# Gravitational collapse in generalized K-essence emergent Vaidya spacetime
via $f(\bar{R},\bar{T})$ gravity

Arijit Panda<EMAIL_ADDRESS>Department of Physics, Raiganj
University, Raiganj, Uttar Dinajpur 733 134, West Bengal, India $\&$
Department of Physics, Prabhat Kumar College, Contai, Purba Medinipur 721404,
West Bengal, India  Goutam Mannaa<EMAIL_ADDRESS>
aCorresponding 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  Saibal Ray<EMAIL_ADDRESS>Centre for Cosmology,
Astrophysics and Space Science (CCASS), GLA University, Mathura 281406, Uttar
Pradesh, 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 Asrtroparticle Physics 10, rue Garreau, 75018 Paris,
France  Md. Rabiul Islam<EMAIL_ADDRESS>Department of Physics, Raiganj
University, Raiganj, Uttar Dinajpur 733134, West Bengal, India.

(Received: date / Accepted: date)

###### Abstract

This study focuses on investigating the collapse possibilities of the
generalized emergent Vaidya spacetime within the framework of
$f(\bar{R},\bar{T})$ gravity, specifically in the setting of K-essence theory.
The Dirac-Born-Infeld type non-standard Lagrangian is used in this work to
determine the emergent metric $\bar{G}_{\mu\nu}$, which is not conformally
equal to the usual gravitational metric. Here, we employ the function
$f(\bar{R},\bar{T})$ to represent the additive character of the emergent Ricci
scalar ($\bar{R}$) and trace of the emergent energy-momentum tensor
($\bar{T}$). In our analysis, it can be shown that certain selections of
$f(\bar{R},\bar{T})$ can lead to the presence of a naked singularity resulting
from gravitational collapse. Additionally, alternative choices of
$f(\bar{R},\bar{T})$ were found to yield an accelerating universe primarily
governed by dark energy. In addition, the investigation revealed the existence
of both positive and negative masses, which can be understood as a
gravitational dipole. It is also found that the K-essence theory has the
potential to serve as both a dark energy framework and a simple gravitational
theory at the same time, allowing for the investigation of a wide range of
cosmological phenomena.

###### pacs:

04.20.-q, 04.20.Dw, 04.50.Kd, 04.70.Bw

##  I Introduction

Gravitational collapse is a well-recognised phenomenon in the fields of
general relativity and astrophysics, as evidenced by the works of Joshi et al.
[1, 2]. It serves as a crucial avenue for comprehending many astrophysical
facets inside our universe. The phenomenon of gravitational collapse offers
valuable insights into several aspects of astrophysics, such as structure
development, stellar characteristics, black hole generation, and the formation
of white dwarfs or neutron stars, among other phenomena. Specifically,
gravitational collapse is an event where a star undergoes the process of
collapse due to its own mass, and depending upon some specific condition of
the initial mass it may end up at different stages of its collapse. Sometimes,
the process of stellar collapse bypasses the intermediate phases, such as the
production of a white dwarf or neutron star, and proceeds directly to the
creation of a singularity known as a black hole (BH), provided that the star
possesses a substantial mass exceeding $20M_{\odot}$, where $M_{\odot}$
denotes the solar mass. In their investigation of gravitational collapse,
Oppenheimer and Snyder employed a spherically symmetric dust cloud as the
fundamental framework. The process of collapse, as depicted in their study,
indicated the existence of a black hole in the final state. This model was
characterized by a static Schwarzschild exterior and Friedmann interior,
serving as an idealized representation of the phenomenon [3]. Subsequently,
extensive research pertaining to gravitational collapse has been conducted
globally, and comprehensive evaluations of these studies can be found in the
references [1, 2, 4].

In the context of this framework, Roger Penrose proposed the renowned Cosmic
Censorship Hypothesis (CCH), which argues that a cosmic singularity will
invariably be surrounded by an event horizon [5, 6]. Upon further examination
of this hypothesis, a novel model has emerged in which the singularity is not
concealed by an event horizon, resulting in the formation of a “naked
singularity” [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. This concept is
considered crucial in the advancement of a viable quantum gravity theory. The
first relativistic line element, which accurately depicted the spacetime of a
plausible star, was introduced by P. C. Vaidya in 1951 [18]. It generalized
the static solution of Schwarzschild by representing the radiation for a non-
static mass. The solution proposed by Schwarzschild represents the spacetime
surrounding a spherically symmetric, cold, black object with a fixed mass.
Hence, it is evident that the model is unable to adequately represent
spacetime beyond the boundaries of a star. The solution proposed by Vaidya
[18], commonly referred to as Vaidya spacetime or the radiating Schwarzschild
metric, was presented as a potential explanatory framework. It is important to
acknowledge that the primary difference between the two metrics lies in the
fact that the Vaidya metric incorporates a time-dependent mass parameter,
whilst the Schwarzschild metric utilizes a constant mass value, hence
resulting in a spacetime that varies with time. The Vaidya metric has been
extensively studied in the literature, and several notable works have
contributed to our understanding of this topic. Relevant works on the Vaidya
metric can be found in [19, 20, 21, 22, 23].

On the other hand, the findings derived from type IA Supernovae, Baryon
Acoustic Oscillator (BAO), Cosmic Microwave Background (WMAP7), and Planck
discoveries have provided compelling evidence indicating that the expansion of
our universe is now speeding at an unprecedented rate [24, 25, 26, 27, 28,
29]. The cause of this acceleration has been attributed to dark energy,
prompting scientists to contemplate modifying the general theory of relativity
by incorporating a curvature term as a variable function of $R$ in the action
($R$ referred to as the Ricci scalar). This culminated in the development of
the modified theory of gravity [30]. The standard theory of gravity has been
modified by various relevant theories, such as $f(R)$, $f(G)$, $f(R,L_{m})$,
and $f(R,G)$, where $L_{m}$ represents the matter Lagrangian and $G$ denotes
the Gauss-Bonnet invariant term [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
97, 43, 44]. As previously mentioned, $f(R)$ gravity is a theoretical
framework in which the action is formulated as a function of the Ricci scalar,
denoted as $R$. One notable aspect of this updated theory is its ability to
account for the acceleration of the cosmos without requiring the addition of a
cosmological constant ($\Lambda$). Previous studies in this subject have
introduced some generalizations by considering the connection between the
Ricci Scalar ($R$) and matter ($L_{m}$), as well as the trace of the energy-
momentum tensor ($T$), among other factors. The $f(R,L_{m})$ theory was
developed by Harko et al. [45], whereby the authors investigate the coupling
between the Ricci scalar ($R$) and the matter Lagrangian ($L_{m}$). The theory
provides the gravitational field equation and the equation of motion for test
particles. Subsequently, Harko et al. [46] extended the $f(R)$ theory by
proposing a connection between the Ricci scalar and the trace of the energy-
momentum tensor, resulting in the formulation of the $f(R,T)$ theory. In their
study, the authors of [46] employed the widely recognized variational
technique to get the field equation of this particular theory. Additionally,
they performed calculations to determine the covariant divergence of the
energy-momentum tensor. In recent years, there has been a rise in the
popularity of this idea, leading to a significant body of research and ongoing
scholarly endeavors in this field. The study conducted by Sahoo et al. [47]
investigates the thermodynamics of Bianchi-III and Bianchi-$VI_{0}$
cosmological models with a string fluid source in the context of $f(R,T)$
gravity. The authors also consider the limitations in the choice of $T$ as
discussed in Alvarenga’s work [48] and the thermodynamics in this theory as
examined by Sharif et. al. [49]. Numerous studies have been conducted in the
present context [51, 52, 53, 54, 55, 56, 57, 50, 58, 59, 60, 61, 62, 63, 64,
65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80].

Typically, our research endeavors involve the exploration of a canonical
theory, whereby the Lagrangian assumes its conventional interpretation as
$L=T-V$. However, it is important to note that the situation in question is
unique and requires specific consideration [81, 82, 83]. The non-canonical
theory can be seen as the general case, as it is possible to derive the
canonical theory from the non-canonical theory by imposing appropriate
restrictions. In scleronomic systems, the canonical Lagrangian is not
permitted to include explicit dependence on time because of the inability to
compute forces from any potential. Furthermore, the Lagrangian can take on
several forms, all of which maintain the preservation of the Euler-Lagrange
equation of motion. In addition, under the framework of special relativistic
dynamics, the classical notion of $L=T-V$ is no longer considered appropriate
[82]. In a recent study conducted by P. Jiroušek et al., [84], it has been
demonstrated in a comprehensive manner that the general form of the Lagrangian
is non-canonical. Moreover, it’s crucial to keep in mind that conventional or
canonical, or standard theories do not offer an all-encompassing knowledge of
everything. The nature of dark matter, dark energy, the mechanisms underlying
the Big Bang, the disparity between matter and antimatter, the cosmological
constant problem, the dimensions and configuration of the universe, cosmic
inflation, the horizon problem, and other pertinent aspects within the field
of cosmos are not sufficiently explained by the information provided. The
unification of gravity and quantum mechanics inside a single theoretical
framework is the main outstanding problem in the subject of fundamental
physics. So, there’s still work to be done. In the present context, we have
opted for the utilization of the K-essence theory as a non-canonical theory
for our investigation. This choice is supported by the works of Visser,
Babichev et al., Vikman, Chimento et al., Picon et al., Scherrer, and others,
as referenced in the citation [85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95].

