_{b}>20$. Let
$t_{b}=21$, then from (133), $\bar{B}(t_{a})\approx 5.1275$. Now if we
consider $t_{0}=0.1$ then for $c_{9}=0.1$, $r_{b}\approx 5.2275$
$\Phi_{b}\approx-5.2275$ and also for $c_{9}=20$, $r_{b}\approx 25.1275$ and
$\Phi_{b}\approx-25.1275$ by using (136) and (137). Figure-(19) shows that for
$c_{9}=0.1$, the Eq. (149) has a solution for $t_{s}$ since the the curves
$y=f(t_{s})$ and $y=g(t_{s})$ intersect each other. Also Figure-(20) shows
that for $c_{9}=20$, the Eq. (149) has a solution for $t_{s}$ since the curves
$y=f(t_{s})$ and $y=g(t_{s})$ intersect each other. So for $c_{9}=0.1$ and
$c_{9}=20$, the particle which starts from $r=r_{b}$ at time $t=t_{b}>20$ and
it will reach the singularity ($r=0$) at $t=t{s}$ as shown in Figure-(21)
using the equations (136) and (137).

Figure 19: $\mu=0.05$, $L=1$, $t_{0}=0.1$ and $c_{9}=0.1$: Blue Colour line
for $f(t_{s})$ and Red colour line for $g(t_{s})$ Figure 20: $\mu=0.05$,
$L=1$, $t_{0}=0.1$ and $c_{9}=20$: Blue Colour line for $f(t_{s})$ and Red
colour line for $g(t_{s})$ Figure 21: Null geodesics for
$r-t_{0}E(t)-\bar{B}-c_{9}=0$: When $\mu=0.05$, $L=1$, $t_{0}=0.1$: Blue
Colour line for $c_{9}=0.1$ and Red colour line for $c_{9}=20$

##  IV Discussion and Conclusion

In the present investigation we have traced time-like geodesics and null
geodesics for two different K-essence Vaidya Mass functions
$\mathcal{M}(t,r)=M+\frac{r}{2}e^{-\frac{t}{t_{0}}}$ and $\mathcal{M}(t,r)=\mu
t+\frac{r}{2}e^{-\frac{t}{t_{0}}}$ on equatorial plane $\theta=\frac{\pi}{2}$
by using Euler-Lagrange equations. For both the cases, we observe that
$\mathcal{M}(t,r)$ can be written in terms of energy $E(t)$. We have noticed
that for the radial time-like and radial null geodesics for both the K-essence
Vaidya masses, a particle having zero angular momentum $(L)$ starting from a
finite distance takes a finite time to reach the singularity. For the first
K-essence Vaidya mass function, there are two types of families for the time-
like case that can be traced one for $M=\sqrt{2}L$ and the other for
$M>\sqrt{2}L$ and similarly for the null case one for $M=L$ and the other for
$M>L$. But a bit of difference we observed for the second K-essence Vaidya
mass function. There does exist only one type of family for the time-like case
when $\mu t>\sqrt{2}L$ and for the null case $\mu t>L$. For all the above-
mentioned cases, for different finite value of arbitrary constants $c$, we
have traced different kinds of orbits for both the generalized K-essence
Vaidya mass function and the conclusions as follows:

Based on Fig. (1), it may be inferred that a future observer will observe the
particle approaching to the singularity within a finite amount of time, as
previously stated. According to Figs. 2 – 4 it is evident that a future
observer can watch the particle reaching at the singularity within a finite
period and thereafter departing from it. Based on the information provided in
Figs. 1 – 4, for non-radial time-like geodesics of the generalized K-essence
Vaidya spacetime (13) with the first mass function (17) or (27), it can be
inferred that there is a worry with the presence of a central singularity
(where both $r$ and $t$ approach to $0$). It is evident that the central
singularity in the generalized Vaidya spacetime is defined by the limits
$r\rightarrow 0$ and $t\rightarrow 0$, as shown in [29, 31, 37, 21, 32].
However, based on our analysis of the given data presented in Figs. 1 – 4, we
may conclude that when $r$ approaches zero, $t$ does not tend to zero. This
phenomenon is also observed in the non-radial null geodesics with the same
mass function but with varying finite time for different situations, as
depicted in Figs. 5 – 8. These phenomena can be explained as follows: a future
observer is expected to see the particle approaching to the singularity and
then moving away from it for a limited period of time (51) or (62) or (67)
(non-radial time-like geodesic) and (80) or (95) or (96) (non-radial null
geodesic) in different situations. This observation could potentially indicate
the presence of a wormhole during the extreme stages of spacetime,
specifically the black hole and white hole, similar to the concept of the
Einstein-Rosen bridge [75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86].
Regarding particles, it is clear that their signature is changed by the
singularity. In this interesting scenario, it seems that the presence of a
wormhole allows for the possibility of gravitational collapse resulting in
either a naked singularity or a black hole and a white hole in the generalized
$K$-essence Vaidya spacetime with the given mass function (27). Based on the
analysis of collapsing scenarios in the generalized Vaidya spacetime, it has
been established that there can be either a naked singularity or a black hole
[29, 37]. This is supported by the potential presence of a dynamical horizon,
as discussed in [87, 88, 89, 90]. On the other hand, Vertogradov [34] has also
addressed the issue and showed the presence of naked singularities as well as
white holes within the framework of the usual generalized Vaidya spacetime in
a different setting. However, in our case, the results are a bit different as
we have already mentioned that there is the possibility of either a naked
singularity or a black hole and a white hole. If we look at all the
aforementioned figures with a closer view then it actually reveals the
presence of a wormhole situation also in between the black hole and white hole
scenarios. But here the conditions being extremal ($r\rightarrow 0$ and $t\neq
0$) we have a membrane-type wormhole instead of a tunnel as has been usually
suggested. Alternatively, the previous explanation can also be elucidated as
follows: Given that the particle reaches the point at $r=0$ and then escapes
from there within a certain period, it may be possible that a quantum
tunneling event occurs at the vicinity of the central singularity. Since there
is a finite time at the typical singular region at $r=0$, it follows that
there is likewise a finite probability of escaping the region at $r=0$. We
also have a finite time (43) and (73) to approach the singularity for a
particle for the radial time-like and null geodesic situations.

On the other hand, we have the characteristics of the radial and non-radial
time-like or null geodesics structures for the case of the second mass
function (97). Through an analysis of the radial properties of the time-like
and null geodesics, we have determined that the particle ultimately approaches
the singularity at $r=0$ within a limited amount of time, as shown by Eqs.
(108) and (130) respectively. To investigate the non-radial geodesics of the
specified spacetime (13) with the mass function (97), we have not found a
finite expression for time for a particle to reach the singularity at $r=0$.
However, we have obtained a transcendental equation for different
possibilities, namely Eqs. (122) or (127) for time-like geodesics and Eqs.
(143) or (149) for null geodesics. But from these transcendental equations, we
have analyzed the non-radial time-like or null geodesics through numerical and
graphical analysis. Based on the Figs. 9 – 14, our investigation of non-radial
time-like geodesics reveals that when $c_{8}=0.1$, the particle is unable to
reach at the singularity at $r=0$ (blue line). This phenomenon may be
explained as follows: The presence of a black hole singularity with an event
horizon might result in the spectator being unable to perceive the particle’s
journey towards the singularity. Based on the aforementioned figures (red
line), it has been observed that when the constant $c_{8}$ is set to 30, the
particle approaches to the singularity at $r=0$ after a finite amount of time
and subsequently moves away from it. Here, we also see the occurrence of a
wormhole-like physical phenomenon or quantum tunneling effect near the
singularity at $r=0$.

Regarding the non-radial null geodesic characteristics of the aforementioned
spacetime and mass function (97), we note that the transcendental Eqs. (143)
and (149) exhibit the same kind of solutions as Eq. (131). In this situation
also, we are yet unable to determine the expression for the time it takes to
reach the singularity at $r=0$. After solving the transcendental equations
using numerical analysis, we find the graphical solutions through the
following figures, i.e. Fig. 15, Fig. 16, Fig. 17, Fig. 18, Fig. 19, Fig. 20
and Fig. 21. From Fig. 17, we have not found any tracing of the non-radial
null geodesic structure for two choices of the constants viz. $c_{9}=0.1$ and
$c_{9}=20$ though available for the choice of $c_{9}=0$, from Fig. 19 – 18. No
traces of the non-radial null geodesic structure were noticed in the Fig. 17
for two different values of the constants, namely $c_{9}=0.1$ and $c_{9}=20$.
However, by setting $c_{9}=0$, as seen in Fig. 18, it is evident that the
particle originates from a specific location and ultimately drops into the
singularity. Alternatively, we have explored an additional solution to Eq.
(131) and its associated Eq. (149). We have analyzed the non-radial null
geodesic structure and depicted it in Fig. 21. In this scenario, we note that
for two specific values of the constants, namely $c_{9}=0.1$ and $c_{9}=20$,
the particle begins its trip from a fixed position and eventually reaches the
singularity within a limited amount of time. After reaching at the
singularity, the particle then escapes from it. This once again indicates the
presence of a wormhole or the quantum tunneling effect.