In the K-essence theory [85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95], the
scalar field is coupled minimally with gravity and involves a non-canonical
Lagrangian of the type $L(X)=-V(\phi)F(X)$, 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 alternative
variant of the Lagrangian, as discussed by Tian [96], may be expressed as
$L=[1+f(y)]X+[1+g(y)]V_{exp}$, 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
arbitrary functions. The use of K-essence model by the authors of this
research is employed in the investigation of primordial dark energy.
Additionally, it is worth noting that there are instances of non-minimally
coupled K-essence theories as discussed by Refs. [97, 98, 99]. However, the
focus of this article is solely on the minimally coupled K-essence theory, as
explored by Refs. [85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95]. Typically, the
Lagrangian possesses the capability to rely on arbitrary functions of $\phi$
and $X$ in a broad sense. The advantage of the K-essence theory is that it can
avoid the fine-tuning problem and produce the negative pressure responsible
for the acceleration of the universe through the field’s kinetic energy only.
The potential term is suppressed by the kinetic term of the field. This may be
a solution for the well-known Cosmic Coincidence problem [100] that arises in
the usual theory of general theory of relativity. In [91] we can find some
attractor solutions in which the evolution of the universe is determined by
the scalar field of the models. Because the K-essence field sub-dominated and
imitated the radiation’s equation of state (EOS), the ratio of the K-essence
field to radiation density stayed constant during the radiation-dominated
phase. Dynamical limitations prevented the K-essence field from reproducing
the dust-like EoS during the dust-dominated period, but it quickly decreased
its energy value by many orders of magnitude and achieved a constant value.
Later, at a time roughly equal to the age of the universe today, the matter
density was suppressed by the K-essence field and the cosmos began to
accelerate. The K-essence theory’s EoS eventually returns to a value between 0
and -1. Though theoretically, it can go beyond $-1$. The possibility of the
K-essence theory to produce a form of dark energy in which sound travels at a
constantly slower speed than light is another fascinating feature of the
theory. The cosmic microwave background (CMB) disturbances on large angular
scales may be lessened by this attribute [101, 102, 103]. The observational
evidence for the K-essence theory and other modified theories have been
mentioned in [27, 28].

In this particular context, Manna et al. [104, 105, 106, 107, 109, 110, 111,
108, 112, 113] have formulated an interesting emergent gravity metric denoted
as $\bar{G}_{\mu\nu}$. This metric exhibits additional characteristics
compared to the conventional gravitational metric $g_{\mu\nu}$ and is derived
based on the principles of the Dirac-Born-Infeld (DBI) type action [114, 115,
116, 117]. Dirac et al. [117] suggested a non-canonical Lagrangian to get rid
of the infinite self-energy of the electron. Later, this type of Lagrangian
found its vast use in the field of cosmology and quantum gravity [118, 119,
120, 121, 122, 123, 124, 125, 126]. In order to construct the $f(R,L(X))$ type
modified theory of gravity and its cosmological implications, Manna et al.
[108] used the emergent metric described in [104, 105, 106]. Manna [110] and
Ray et al. [111] conducted research on the characteristics of singularity
within the framework of the generalized Vaidya metric, employing the
aforementioned non-canonical method in different settings. The researchers
found that under certain precise circumstances, the singularity might manifest
as a naked singularity. The research has also examined the strength of
uniqueness in their study. Panda et al. [127] conducted a study on the
$f(\bar{R},\bar{T})$ gravity theory, employing the emergent metric
$(\bar{G}_{\mu\nu})$ inside a non-canonical geometry. Their findings indicate
a strong agreement between the equation of state (EOS) parameter and the
observational data. On the other end of the spectrum, the K-essence idea is
commonly employed for the purpose of investigating dark energy as a model.
However, it can be utilized solely from a gravitational or geometrical
perspective [109, 110, 111, 112, 113] due to the ongoing debate about the
presence of dark energy [128] based on current observations [129].

This study aims to investigate the fate of singularity within the framework of
$f(\bar{R},\bar{T})$ gravity in generalized emergent Vaidya spacetime. To do
this, we employ the non-canonical technique known as the $K$-essence. The
field equation of $f(\bar{R},\bar{T})$ gravity, derived from emergent
geometry, is studied in the present research [127]. The investigation focuses
on the fate of the singularity by taking the emergent Vaidya metric as the
primary metric [110]. In section II, we provide a concise overview of the
K-essence model in connection with the associated $f(\bar{R},\bar{T})$
gravity.

The revised emergent field equation has a different geometric nature compared
to the equation developed by Harko et al. [46]. The solution for the emergent
Vaidya metric within the framework of $f(\bar{R},\bar{T})$ gravity is
presented in Section III. In this section, the major measure considered is the
generalized emergent Vaidya metric. By analyzing the given information, we are
able to derive the distinct components of the modified emergent field
equation. We obtain the various mass functions for various choices of the
function $f(\bar{R},\bar{T})$. In section IV, we conducted an extensive study
of the collapsing prospects of the given spacetime, considering various
options of the function $f(\bar{R},\bar{T})$. In this analysis, a global naked
singularity has been found, which is associated with the gravitational
collapse under certain selections of the function $f(\bar{R},\bar{T})$.
Conversely, our observations have shown examples of the universe exhibiting
acceleration, mostly driven by the presence of dark energy. Additionally, we
have detected the presence of both positive and negative masses, which give
rise to gravitational dipoles. The last section is our conclusion of the work.

##  II Brief review of K-essence theory and the corresponding
$f(\bar{R},\bar{T})$ gravity

This section provides a concise overview of K-essence geometry and the
associated $f(\bar{R},\bar{T})$ gravity. Firstly, we provide a concise
overview of the geometry associated with the K-essence, as discussed in many
academic sources [85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95]. The action of
this geometry is

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

where $X=\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$ is the
canonical kinetic term and $L(X,\phi)$ is the non-canonical Lagrangian. Here,
the typical gravitational metric $g_{\mu\nu}$ has minimally coupled with the
K-essence scalar field ($\phi$).

The corresponding energy-momentum tensor associated with the K-essence scalar
field only is:

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

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

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

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

where

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

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

The inverse metric is

$\displaystyle
G_{\mu\nu}=\frac{L_{X}}{c_{s}}[g_{\mu\nu}-{c_{s}^{2}}\frac{L_{XX}}{L_{X}}\nabla_{\mu}\phi\nabla_{\nu}\phi].$
(5)

After a conformal transformation [104, 105]
$\bar{G}_{\mu\nu}\equiv\frac{c_{s}}{L_{X}}G_{\mu\nu}$ we have

$\displaystyle\bar{G}_{\mu\nu}=g_{\mu\nu}-\frac{L_{XX}}{L_{X}+2XL_{XX}}\nabla_{\mu}\phi\nabla_{\nu}\phi.$
(6)

The Eqs. (4)–(6) hold physical significance under the condition that $L_{X}$
is not equal to zero, given a positive definite $c_{s}^{2}$. Eq. (6) asserts
that the emergent metric, denoted as $\bar{G}_{\mu\nu}$, is conformally
different from the metric $g_{\mu\nu}$ when considering non-trivial
configurations of the scalar field $\phi$. Similar to canonical scalar fields,
the variable $\phi$ has varied local causal structural characteristics. It is
also different from those that are defined with $g_{\mu\nu}$. The equation of
motion, as expressed in Eq. (3), remains applicable even when considering the
implicit dependence of $L$ on $\phi$. Then the EOM Eq. (3) is:

$\displaystyle\frac{1}{\sqrt{-g}}\frac{\delta
S_{k}}{\delta\phi}=\bar{G}^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=0.$ (7)

In this study, we look into the Dirac-Born-Infeld (DBI) type non-canonical
Lagrangian denoted as $L(X,\phi)\equiv L(X)$ [104, 105, 106, 114, 115, 116,
117, 127]:

$\displaystyle L(X)=1-\sqrt{1-2X}.$ (8)

The K-essence framework argues that the dominance of kinetic energy over
potential energy leads to the omission of the potential term in the Lagrangian
equation (8) [114, 127]. Consequently, the squared speed of sound, denoted as
$c_{s}^{2}$, is given by $(1-2X)$. Thus, the effective emergent metric Eq. (6)
becomes

$\displaystyle\bar{G}_{\mu\nu}=g_{\mu\nu}-\nabla_{\mu}\phi\nabla_{\nu}\phi=g_{\mu\nu}-\partial_{\mu}\phi\partial_{\nu}\phi,$
(9)

since $\phi$ is a scalar.

Following [104, 105], the Christoffel symbol associated with the emergent
gravity metric Eq. (9) is:

$\displaystyle\bar{\Gamma}^{\alpha}_{\mu\nu}$
$\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\ $ (10)

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

Therefore, the geodesic equation for the K-essence geometry becomes:

$\displaystyle\frac{d^{2}x^{\alpha}}{d\lambda^{2}}+\bar{\Gamma}^{\alpha}_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}=0,$
(11)

where $\lambda$ is an affine parameter.

The covariant derivative $D_{\mu}$ [86] linked with the emergent metric
$\bar{G}_{\mu\nu}$ $(D_{\alpha}\bar{G}^{\alpha\beta}=0)$ yields

$\displaystyle
D_{\mu}A_{\nu}=\partial_{\mu}A_{\nu}-\bar{\Gamma}^{\lambda}_{\mu\nu}A_{\lambda},$
(12)

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

Hence, when taking into account the comprehensive action that characterizes
the dynamics of K-essence and general relativity [88], the Emergent Einstein’s
Equation (EEE) may be expressed as follows:

$\displaystyle\mathcal{\bar{G}}_{\mu\nu}=\bar{R}_{\mu\nu}-\frac{1}{2}\bar{G}_{\mu\nu}\bar{R}=\kappa\bar{T}_{\mu\nu},$
(13)

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. Additionally, the energy-momentum tensor $\bar{T}_{\mu\nu}$ is
associated with this emergent spacetime. In order to establish a comprehensive
understanding of the energy-momentum tensor associated with this particular
geometry, it is important to proceed with its definition as [127]

$\displaystyle\bar{T}_{\mu\nu}=-\frac{2}{\sqrt{-\bar{G}}}\frac{\partial\Big{(}\sqrt{-\bar{G}}L(X)\Big{)}}{\partial\bar{G}_{\mu\nu}},$
(14)

where
$\big{(}-\bar{G}\big{)}^{1/2}=\big{(}-det({\bar{G}_{\mu\nu}})\big{)}^{1/2}$.

It is worth noting that there are two distinct methods for deriving the
emergent energy-momentum tensor ($\bar{T}_{\mu\nu}$). There are two options
available for us to consider. The first option is to utilize the left-hand
side of the EEE equation, as referenced in Eq. (13). Alternatively, we can opt
to employ the definition of the emergent energy-momentum tensor
$\bar{T}_{\mu\nu}$, as stated in Eq. (14). In [127], they have opted to
utilize the second approach due to its ability to include the action principle
of $f(\bar{R},\bar{T})$ gravity. The first methodology employed in obtaining
the emergent energy-momentum tensor is likewise utilized by Manna et al. [109,
110, 111, 113] in their scholarly works pertaining to the emergent Vaidya
geometry. In the field of gravitational or cosmological investigations
pertaining to the cosmos, researchers have the option to utilize either the
first strategy or a second approach in order to figure out the energy-momentum
tensor.

Now, we will provide a concise overview of the $f(\bar{R},\bar{T})$ gravity
theory within the framework of K-essence geometry, drawing inspiration from
the work of Harko et al. [46]. The phenomenon of modified gravity within the
context of K-essence geometry has been discussed in a study by Panda et al.
[127].

The action ($\kappa=1$) of this type of modified gravity is [127]:

$\displaystyle S=\int
d^{4}x\sqrt{-\bar{G}}\Big{[}f(\bar{R},\bar{T})+L(X)\Big{]},$ (15)

where the function $f(\bar{R},\bar{T})$ represents an arbitrary function of
the Ricci scalar ($\bar{R}$) and the trace of the energy-momentum tensor
($\bar{T}=\bar{T}^{\mu\nu}\bar{G}_{\mu\nu}$). On the other hand, $L(X)$
denotes the non-canonical Lagrangian associated with the K-essence geometry.

The updated action (15) reveals a clear dependence on the variables $\bar{R}$,
$\bar{T}$, and $X(=\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi)$,
rather than on the explicit dependence of the K-essence scalar field ($\phi$).
Considering $\bar{G}_{\mu\nu}$ as in Eq. (9), the emergent energy-momentum
tensor (14) can also be written as

$\displaystyle\bar{T}_{\mu\nu}=\bar{G}_{\mu\nu}L(X)-2\frac{\partial
L(X)}{\partial\bar{G}^{\mu\nu}}.$ (16)

Varying the action and following [127] we get the modified field equation of
emergent $f(\bar{R},\bar{T})$ gravity for $L(X)$ taken as Eq. (8)

$\displaystyle
F\bar{R}_{\mu\nu}-\frac{1}{2}f(\bar{R},\bar{T})\bar{G}_{\mu\nu}+(\bar{G}_{\mu\nu}\bar{\square}-D_{\mu}D_{\nu})F$
$\displaystyle=\frac{1}{2}\bar{T}_{\mu\nu}-f_{\bar{T}}\bar{T}_{\mu\nu}-f_{\bar{T}}\bar{\Theta}_{\mu\nu}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (17)

where

$\displaystyle\bar{\Theta}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\bar{G}^{\alpha\beta}\frac{\partial\bar{T}_{\alpha\beta}}{\partial\bar{G}^{\mu\nu}}$
(18) $\displaystyle=$
$\displaystyle\bar{G}_{\mu\nu}L(X)-2\bar{T}_{\mu\nu}-2\bar{G}^{\alpha\beta}\frac{\partial^{2}L(X)}{\partial\bar{G}^{\mu\nu}\partial\bar{G}^{\alpha\beta}},$

with $\bar{\square}=D_{\mu}D^{\mu}$, $F(\bar{R},\bar{T})=\partial
f(\bar{R},\bar{T})/\partial\bar{R}\equiv F$ and
$f_{\bar{T}}(\bar{R},\bar{T})=\partial
f(\bar{R},\bar{T})/\partial\bar{T}\equiv f_{\bar{T}}$ respectively. It is
important to note that the modified field equation of emergent
$f(\bar{R},\bar{T})$ gravity [127] looks like the equation obtained by Harko
et al. [46] in their article. However, it is crucial to recognize that there
is a significant geometric distinction between our findings and theirs. This
disparity arises from the fact that the emergent geometry ($\bar{G}_{\mu\nu}$)
is not conformally equivalent to the conventional gravitational geometry
($g_{\mu\nu}$).

##  III Solution of emergent Vaidya Metric in $f(\bar{R},\bar{T})$ gravity

The purpose of this section is to investigate the emergent Vaidya metric in
the context of $f(\bar{R},\bar{T})$ gravity and evaluate its characteristics.
In this study, we focus on the primary metric known as the emergent Vaidya
type, as introduced by Manna et al. in their work [110], utilizing the
K-essence geometry. The K-essence emergent Vaidya line element is

$\displaystyle
dS^{2}=-\Big{[}1-2\mathcal{M}(v,r)\Big{]}dv^{2}+2dvdr+r^{2}d\Omega^{2},$ (19)

where $\mathcal{M}$ is the K-essence emergent Vaidya mass function having the
form

$\displaystyle\mathcal{M}(v,r)=m(v,r)+\frac{r}{2}\phi_{v}^{2},$ (20)

where $d\Omega^{2}=d\theta^{2}+sin^{2}\theta d\Phi^{2}$,
$\phi_{v}=\frac{\partial\phi}{\partial v}$ and $\phi_{v}^{2}$ denote the non-
zero kinetic energy associated with the K-essence scalar field.

The mass function $m(v,r)$ is referred to as the mass function for the
conventional generalized Vaidya spacetime, which serves as the underlying
gravitational metric. Alternatively, the function $m(v,r)$ represents the mass
distribution associated with the gravitational energy within a certain radius
$r$. The variable $v$ corresponds to the null coordinate that corresponds to
the Eddington advanced time, while $r$ decreases towards the future along a
ray defined by $v=constant$ [130, 132, 131]. According to the metric (19), the
values of $\phi_{v}^{2}$ are between $0$ and $1$, as well as a non-decreasing
function of $v$ under the energy conditions [109]. This is a somewhat
significant constraint on the types of K-essence scalar field configurations
that can be used in the emergent generalized Vaidya solution. The authors in
the cited work [110] have employed the emergent gravity metric, as shown by
Eq. (9), in conjunction with the Lagrangian of the DBI type, as represented by
Eq. (8). It is important to acknowledge that within the framework of
K-essence, the acceptance of Lorentz violation arises from the fact that the
solutions of the K-essence equation of motion (EoM) exhibit spontaneous
breaking of Lorentz invariance. Furthermore, it alters the metric pertaining
to the perturbation in the vicinity of these solutions [86, 87]. Also, the
selection of $\phi$ as a function of $v$ in this particular scenario violates
the principle of local Lorentz invariance. In addition to possessing spherical
symmetry, denoted as $\phi(x)=\phi(v,r)$, the aforementioned supplementary
assumption implies that outside of this emergent framework, $\phi$ may vary as
a function of both $v$ and $r$. The K-essence theory permits the occurrence of
Lorentz violation due to the inherent nature of the dynamical solutions of the
K-essence equation of motion. These solutions exhibit spontaneous breaking of
Lorentz invariance and induce alterations in the metric for the perturbations
around these solutions.

In the study conducted by Manna et al. [110], the emergent energy-momentum
tensor ($\bar{T}_{\mu\nu}$) was calculated for the first process mentioned
above. This calculation took into account the specific form proposed by Husain
et al. [130], Mkenyeleye et al. [131] and Wang et al. [132] which states that

$\displaystyle{\bar{T}}_{\mu\nu}={\bar{T}}_{\mu\nu}^{(n)}+{\bar{T}}_{\mu\nu}^{(m)},$
(21)

with $\bar{T}_{\mu\nu}^{(n)}=\bar{\gamma}l_{\mu}l_{\nu}$ being the
contribution due to Vaidya null radiation and
$\bar{T}_{\mu\nu}^{(m)}=(\bar{\rho}+\bar{p})(l_{\mu}n_{\nu}+l_{\nu}n_{\nu})+\bar{G}_{\mu\nu}\bar{p}$,
where $\bar{\rho}$ and $\bar{p}$ are the energy density and pressure of the
emergent perfect fluid, respectively. The vectors $l_{\mu}$ and $n_{\mu}$ are
two null vectors. The symbol $\bar{\gamma}$ represents the energy density
associated with the emergent Vaidya null radiation. It is noteworthy to point
out that within the K-essence model, using the DBI type non-canonical
Lagrangian (8), there exists a similarity to ideal fluid models without
vorticity [88, 86]. In this setting, we have employed the energy-momentum
tensor $(\bar{T}_{\mu\nu})$ in a way consistent with a perfect fluid. These
null vectors are defined as: $l_{\mu}=(1,0,0,0)$,
$n_{\mu}=(\frac{1}{2}(1-\frac{\mathcal{M}(v,r)}{r}),-1,0,0)$ with
$l_{\mu}l^{\mu}=n_{\mu}n^{\mu}=0$ and $l_{\mu}n^{\mu}=-1$. With these
definitions of null vectors, we can write the emergent energy-momentum tensor
as [22]

$\displaystyle{\bar{T}}_{\mu\nu}=\left[\begin{array}[]{cccc}[\bar{\gamma}+\bar{\rho}(1-\frac{2\mathcal{M}(v,r)}{r})]&-\bar{\rho}&0&0\\\
-\bar{\rho}&0&0&0\\\ 0&0&\bar{p}r^{2}&0\\\
0&0&0&\bar{p}r^{2}sin^{2}\theta\end{array}\right].$ (26)