Based on the aforementioned discussion, we can conclude that in our specific
model, we have investigated the existence of a wormhole during the extreme
stages of spacetime, specifically the black hole and white hole, similar to
the concept of the Einstein-Rosen bridge or quantum tunneling effect near the
central singularity. This investigation was carried out in the comoving frame
for the generalized K-essence Vaidya spacetime using two types of K-essence
Vaidya mass functions. It is possible to argue that extra interactions between
the K-essence scalar field and the usual gravity are the cause of this
occurrence.

Based on the investigation of the present work, especially observations on the
Figs. 2 – 4, one may raise the issue of the black holes and baby universes
[92] that whether there is any possibilities of such situations under the
generalized $K$-essence Vaidya spacetime. This obviously draws special
attention and can be considered as a potential topic in the near future.

## Acknowledgments

The research by M.K. was carried out in Southern Federal University with
financial support of the Ministry of Science and Higher Education of the
Russian Federation (State contract GZ0110/23-10-IF). S.R. thanks the Inter-
University Centre for Astronomy and Astrophysics (IUCAA), Pune, Government of
India for providing visiting associateship and also acknowledges the facility
availed under ICARD at CCASS, GLA University, Mathura.

## References

  * [1] S. Chandrasekhar, The Mathematical Theory of Black Holes, Indian Edition 2010, Oxford University Press, 1992.
  * [2] N. Cruz, M. Olivares, J. Saavedra and J. R. Villanueva, Class. Quantum Grav. 22, (2005), 1167–1190.
  * [3] E. Berti, A Black-Hole Primer: Particles, Waves, Critical Phenomena and Superradiant Instabilities, arXiv:1410.4481, (2014).
  * [4] G. W. Gibbons, Class. Quantum Grav. 33, 025004, (2016),
  * [5] S. Chanda, G. W. Gibbons, and P. Guha, Jour. Math. Phys., 58, 032503 (2017),
  * [6] L. P. Eisenhart, Annals of Mathematics, Second Series, 30, 591-606, (1928),
  * [7] C. Duval, G. Burdet, H. P. Künzle, and M. Perrin., Phy. Rev. D31, 1841, (1985)
  * [8] B. Majumder, G. Manna and A. Das, Class. Quantum Grav. 37, 115002, (2020)
  * [9] Barriola M. and Vilenkin A., Phys. Rev. Lett., 63, 341, (1989)
  * [10] D. Gangopadhyay and G. Manna, EPL, 100, 49001, (2012)
  * [11] P. C. Vaidya, Proc. Indian Acad. Sci. Sect. A 33 (1951), 264.
  * [12] P. S. Joshi, Pramana J. Phys. 55 (2000), 529.
  * [13] P. S. Joshi, Global Aspects in Gravitation and Cosmology, Clarendon, Oxford (1993).
  * [14] P. S. Joshi and I. H. Dwivedi, Phys. Rev. D 47 (1993), 5357.
  * [15] P. S. Joshi and I. H. Dwivedi, Commun. Math. Phys. 146 (1992), 333.
  * [16] P. S. Joshi and T. P. Singh, Phys. Rev. D 51 (1995), 6778.
  * [17] J. R. Oppenheimer and H. Snyder, Phys. Rev. 56 (1939), 455.
  * [18] D. Malafarina, Universe 3 (2017), 48.