The corresponding values of $\bar{\gamma}$, $\bar{\rho}$ and $\bar{p}$ are
($\kappa=1$) [110]:
$\bar{\gamma}=\frac{2\mathcal{M}_{v}}{r^{2}}=\frac{2m_{v}}{r^{2}}+\frac{2\phi_{v}\phi_{vv}}{r}$;
$\bar{\rho}=\frac{2\mathcal{M}_{r}}{r^{2}}=\frac{2m_{r}}{r^{2}}+\frac{\phi_{v}}{r^{2}}$;
and $\bar{p}=-\frac{\mathcal{M}_{rr}}{r}=-\frac{m_{rr}}{r}$. The author [110]
achieved the following energy conditions for the energy-momentum tensor
${\bar{T}}_{\mu\nu}$ combination of Type-I and Type-II fluids, according to
the definition in [133]: (a) the weak and strong energy conditions
$\bar{\gamma}\geq 0\leavevmode\nobreak\ ,\leavevmode\nobreak\ \bar{\rho}\geq
0\leavevmode\nobreak\ ,\leavevmode\nobreak\ \bar{p}\geq 0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (\bar{\gamma}\neq 0)$ and (b) the
dominant energy conditions $\bar{\gamma}\geq 0\leavevmode\nobreak\
,\leavevmode\nobreak\ \bar{\rho}\geq\bar{p}\geq 0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (\bar{\gamma}\neq 0)$ provided that the conditions
imposed on ${\bar{T}}_{\mu\nu}$ as $\bar{\gamma}>0\Rightarrow
m_{v}+r\phi_{v}\phi_{vv}>0$; $\bar{\rho}>0\Rightarrow 2m_{r}+\phi_{v}^{2}>0$;
$\bar{p}>0\Rightarrow m_{rr}<0$.

To explore the phenomenon of gravitational collapse in the framework of the
non-canonical $f(\bar{R},\bar{T})$ theory of gravity, we are now focusing on
the emergent line element (19) with the emergent Vaidya mass function (20).
Additionally, we want to determine the ultimate outcome of the singularity.
Here, we utilize the suitable energy-momentum tensor (21) to solve the
modified field Eq. (17) together with the Eq. (18). In this study, it is
necessary to select a specific form of the function $f(\bar{R},\bar{T})$ [22],
which may be expressed as $f(\bar{R},\bar{T})=f_{1}(\bar{R})+f_{2}(\bar{T})$.
The choice of these functions will vary depending on the specific kinds of
$f_{1}(\bar{R})$ and $f_{2}(\bar{T})$ being considered. Then the modified
field Eq. (17) becomes

$\displaystyle
f_{1}^{\prime}(\bar{R})\bar{R}_{\mu\nu}-\frac{1}{2}f_{1}(\bar{R})\bar{G}_{\mu\nu}+\Big{(}\bar{G}_{\mu\nu}\bar{\square}-D_{\mu}D_{\nu}\Big{)}f_{1}^{\prime}(\bar{R})$
$\displaystyle=\frac{1}{2}\bar{T}_{\mu\nu}+\frac{1}{2}f_{2}(\bar{T})\bar{G}_{\mu\nu}-f_{2}^{\prime}(\bar{T})\Big{[}\bar{T}_{\mu\nu}+\bar{\Theta}_{\mu\nu}\Big{]},$
(27)

where $f_{1}^{\prime}(\bar{R})=\frac{\partial
f(\bar{R},\bar{T})}{\partial\bar{R}}\equiv\frac{\partial
f_{1}(\bar{R})}{\partial\bar{R}}$ and $f_{2}^{\prime}(\bar{T})=\frac{\partial
f(\bar{R},\bar{T})}{\partial\bar{T}}\equiv\frac{\partial
f_{2}(\bar{T})}{\partial\bar{T}}$.

The non-vanishing components of the emergent field Eq. (27) can be found as:
(i) The $(00)$ component is:

$\displaystyle-\bar{\gamma}+\frac{\Big{(}f_{1}(\bar{R})+f_{2}(\bar{T})\Big{)}\Big{(}r-2\mathcal{M}(v,r)\Big{)}}{2r}+\bar{\rho}\Big{(}\frac{2\mathcal{M}(v,r)}{r}-1\Big{)}$
(28) $\displaystyle+$ $\displaystyle
f_{2}^{\prime}(\bar{T})\Bigg{[}\bar{\gamma}+\bar{\rho}-\frac{2\bar{\rho}\mathcal{M}(v,r)}{r}-2\Big{(}\bar{\gamma}+\bar{\rho}-\frac{2\bar{\rho}\mathcal{M}(v,r)}{r}\Big{)}$
$\displaystyle+$
$\displaystyle\frac{\phi_{v}^{4}\Big{(}-5-8\phi_{v}^{2}+4\phi_{v}^{4}\Big{)}}{2\sqrt{1-\phi_{v}^{2}}}+\frac{\Big{(}r-2\mathcal{M}(v,r)\Big{)}\Big{(}-1+\sqrt{1-\phi_{v}^{2}}\Big{)}}{r}\Bigg{]}$
$\displaystyle+f_{1}^{\prime}(\bar{R})\frac{2\dot{\mathcal{M}}(v,r)-(r-2\mathcal{M}(v,r))\mathcal{M}^{\prime\prime}(v,r)}{r^{2}}=0,$

(ii) the $(01)$ component is:

$\displaystyle-r\Big{[}f_{1}(\bar{R})+f_{2}(\bar{T})-2\bar{\rho}\Big{(}1+f_{2}^{\prime}(\bar{T})\Big{)}-2f_{2}^{\prime}(\bar{T})\Big{(}1-\sqrt{1-\phi_{v}^{2}}\Big{)}\Big{]}$
$\displaystyle+2f_{1}^{\prime}(\bar{R})\mathcal{M}^{\prime\prime}(v,r)=0,$
(29)

(iii) and the $(22)$ or $(33)$ component of the field equation is:

$\displaystyle
r^{2}\Big{[}f_{1}(\bar{R})+f_{2}(\bar{T})+2\omega\bar{\rho}-2f_{2}^{\prime}(\bar{T})\Big{(}-1+\bar{\rho}\omega+\sqrt{1-\phi_{v}^{2}}\Big{)}\Big{]}$
$\displaystyle-4f_{1}^{\prime}(\bar{R})\mathcal{M}^{\prime}(v,r)=0.$ (30)

It should be mentioned that the (00) component of the emergent field Eq. (27)
is not only too large it also contains both the derivatives of
$\mathcal{M}(v,r)$ with respect to $v$ and $r$. The solution of this specific
component of the field equation is a challenging task. We will now focus on
the remaining two components of the emergent field Eq. (27), namely the (01)
and (22) or (33) components, designated as Eq.s (29) and (30) respectively.

Now we look into several forms of $f_{1}(\bar{R})$ and $f_{2}(\bar{T})$ in the
following manner:

Case 1: First, we consider the form of $f(\bar{R},\bar{T})$ [22] as

$\displaystyle
f(\bar{R},\bar{T})=f_{1}(\bar{R})+f_{2}(\bar{T})=g_{1}\bar{R}^{\beta_{1}}+g_{2}\bar{T}^{\beta_{2}},$
(31)

where $g_{1},\leavevmode\nobreak\ \beta_{1},\leavevmode\nobreak\
g_{2},\leavevmode\nobreak\ \beta_{2}$ are constants. Note that for this choice
of $f(\bar{R},\bar{T})$ (31), the field Eq. (27) can only be solved for the
choices of $\beta$ as $\beta_{1}=\beta_{2}=1$.

Using the generalized emergent Vaidya line element (19) with the emergent
Vaidya mass function (20) and the emergent energy-momentum tensor (26), we
evaluate the $(22)$ or $(33)$ component of the field equation (30) as:

$\displaystyle
g_{1}\mathcal{M}^{\prime\prime}(v,r)+nr\Big{(}g_{2}(2\omega-1)+\omega\Big{)}\mathcal{M}(v,r)$
$\displaystyle+g_{2}r\Big{(}\sqrt{1-\phi_{v}^{2}}-1\Big{)}=0,$ (32)

where $\omega$ is the EoS parameter given by the relation
$\bar{p}=\omega\bar{\rho}$ and the density can be written in terms of the mass
parameter as $\bar{\rho}=n\mathcal{M}(v,r)$, $n(>0)$ being the particle number
density [22, 134] and
$\mathcal{M}^{\prime}(v,r)=\frac{\partial\mathcal{M}(v,r)}{\partial r}$,
$\mathcal{M}^{\prime\prime}(v,r)=\frac{\partial^{2}\mathcal{M}(v,r)}{\partial
r^{2}}$. Mention that the relation $\bar{p}=\omega\bar{\rho}$ may be used to
get the values of pressure $(\bar{p})$ and energy density $(\bar{\rho})$. The
above Eq. (32) can also be written in a simple form as:

$\displaystyle\mathcal{M}^{\prime\prime}(v,r)+\frac{A}{g_{1}}\mathcal{M}(v,r)r=kr,$
(33)

where $A=n\Big{(}w+g_{2}(2w-1)\Big{)}$ and
$k=\frac{g_{2}}{g_{1}}\Big{(}1-\sqrt{1-\phi_{v}^{2}}\Big{)}$.

The solution of the above differential Eq. (33) is given by

$\displaystyle\mathcal{M}(v,r)=$ $\displaystyle\frac{1}{A}\Big{[}\pi
g_{1}kAiryAi\Big{(}-\frac{Ar}{(-\frac{A}{g_{1}})^{2/3}g_{1}}\Big{)}AiryBi^{\prime}\Big{(}\sqrt[3]{-\frac{A}{g_{1}}}r\Big{)}-\pi
g_{1}kAiryAi^{\prime}\Big{(}\sqrt[3]{-\frac{A}{g_{1}}}r\Big{)}AiryBi\Big{(}-\frac{Ar}{(-\frac{A}{g_{1}})^{2/3}g_{1}}\Big{)}\Big{]}$
(34)
$\displaystyle+c_{1}(v)AiryAi\Big{(}-\frac{Ar}{(-\frac{A}{g_{1}})^{2/3}g_{1}}\Big{)}+c_{2}(v)AiryBi\Big{(}-\frac{Ar}{(-\frac{A}{g_{1}})^{2/3}g_{1}}\Big{)}$

where $AiryAi$, $AiryAi^{\prime}$, $AiryBi$ and $AiryBi^{\prime}$ are the four
Airy functions defined in [22] and the functions $c_{1}(v)$ and $c_{2}(v)$ are
arbitrary functions of time that result from the process of integration. It is
important to acknowledge that the Airy function is a special mathematical
function that has been called in honor of the renowned British astronomer
George Biddell Airy (1801-1892). There are two distinct Airy functions,
denoted as $Ai(x)$ (Airy function of the first kind) and $Bi(x)$ (Airy
function of the second kind). These functions are considered to be linearly
independent solutions to the Airy differential equation, which may be
expressed as $\frac{d^{2}y}{dx^{2}}-xy=0$.

The $(01)$ component (29) of the field Eq. (27) can be expressed as:

$\displaystyle
2g_{1}\Big{(}2\mathcal{M}^{\prime}(v,r)+r\mathcal{M}^{\prime\prime}(v,r)\Big{)}+2r^{2}g_{2}n(\omega-1)\mathcal{M}(v,r)=0$
(35)

and the solution of this Eq. (35) becomes

$\displaystyle\mathcal{M}(v,r)$
$\displaystyle=\frac{1}{r}\Bigg{[}c_{3}(v)AiryAi\Big{(}\frac{r(g_{2}n-g_{2}n\omega)}{g_{2}\Big{(}\frac{g_{2}n-g_{2}n\omega}{g_{1}}\Big{)}^{2/3}}\Big{)}+c_{4}(v)AiryBi\Big{(}\frac{r(g_{2}n-g_{2}n\omega)}{g_{1}\Big{(}\frac{g_{2}n-g_{2}n\omega}{g_{1}}\Big{)}^{2/3}}\Big{)}\Bigg{]}.$
(36)

It is evident that between these two solutions the first solution (34) is much
more general than the second one (36). It has also been verified that for some
particular choices of initial conditions, we can get back Eq. (36) from Eq.
(34). Most importantly the contributions of the K-essence scalar field are
absent in the second solution. Therefore, we have decided to work with the
first solution of the field equation to check the fate of singularity. It
should also be highlighted that for every other option, we solve just Eq. (30)
for the same reasons indicated above.

Case 2:

The form $f(\bar{R},\bar{T})$ in this case is

$\displaystyle
f(\bar{R},\bar{T})=f_{1}(\bar{R})+f_{2}(\bar{T})=g_{1}\bar{R}^{\beta_{1}}+g_{2}e^{\beta_{2}\bar{T}}.$
(37)

The $(22)$ and $(33)$ component of field equation becomes

$\displaystyle-\frac{1}{2}r^{2}\Big{(}g_{1}\Big{(}\frac{4\mathcal{M}^{\prime}(v,r)+2r\mathcal{M}^{\prime\prime}(v,r)}{r^{2}}\Big{)}^{\beta_{1}}+g_{2}e^{2\beta_{2}n(\omega-1)\mathcal{M}(v,r)}\Big{)}+2\beta_{1}g_{1}\mathcal{M}^{\prime}(v,r)\Big{(}\frac{4\mathcal{M}^{\prime}(v,r)+2r\mathcal{M}^{\prime\prime}(v,r)}{r^{2}}\Big{)}^{\beta_{1}-1}$
$\displaystyle-\beta_{2}g_{2}r^{2}e^{2\beta_{2}n(\omega-1)\mathcal{M}(v,r)}\Big{(}n\omega\mathcal{M}(v,r)+\sqrt{1-\phi_{v}^{2}}-1\Big{)}-nr^{2}\omega\mathcal{M}(v,r)=0.$
(38)

The solution for $\mathcal{M}(v,r)$ can be achieved for the condition
$\beta_{1}=1$ and $\beta_{2}=0$. Hence, we get

$\displaystyle\mathcal{M}(v,r)=$
$\displaystyle\frac{1}{2n\omega}\Bigg{[}AiryAi\Big{(}r\sqrt[3]{-\frac{n\omega}{g_{1}}}\Big{)}\Big{(}-\pi
g_{2}AiryBi^{\prime}\Big{(}r\sqrt[3]{-\frac{n\omega}{g_{1}}}\Big{)}+2d_{1}(v)n\omega\Big{)}+\pi
g_{2}AiryAi^{\prime}\Big{(}r\sqrt[3]{-\frac{n\omega}{g_{1}}}\Big{)}AiryBi\Big{(}r\sqrt[3]{-\frac{n\omega}{g_{1}}}\Big{)}$
(39) $\displaystyle+2d_{2}(v)n\omega
AiryBi\Big{(}r\sqrt[3]{-\frac{n\omega}{g_{1}}}\Big{)}\Bigg{]}.$

In this specific case, the mass function $\mathcal{M}(v,r)$ exhibits also
dependence on Airy functions as well as arbitrary functions of time, namely
$d_{1}(v)$ and $d_{2}(v)$, which arise through the process of integration.

Case 3:

In this case we take the function $f(\bar{R},\bar{T})$ as

$\displaystyle
f(\bar{R},\bar{T})=f_{1}(\bar{R})+f_{2}(\bar{T})=g_{1}e^{\beta_{1}\bar{R}}+g_{2}e^{\beta_{2}\bar{T}}$
(40)

where $g_{1},\leavevmode\nobreak\ g_{2},\leavevmode\nobreak\
\beta_{1},\leavevmode\nobreak\ \&\leavevmode\nobreak\ \beta_{2}$ are
constants. This special type of choice is often known as, the double-
exponential (DE) model. The $(22)$ and $(33)$ component of the field equation
(30) becomes

$\displaystyle-$
$\displaystyle\frac{1}{2}r^{2}\Big{[}g_{1}e^{\frac{1}{r^{2}}\Big{(}2\beta_{1}(2\mathcal{M}^{\prime}(v,r)+r\mathcal{M}^{\prime\prime}(v,r))\Big{)}}+g_{2}\Big{(}2\beta_{2}(\sqrt{1-\phi_{v}^{2}}-1)+1\Big{)}e^{2\beta_{2}n(\omega-1)\mathcal{M}(v,r)}\Big{]}$
(41) $\displaystyle+$ $\displaystyle
2\beta_{1}g_{1}\mathcal{M}^{\prime}(v,r)\epsilon^{\frac{1}{r^{2}}\Big{(}2\beta_{1}(2\mathcal{M}^{\prime}(v,r)+r\mathcal{M}^{\prime\prime}(v,r))\Big{)}}-nr^{2}\omega\mathcal{M}(v,r)\Big{(}\beta_{2}g_{2}e^{2\beta_{2}n(\omega-1)\mathcal{M}(v,r)}+1\Big{)}=0.$

We have exponential terms that include the first and second-order derivatives
of $\mathcal{M}(v,r)$. So rather than finding a general solution, we look for
an approximate solution taking the consideration $\beta_{1}=0$ and expanding
the second and fourth exponential in Taylor’s series. Other options for the
aforementioned constants, which are easily verified, cannot be solvable. Thus
we obtain the solution of $\mathcal{M}(v,r)$ as

$\displaystyle\mathcal{M}(v,r)=\frac{g_{1}\Big{[}-1+2\beta_{2}(1+\sqrt{1-\phi_{v}^{2}})\Big{]}+g_{2}\Big{[}-1+2\beta_{2}(1-2n+\sqrt{1-\phi_{v}^{2}})\Big{]}}{2g_{2}\beta_{2}n(\omega-1)\Big{[}1+\beta_{2}\Big{(}n-2(1+\sqrt{1-\phi_{v}^{2}})\Big{)}\Big{]}}.$
(42)

The solution of $\mathcal{M}(v,r)$ does not depend on $r$ anymore. However,
there is an explicit dependence on $v$ through $\phi_{v}^{2}$, the values of
this kinetic part of the K-essence field range between $0$ and $1$.
Consequently, the collapse is not feasible in this particular case, as we
shall elaborate on in the next section.

Case 4: In this case we choose the function $f(\bar{R},\bar{T})$ as

$\displaystyle
f(\bar{R},\bar{T})=f_{1}(\bar{R})+f_{2}(\bar{T})=g_{1}e^{\beta_{1}\bar{R}}+g_{2}\bar{T}^{\beta_{2}}.$
(43)

The $(22)$ or $(33)$ component of the field Eq. (30) is:

$\displaystyle-\frac{1}{2}r^{2}\Big{(}g_{1}\exp\Big{(}\frac{2\beta_{1}\Big{(}2\mathcal{M}^{\prime}(v,r)+r\mathcal{M}^{\prime\prime}(v,r)\Big{)}}{r^{2}}\Big{)}+2^{\beta_{2}}g_{2}(n(\omega-1)\mathcal{M}(v,r))^{\beta_{2}}\Big{)}$
$\displaystyle+2\beta_{1}g_{1}\mathcal{M}^{\prime}(v,r)\exp\Big{(}\frac{2\beta_{1}\Big{(}2\mathcal{M}^{\prime}(v,r)+r\mathcal{M}^{\prime\prime}(v,r)\Big{)}}{r^{2}}\Big{)}-2^{\beta_{2}-1}\beta_{2}g_{2}r^{2}(n(\omega-1)\mathcal{M}(v,r))^{\beta_{2}-1}\Big{(}n\omega\mathcal{M}(v,r)+\sqrt{1-\phi_{v}^{2}}-1\Big{)}$
$\displaystyle-nr^{2}\omega\mathcal{M}(v,r).$ (44)

The solvability of $\mathcal{M}(v,r)$ is determined by the choice of the
constants $\beta_{1}=0$ and $\beta_{2}=1$. It should be noted that for other
choices of constants there do not yield solvable solutions.