  * [19] A. Papapetrou, ”A Random Walk in Relativity and Cosmology”, edited by N. Dadhich, J. K. Rao, J. V . Narlikar, and C. V . Vishveshwara, John Wiley and Sons, New York, P-184, (1985).
  * [20] R. Penrose, Riv. Nuovo Cim. 1, 252-276, (1969); Gen. Rel. Grav. 34, 1141-1165, (2002)
  * [21] I H Dwivedi and P S Joshi, Class. Quantum Grav. 6, 1599-1606 (1989).
  * [22] V. Vertogradov, Int. Jour. of Mod. Phys.: Conference Series, 41, 1660124 (2016)
  * [23] P.C.Vaidya, Phys. Rev., 83 1, (1951)
  * [24] P.C. Vaidya and I.M. Pandya, Prog. of Theo. Phys., 35 1, (1966)
  * [25] P.C.Vaidya, Phys. Rev., 174 5, (1968)
  * [26] P.C.Vaidya, APJ, 144 3, (1966)
  * [27] P.C.Vaidya, Gen. Rel. Grav., 31 1, (1999).
  * [28] J. B. Griffiths and J. Podolsky, Exact Space-Times in Einstein’s General Relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, (2009).
  * [29] V. Husain, Phys. Rev. D 53 4, (1996).
  * [30] A. Wang and Y. Wu, Gen. Rel. Grav., 31 1, (1999).
  * [31] M. D. Mkenyeleye, R. Goswami and S. D. Maharaj, Phys. Rev. D90, 064034 (2014)
  * [32] K. D Patil, Pramana-jour. of Phys., 60, 3, 423 (2003)
  * [33] A. Coudray and JP. Nicolas, Gen. Rel. Gravit., 53, 73, (2021).
  * [34] V. Vertogradov, Int. Jour. Mod. Phys. A 37, 28n29, 2250185 (2022)
  * [35] J. Solanki and V. Perlick, Phys. Rev. D, 105, 064056, (2022)
  * [36] G. Manna, P. Majumdar and B. Majumder, Phys. Rev. D 101, 124034, (2020)
  * [37] G. Manna, Eur. Phys. J. C, 80, 813, (2020)
  * [38] B. Majumder, S. Ray and G. Manna, Fortschr. Phys., 2300133, (2023)
  * [39] S. Sawayama, Phys. Rev. D, 73, 064024, (2006)
  * [40] M. Visser, C. Barcelo, S. Liberati, Gen. Rel. Grav. 34, 1719, (2002).
  * [41] E. Babichev, V. Mukhanov, A. Vikman, J. High Energy Phys. 02, 101, (2008).
  * [42] E. Babichev, V. Mukhanov, A. Vikman, J. High Energy Phys. 0802, 101, (2008).
  * [43] A. Vikman, K-essence: Cosmology, causality and Emergent Geometry, Dissertation an der Fakultatfur Physik,Arnold Sommerfeld Center for Theoretical Physics, der Ludwig-Maximilians-Universitat Munchen, Munchen (2007).
  * [44] E. Babichev, V. Mukhanov, A. Vikman, Looking beyond the Horizon, WSPC Proceedings (2008).
  * [45] L. P. Chimento, Phys. Rev. D 69, 123517, (2008).
  * [46] C. Armendariz-Picon, V. Mukhanov, P. J. Steinhardt, Phys. Rev. D 63, 103510, (2001).
  * [47] R. J. Scherrer, Phys. Rev. Lett. 93, 011301, (2004).
  * [48] L. P. Chimento, Phys. Rev. D 69, 123517, (2004).
  * [49] C. Armendariz-Picon, V. Mukhanov, P. J. Steinhardt, Phys. Rev. Lett. 85, 4438, (2000).
  * [50] C. Armendariz-Picon, T. Damour, V. Mukhanov, Phys. Lett. B 458, 209, (1999).
  * [51] S. X. Tian and Z.H. Zhu, Phys. Rev. D, 103, 043518, (2021).
  * [52] R. Myrzakulov and L. Sebastiani, Symmetry 8, 57, (2016).
  * [53] A. A. Sen and N. C. Devi, Gen. Relativ. Gravit. 42, 821, (2010).
  * [54] A. Chatterjee, S. Hussain, K. Bhattacharya, Phys. Rev. D 104, 103505, (2021).