Now

$\displaystyle\mathcal{M}(v,r)=\frac{-2g_{2}\sqrt{1-\phi_{v}^{2}}-g_{1}+2g_{2}}{2n\Big{(}2g_{2}\omega-
g_{2}+\omega\Big{)}}.$ (45)

The result bears similarity to case-3, as the mass function $M(v,r)$ remains
unaffected by the radial component but is dependent upon time due to the
presence of the kinetic portion of the K-essence scalar field.

##  IV Gravitational Collapse

To study the type of singularity formed in our case we follow the approach of
[22, 20, 23, 131, 110, 111]. The authors [22, 135, 136], studied gravitational
collapse through the study of the radial null geodesic emerging from the
singularity in the case of usual $f(R,T)$ gravity. In this analysis, we will
look at a system undergoing spherical collapse. Specifically, we will focus on
the physical radius, denoted as $\mathcal{R}(v,r)$, of the $r-th$ shell of the
star at time $v$. An appropriate starting condition can be defined as follows:
at $v=0$, $\mathcal{R}(0,r)=r$. It is evident that in the case of an
inhomogeneous collapse, distinct collapsing shells may reach singularity at
varying points in time. Our focus lies on the light particles, known as
photons, that emanate from the singularity and traverse through the geodesics,
ultimately reaching an observer situated outside the singularity. The presence
of an event horizon poses a barrier to the passage of photons, hence
preventing their arrival to the observer. In this study, we will investigate
the presence of outgoing non-spacelike geodesics. Theoretically, if such
geodesics possess well-defined tangent at the singularity, the quantity
$\frac{d\mathcal{R}}{dr}$ will definitely tend towards a finite limit with the
geodesics approaching the singularity in the past following the trajectories.
At the instances when these trajectories intersect the locations
$(v_{0},r)=(v_{0},0)$, an extensive breakdown in mathematical and physical
principles takes place, leading to the emergence of a singularity at
$\mathcal{R}(v_{0},0)=0$. Ideally, at these specific locations, the matter
shells undergo a process of collapse, ultimately reaching a radius of zero.
This collapse leads to the emergence of a central singularity. This is a very
compact object because a large quantity of mass is packed into a very small
volume. The equation for outgoing radial null geodesic from the emergent
Vaidya metric (19) taking $dS^{2}=0$ and $d\Omega^{2}=0$ as (using Eq. (20))
[110]

$\displaystyle\frac{dv}{dr}=\frac{2}{\Big{(}1-\frac{2\mathcal{M}(v,r)}{r}\Big{)}}=\frac{2}{\Big{(}1-\frac{2m(v,r)}{r}+\phi_{v}^{2}\Big{)}}.$
(46)

Clearly, the above equation has a singularity at $r\rightarrow 0$,
$v\rightarrow 0$. So, we must study limiting behavior as we approach
singularity. To do this let us introduce a parameter, $Y=v/r$. Now we will
examine the nature of $Y$ at the limiting point, $r\rightarrow 0$,
$v\rightarrow 0$. Let us consider at $r\rightarrow 0$, $v\rightarrow 0$ the
value of $Y$ is $Y_{0}$. Then we can write

$\displaystyle Y_{0}$ $\displaystyle=$
$\displaystyle\lim\limits_{\begin{subarray}{c}v\to 0\\\ r\to
0\end{subarray}}Y=\lim\limits_{\begin{subarray}{c}v\to 0\\\ r\to
0\end{subarray}}\frac{v}{r}=\lim\limits_{\begin{subarray}{c}v\to 0\\\ r\to
0\end{subarray}}\frac{dv}{dr}=\lim\limits_{\begin{subarray}{c}v\to 0\\\ r\to
0\end{subarray}}\frac{2}{1-(\frac{2\mathcal{M}(v,r)}{r})}$ (47)
$\displaystyle=$ $\displaystyle\lim\limits_{\begin{subarray}{c}v\to 0\\\ r\to
0\end{subarray}}\frac{2}{\Big{(}1-\frac{2m(v,r)}{r}+\phi_{v}^{2}\Big{)}}.$

The above-limiting condition gives rise to a quadratic equation of $Y_{0}$
[22, 110, 131]. The solution of the quadratic equation represents the slope of
the tangent to the null geodesics at the singularity. As we are interested in
the realistic gravitational collapse, we will consider the real roots of the
equation. The positivity of the root indicates the nakedness of the
singularity whereas, the negativity of the roots represents the formation of
the Black Hole in the future [23, 110]. We know that a ray released from the
singularity would reach the outside observer if a single null geodesic in the
$(v,r)$ plane escapes it and the observer would be able to observe the
singularity, which makes the singularity locally naked for an instant of time.
Observing the singularity by the observer such that the observer extracts some
information is necessary for a bunch of geodesics to escape the $(v,r)$ plane.
In other words, this phenomenon excludes the existence of an event horizon.
The escaping of the family of null geodesic makes a globally naked singularity
[11, 12, 13]. The comprehensive explanation and interpretation of $Y_{0}$,
together with the associated emergent mass function (20), its derivatives, and
the K-essence scalar field, as well as its derivatives at the central
singularity $(r\rightarrow 0,\leavevmode\nobreak\ v\rightarrow 0)$, may be
found in the Ref. [110]. It is essential to keep in mind that in this
particular section, we undertake an investigation of the characteristics of
various mass functions. These functions are obtained from the various choices
of $f(\bar{R},\bar{T})$ within the framework of K-essence geometry,
specifically in connection to the emergent Vaidya spacetime.

Case 1: Following [22], using Eq. (34) in Eq. (47) at the central singularity
and considering the value of $AiryAi(0)$, $AiryBi(0)$, $AiryAi^{\prime}(0)$
and $AiryBi^{\prime}(0)$ we get the equation of $Y_{0}$ after some algebraic
calculations

$\displaystyle
Y_{0}^{2}\Bigg{[}\frac{\xi_{1}}{3^{2/3}\Gamma(2/3)}+\frac{\xi_{2}}{3^{1/6}\Gamma(2/3)}\Bigg{]}-Y_{0}\Bigg{[}\frac{\phi_{v}^{4}}{4}+\frac{1}{2}\Bigg{]}+1=0,$
(48)

where the integration constants $c_{1}(v)$ and $c_{2}(v)$ in Eq.(34) are
chosen as $c_{1}(v)=\xi_{1}v$ and $c_{2}(v)=\xi_{2}v$. The values of Airy
function $(Ai(x),\leavevmode\nobreak\ Bi(x))$ and its derivatives
$(Ai^{\prime}(x),\leavevmode\nobreak\ Bi^{\prime}(x))$ at $x=0$ are given by,
$Ai(0)=\frac{1}{3^{2/3}\Gamma(2/3)}$, $Bi(0)=\frac{1}{3^{1/6}\Gamma(2/3)}$,
$Ai^{\prime}(0)=-\frac{1}{3^{1/3}\Gamma(1/3)}$ and
$Bi^{\prime}(0)=\frac{3^{1/6}}{\Gamma(1/3)}$. Here, $\Gamma$ denotes the usual
gamma function. The solutions of the above quadratic Eq. (48) can be written
as:

$\displaystyle Y_{0{\pm}}=\frac{1}{8\leavevmode\nobreak\
3^{1/3}(\xi_{1}+\sqrt{3}\xi_{2})}\Bigg{[}3\Gamma(\frac{2}{3})(2+\phi_{v}^{2})\pm\sqrt{\Gamma(-\frac{1}{3})\Big{(}64\leavevmode\nobreak\
3^{1/3}(\xi_{1}+\sqrt{3}\xi_{2})+\Gamma(-\frac{1}{3})(2+\phi_{v}^{2})^{2}\Big{)}}\Bigg{]}.$
(49)

Here we denote $Y_{0+}$ for the positive root and $Y_{0-}$ for the negative
root. As we have mentioned earlier we are interested in real solutions only,
for this, the condition is

$\displaystyle\Gamma(-\frac{1}{3})\Big{(}64\leavevmode\nobreak\
3^{1/3}(\xi_{1}+\sqrt{3}\xi_{2})+\Gamma(-\frac{1}{3})(2+\phi_{v}^{2})^{2}\Big{)}\geq
0$ (50)

must hold true.

Now to achieve a locally naked singularity we must have one of the solutions
to be positive but for globally naked singularity we should have both the
solutions to be positive, i.e., $Y_{0\pm}>0$. The condition in terms of
$\phi_{v}^{2}$ can be written as

$\displaystyle(2+\phi_{v}^{2})^{2}\geq-64\leavevmode\nobreak\
\frac{3^{1/3}}{\Gamma(-\frac{1}{3})}\Big{(}\xi_{1}+\sqrt{3}\xi_{2}\Big{)}.$
(51)

Let us now check the occurrence of naked singularity through graphical
analysis. As we know the value of $\phi_{v}^{2}$ should lie in between
$0\leavevmode\nobreak\ \&\leavevmode\nobreak\ 1$ we vary it accordingly. The
arbitrary constants $\xi_{1}\leavevmode\nobreak\ \&\leavevmode\nobreak\
\xi_{2}$ can have both positive and negative values. So, we plot two graphs.
First, we consider $\xi_{2}=1$, i.e., a positive value and then vary $\xi_{1}$
from negative to positive value (Fig. 1(a). Second, we chose $\xi_{2}=-1$,
i.e., a negative value and again vary $\xi_{1}$ from positive to negative
value (Fig. 1(b). Next, we interchange the position of
$\xi_{1}\leavevmode\nobreak\ \&\leavevmode\nobreak\ \xi_{2}$ and plot two more
graphs (Fig. 2(a) and Figure 2(b)). Based on the aforementioned graphs, it is
evident that the value of $Y_{0}$ consistently exhibits positivity, hence
indicating the presence of a global naked singularity. In Case 1, the presence
of a naked singularity is seen instead of a black hole, which is mostly
explained by the gravitational standpoint rather than being influenced by
K-essence as a dark energy theory.

(a) (Color Online) $\xi_{2}=1$

(b) (Color Online) $\xi_{2}=-1$

Figure 1: Variation of $Y_{0}$ with $\phi_{v}^{2}$ with $\xi_{2}=$ constant
and $\xi_{1}$ varying

(a) (Color Online) $\xi_{1}=1$

(b) (Color Online) $\xi_{1}=-1$

Figure 2: Variation of $Y_{0}$ with $\phi_{v}^{2}$ with $\xi_{1}=$ constant
and $\xi_{2}$ varying

Case 2:

It has been noted that, in the limiting case, i.e., at $v\rightarrow 0$ and
$r\rightarrow 0$ the value of $Y_{0\pm}$ becomes exactly the same as in case
1. We would get similar conditions (48) at the central singularity. The
condition for the real solution of $Y_{0}$ becomes the same as mentioned in
(51). Clearly, this is because of the choice of the constants $\beta_{1}$ and
$\beta_{2}$. As a consequence, in this case, also we get a global naked
singularity rather than a black hole.

Case 3:

Following the similar approach of Case 1, we compute the value of $Y(=v/r)$ at
$r\rightarrow 0,v\rightarrow 0$. In this case, we get

$\displaystyle\frac{2}{Y_{0}}=\lim\limits_{\begin{subarray}{c}v\to 0\\\ r\to
0\end{subarray}}\Bigg{(}1-\frac{2g_{1}\Big{[}-1+2\beta_{2}(1+\sqrt{1-\phi_{v}^{2}})\Big{]}+g_{2}\Big{[}-1+2\beta_{2}(1-2n+\sqrt{1-\phi_{v}^{2}})\Big{]}}{2rg_{2}\beta_{2}n(\omega-1)\Big{[}1+\beta_{2}\Big{(}n-2(1+\sqrt{1-\phi_{v}^{2}})\Big{)}\Big{]}}\Bigg{)}.$
(52)

The limit of the expression $\frac{2}{Y_{0}}$ diverges as $r$ approaches zero.
To obtain an accurate solution for the aforementioned equation, it is
necessary for the mass function to explicitly depend on both $v$ and $r$.
However, in this particular scenario, this condition is not satisfied. Thus
far, we have not attained any instances of a collapsing situation. However,
variation of a mass parameter may be seen. Based on the variation of the mass
parameter with $\phi_{v}^{2}$ and time ($v$), we may conclude that it is
possible to encounter either a bouncing or accelerating scenario, as stated
below.

Case 4:

Similar to case 3 here we can not study the gravitational collapse as the mass
parameter (45) does not depend on $r$. But the variation of a mass parameter
can be observed here too.

Now, using graphical analysis we shall discuss about the variations in the
mass parameters Eqs. (42) and (45) for Cases 3 and 4. Basically, Cases 3 and 4
present fascinating scenarios which can be investigated as follows. There is a
clear dependence of $\phi_{v}^{2}$ on the mass parameter $\mathcal{M}(v,r)$ in
(42) and (45). When the mass parameter is plotted against $\phi_{v}^{2}$ for
various values of $\omega$, the resulting graphs are denoted as Figs. 3(a) and
4(a) for instances Cases 3 and 4 respectively. We can observe that for every
EOS parameter ($\omega$), the mass function reduces as $\phi_{v}^{2}$
increases, and at a particular value of $\phi_{v}^{2}\leavevmode\nobreak\
(=0.75)$, the mass function coincides with zero in two cases and varies again.
It is important to note that the values of the kinetic energy of the K-essence
scalar field ($\phi_{v}^{2}$) are $0<\phi_{v}^{2}<1$. The above feature can be
explained as follows:

(a) (Color Online) $\mathcal{M}(v,r)$ vs $\phi_{v}^{2}$

(b) (Color Online) $\mathcal{M}(v,r)$ vs $v/v_{0}$

Figure 3: Variation of mass parameter (Case 3: $\mathcal{M}(v,r)$) for
different EOS parameter ($\omega$)

(a) (Color Online) $\mathcal{M}(v,r)$ vs $\phi_{v}^{2}$

(b) (Color Online) $\mathcal{M}(v,r)$ vs $v/v_{0}$

Figure 4: Variation of mass parameter (Case 4: $\mathcal{M}(v,r)$) for
different EoS parameter ($\omega$)

The K-essence idea is also used in the study of the impact of dark energy
throughout the cosmos [92, 88, 146]. Furthermore, it has been shown by several
observations, such as those conducted by WMAP, Planck collaborations, and IA
Supernovae, etc. [24, 25, 26, 27, 28, 29], that the estimated present value of
dark energy density is around $0.74$. The kinetic energy of the K-essence
scalar field ($\phi_{v}^{2}$) can be employed as a representation of dark
energy density in the unit of critical density [104, 105, 106, 107, 108].

So, at this point ($\phi_{v}^{2}=0.75$), the mass has been completely
transformed into energy, which has caused spacetime to change into Minkowski
spacetime. As a result, it is possible to argue that the universe is dominated
by dark energy and that spacetime is of the Minkowski type at the point
$0.75$. This phenomenon occurs exclusively when observing the cosmos from an
asymptotically flat perspective. It is evident that we are now studying the
emergent Vaidya spacetime as our fundamental metric, which possesses a
dynamical horizon [137, 138, 139, 140], detail explained by Manna et al. [109,
110, 113]. This dynamical horizon comes through the mass function
$\mathcal{M}(v,r)$ affected by the kinetic energy of the K-essence scalar
field ($\phi_{v}^{2}$). The existence of dark energy implies that the
expansion of the universe is accelerating, as opposed to undergoing a
collapse.

Once again, there is a constraint on the value of $\phi_{v}^{2}$ where
$0<\phi_{v}^{2}<1$. The variable $v$ may be represented as
$\phi_{v}^{2}=e^{-v/v_{0}}\theta(v)$, as stated in the Ref. [109]. When the
mass parameter is plotted against the ratio of time ($v$) to the reference
time ($v_{0}$), we find Figs. 3(b) and 4(b) for Cases 3 and 4, respectively.
The constant $v_{0}$ is chosen to parameterize the unit of $v$. Additionally,
it is stated that Figs. 4(a) and 4(b) illustrate the variations in
$\mathcal{M}(v,r)$ as a function of $\phi_{v}^{2}$ and $v/v_{0}$ respectively,
assuming that $g_{1},\leavevmode\nobreak\ g_{2},\leavevmode\nobreak\ n$ are
set to a value of unity for the sake of simplicity. The variation of the mass
parameters with respect to time ($v$) in units of $v_{0}$ may be observed from
Figs. 3(a) and 4(b). Based on the analysis of Figs. 3(b) and 4(b), it is
observed that the mass parameter undergoes a growth, either positive or
negative, depending on the values of the EoS parameter $\omega$, subsequent to
the reaching the value $v/v_{0}=0.287682$.

Based on the aforementioned graphs, specifically Figs. 3(a), 4(a), 3(b) and
4(b), we generate two tables, denoted as Tables 1 and 2. Table 1 presents an
overview of the variations in the mass parameters $\mathcal{M}(v,r)$ under
different values of $\omega$ for Cases 3 and 4, specifically at
$\phi_{v}^{2}=0.5$ and $\phi_{v}^{2}=0.9$. Table 2 presents the mass
parameters $\mathcal{M}(v,r)$ for various values of $\omega$ in Cases 3 and 4,
specifically when $\frac{v}{v_{0}}=2$.

$\omega$ |  Epoch | $\mathcal{M}(v,r)$
---|---|---
$\phi_{v}^{2}=0.5$ | $\phi_{v}^{2}=0.9$
Case 3 | Case 4 | Case 3 | Case 4
-2 | Dark Energy era | 0.0976 | 0.0296 | -0.1937 | -0.0262
-1.5 | Dark Energy era | 0.1171 | 0.0376 | -0.2324 | -0.0334
-1.0 | Dark Energy era | 0.1464 | 0.0518 | -0.2905 | -0.0459
-0.5 | Dark Energy era | 0.1953 | 0.0828 | -0.3874 | -0.0735
0 | Dust era | 0.2929 | 0.2071 | -0.5811 | -0.1838
0.5 | Early Universe | 0.5858 | -0.4142 | -1.16223 | 0.3675
1.0 | Stiff Fluid | Diverging | -0.1035 | Diverging | 0.0919
1.5 |  | -0.5858 | -0.0592 | 1.1623 | 0.0525

Table 1: Table of $\mathcal{M}(v,r)$ for different $\omega$ for Cases 3 and 4
at $\phi_{v}^{2}=0.5$ and $\phi_{v}^{2}=0.9$.