  * [55] H. E. S. Velten, R. F. Vom Marttens, W Zimdahl, The Eur. Phys. J. C. 74, 3160, (2014).
  * [56] J. K. Erickson, R. R. Caldwell, P. J. Steinhardt, C. Armendariz-Picon, V. Mukhanov, Phys. Rev. Lett. 88, 121301, (2002).
  * [57] S. DeDeo, R. R. Caldwell, P. J. Steinhardt, Phys. Rev. D 67, 103509, (2003).
  * [58] R. Bean and O. Dore, Phys. Rev. D 69, 083503, (2004).
  * [59] P. A. R. Ade et al. (Planck Collaboration), A $\&$ A 594, A14, (2016).
  * [60] N. Aghanim et al. (Planck Collaboration), A $\&$ A 641, A6, (2020).
  * [61] N. Aghanim et al. (Planck Collaboration), A $\&$ A 641, A1, (2020).
  * [62] G. Manna and D. Gangopadhyay, Eur. Phys. J. C 74, 2811, (2014).
  * [63] G. Manna and B. Majumder, Eur. Phys. J. C 79, 553, (2019).
  * [64] G. Manna, B. Majumder, A. Das, Eur. Phys. J. Plus 135, 107, (2020).
  * [65] G. Manna, A. Panda, A. Karmakar, S. Ray, Chinese Phys. C 47, 025101, (2023).
  * [66] S. Ray, A. Panda, B. Majumder, Md. R. Islam, G. Manna, Chinese Phys. C 46, 12, 125103, (2022).
  * [67] S. Das, A. Panda, G. Manna, S. Ray, Fortschr. Phys. 71, 2200193, (2023).
  * [68] A. Panda et al., arXiv: 2308.13574v1, (2023)
  * [69] A. Panda et al., arXiv: 2206.14808v2, (2023)
  * [70] S. Mukohyama, R. Namba, Y. Watanabe, Phys. Rev. D 94, 023514, (2016).
  * [71] M. Born and L. Infeld, Proc. Roy. Soc. Lond. A 144, 425, (1934).
  * [72] W. Heisenberg, Zeit. Phys. 113, 61, (1939).
  * [73] P. A. M. Dirac, Proc. Roy. Soc. Lond. A 268, 57, (1962).
  * [74] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey and D.E. Knuth, Adv. Comp. Math., 5, 329-359, (1996).
  * [75] Einstein, A., Rosen, N., 1935, Phys. Rev. 48, 73
  * [76] Einstein, A., Rosen, N., 1935, Ann. Phys. 2, 242
  * [77] Fuller, R. W. & Wheeler, J. A., 1962, Phys. Rev. 128, 919
  * [78] Ellis, H. G., 1973, J. Math. Phys. 14, 104
  * [79] Morris, M. S. & Thorne, K. S., 1988, Am. J. Phys. 56, 395
  * [80] Visser, M., 1989, Phys. Rev. D 39, 3182
  * [81] Hayward, S. A., 2009, Phys. Rev. D 79, 124001
  * [82] Bronnikov, K. A., et al., 2013, Gravit. Cosmol. 19, 269
  * [83] Garattini, R., 2019, Eur. Phys. J. C 79, 951
  * [84] Mishra, B., Agrawal, A. S., Tripathy, S. K. & Ray, S., 2021, Int. J. Mod. Phys. D 30, 2150061
  * [85] Mishra, B., Agrawal, A. S., Tripathy, S. K. & Ray, S., 2022, Int. J. Mod. Phys. A 37, 2250010
  * [86] G. Mustafa1, S.K. Maurya and S. Ray, Astrophys. J. 941, 170 (2022)
  * [87] A. Ashtekar and B. Krishnan, Phys. Rev. D 68, 104030, (2003).
  * [88] A. Ashtekar and B. Krishnan, Phys. Rev. Lett. 89, 261101, (2002).
  * [89] A. Ashtekar and B. Krishnan, Living Rev. Relativity 7, 10, (2004).
  * [90] S. A. Hayward, Phys. Rev. D 49, 6467 (1994).
  * [91] M. Blau, Lecture Notes on General Relativity [http://www.blau.itp. unibe.ch/GRLecturenotes.html], (8th October, 2022)
  * [92] S. Hawking, Black Holes and Baby Universes and other Essays, Bantam Dell Publishing Group (1993).