$\omega$ |  Epoch | $\mathcal{M}(v,r)$
---|---|---
$\frac{v}{v_{0}}=2$
Case 3 | Case 4
-2 | Dark Energy era | 0.1541 | 0.0614
-1.5 | Dark Energy era | 0.1849 | 0.0781
-1.0 | Dark Energy era | 0.2311 | 0.1075
-0.5 | Dark Energy era | 0.3082 | 0.1719
0 | Dust era | 0.4622 | 0.4299
0.5 | Early Universe | 0.9246 | -0.8597
1.0 | Stiff Fluid | Diverging | -0.2149
1.5 |  | -0.9246 | -0.1228

Table 2: Table of $\mathcal{M}(v,r)$ for different $\omega$ for Cases 3 and 4
at $\frac{v}{v_{0}}=2$.

According to the data shown in Table 1, it is evident that all the mass
parameters exhibit positive values until the dust period, specifically at
$\phi_{v}^{2}=0.5$. After that for Case 4, it is seen all values of the mass
parameter are negative and for Case 3 at $\omega=0.5$, the mass parameter is
positive, at $\omega=1.0$, it is diverging, and at $\omega=1.5$, it is again
negative. On the other hand, when $\phi_{v}^{2}=0.9$, the majority of the mass
parameter values exhibit negativity in both scenarios, but a subset of values
demonstrate positivity. In both scenarios, namely before and subsequent to the
current value of the dark energy density ($\phi_{v}^{2}\leavevmode\nobreak\
(=0.74)$), some mass parameters exhibit negativity. The occurrence of
unforeseen properties in our scenarios can be ascribed to the presence of
negative masses, which may indicate the probable existence of a gravitational
dipole [143, 144] within our universe, with signs of future expansion. In this
discourse, we explore the concepts of positive and negative mass in
contemporary contexts.

The concept of negative mass has been explored in several scholarly
publications [141, 144, 142, 143, 145] within various contexts. It is pointed
out that Bondi and Bonnor [143, 145] in their articles elaborately discussed
the existence of negative mass along with the positive mass. In his work,
Miller [144] provides a description of negative-mass lagging cores within the
context of the big bang. The author stated that for very high values of the
geometrically specified coordinate $r$, the mass is positively valued,
whereas, at considerably small values of $r$, the mass is negatively valued.
The area of spacetime characterized by negative mass has local features that
resemble those observed in the negative-mass Vaidya solution. The author may
endeavor to establish a nonsingular past for the given hypersurface. Consider
a possible scenario in which we have a celestial object composed exclusively
of particles that undergo decay, resulting in the emission of photons that
subsequently dissipate. The description of the spacetime outside the
contracting star is effectively explained by a Vaidya solution, whereby the
relationship between the mass and the outgoing null coordinate $v$ is
established based on the rate of particle disintegration.

In his article, the occurrence of the big bang is characterized by a
combination of spacelike and timelike properties, resulting in spacetimes that
do not adhere to the stringent version of the cosmic censorship hypothesis. In
the context of general-relativistic cosmology, it is explicitly acknowledged
that the big bang might potentially possess both spacelike and timelike
characteristics. This implies that a portion of the big bang event may have
been influenced by factors originating inside spacetime itself. In these basic
spherically symmetric models, the timelike section of the big bang is
characterized by a negative mass and serves as an inexhaustible reservoir of
energy, manifesting as gravitational and electromagnetic radiation. If compact
and numerous sources of energy were to exist, they may potentially contribute
to the understanding of several cosmic events characterized by high levels of
energy. On the other hand, in his article [144], in the negative-mass
Schwarzschild solution, it can be observed that the singularity located at
$r=0$ is intersected by both the future and past light cones at any given
position. This implies that the singularity possesses a timelike nature.
Therefore, it may be concluded that spacetime does not adhere to the stringent
version of the cosmic censorship theory, which posits the absence of timelike
singularities, whether they originated from primordial conditions or emerged
from initially non-singular circumstances. Bondi [143] has identified a
notable phenomenon known as the “gravitational dipole,” which is defined by
the presence of both a positive mass and a negative mass. Positive mass
exhibits attractive behavior towards negative mass, whereas negative mass
demonstrates repulsive behavior towards the positive mass. Consequently, both
masses experience acceleration in a corresponding direction, resulting in the
negative mass pursuing the positive mass, leading to the attainment of
exceptionally high velocities by the dipole. Also, it should be noted that in
our generalized emergent Vaidya spacetime (19), the removal of the
$\phi_{v}^{2}$ term and the substitution $\mathcal{M}(v,r)\equiv m(v)$ as well
as not considering $f(\bar{R},\bar{T})$ gravity, which corresponds to the
conventional Vaidya spacetime, leads us back to the investigation conducted by
Miller in his published work [144], which established the existence of both
the positive and negative masses.

Table 2 also presents the positive and negative aspects of the mass parameter,
contingent upon the values of the EOS parameter. This observation further
suggests the presence of dark energy and gravitational dipole across the many
epochs of the cosmos, as shown by the presence of both negative and positive
mass parameters, which are contingent upon the specific value of the EOS
parameter. It should be also noted that a negative value of the EOS parameter
indicates a world that is predominantly dominated by dark energy. Typically,
the EOS parameter value for a dark energy-dominated universe is considered as
-1 and a larger negative value signifies a greater dominance of dark energy in
the universe. The EOS parameter zero is often known as the dust or matter-
dominated epoch. A value of $0.5$ indicates scenarios pertaining to the early
universe, whereas a value of $1.0$ corresponds to the stiff fluid era. The
presence of a negative mass parameter, coupled with a positive EOS parameter,
suggests the possibility of the occurrence of negative mass during the early
stages of the cosmos. On the other hand, the appearance of a gravitational
dipole and dark energy characterized by negative EoS parameters demonstrate
the phenomenon of late-time acceleration in the cosmos. According to Bondi
[143], it can be asserted that in our particular model, the gravitational
dipole is also present. This implies that in our model, both negative and
positive masses simultaneously exist, given the specific choices of
$f(\bar{R},\bar{T})$ as outlined in Eqs. (40) and (43), subject to certain
fixed constants. So, we can say that under the concept of a gravitational
dipole, there exists a dark energy-dominated universe, implying that the world
is accelerating.

##  V Conclusion

In this section, we provide an overview of our research findings, which may be
summarised as follows: The emergent $f(\bar{R},\bar{T})$ gravity in the
K-essence geometry was initially taken into consideration. The Lagrangian
employed in this study is a non-canonical Lagrangian of the DBI type, as shown
by Eq. (8). Additionally, the modified field equation associated with this
theory, represented by Eq. (17), corresponds to the emergent energy-momentum
tensor ($\bar{T}_{\mu\nu}$) as indicated in Eq. (14). This part has been
summarised in section II. On the other hand, we have also taken into account
the emergent Vaidya spacetime as our primary metric (19), while specifying the
mass function in (20) and the energy-momentum tensor in (26). In accordance
with the reference [110] , several matter-dependent parameters, namely
$\bar{\gamma}$, $\bar{\rho}$, $\bar{p}$, have been introduced. These
parameters correspond to the energy density associated with the emergent
Vaidya null radiation, as well as the energy density and pressure of the
emergent perfect fluid, respectively. Note that the mass parameter in this
context has been considered as a function of $r$ and $v$, where $v$ represents
the null coordinate associated with the Eddington advanced time. The value of
$r$ decreases towards the future along a ray that is associated with a
constant value of $v$. The temporal variation of the emergent mass function
($\mathcal{M}(v,r)$) arises from the generalized Vaidya mass ($m(v,r)$) in the
background, as well as the kinetic energy associated with the K-essence scalar
field ($\phi_{v}^{2}$). The functional form of $f(\bar{R},\bar{T})$ is seen as
an additive composition of two functions, namely $f_{1}(\bar{R})$ which only
relies on the Ricci scalar ($\bar{R}$), and $f_{2}(\bar{T})$ which solely
relies on the trace of the energy-momentum tensor ($\bar{T}$). By referencing
the field equations in (28), (29), and (30), we go to the subsequent stage of
our study. In our analysis, we focus on the $(22)$ or $(33)$ component of the
field equation in order to create a comprehensive model.

In the subsequent section, section III, we analyze four models of the
$f(\bar{R},\bar{T})$ theory and determine the solution for the generalized
mass parameter, denoted as $\mathcal{M}(v,r)$, for each of the above cases:
Case 1 (Eq. 34), Case 2 (Eq. 39), Case 3 (Eq. 42) and Case 4 (Eq. 45). The
solutions for Cases 1 and 2 give identical findings given the specified
conditions on $\beta_{1}$, $\beta_{2}$, $g_{1}$, and $g_{2}$. Whereas the
solutions in Cases 3 and 4 exhibit similarities under certain particular
circumstances on the parameters. In our proposed model, the mass functions
exhibit an additional K-essence-related component, denoted as
$(\phi_{v}^{2})$, which distinguishes it from the general case. This unique
characteristic encourages us to investigate the remarkable behavior of the
mass function, as discussed in Section IV.

In Section IV we study the underlying physics of the mass parameters obtained
in the previous section. The mass functions in Cases 1 and 2 have an explicit
dependency on $r$, allowing us to study the gravitational collapse and the
fate of singularity. The limiting conditions on the tangent of the radial null
geodesic (47) tell us that the singularity formed in our case is naked for any
value of $\phi_{v}^{2}$. Figures (1(a)), (1(b)), (2(a)) and (2(b)) supports
the statement. In this procedure, we check all the possible values of the
constants $\xi_{1}$ and $\xi_{2}$. On the way, we have established the
condition (51) for which the solution of Eq. (49) is real.

The study of the mass function pertaining to Cases 3 and 4 yields surprising
results. Initially, it should be noted that the solutions of the mass function
do not explicit dependence on the variables $r$ or $v$. Consequently, these
solutions are unable to offer any information about the singularity. However,
their dependence relies on the K-essence term $\phi_{v}^{2}$, which exhibits a
temporal dependency on the variable $v$. In the aforementioned scenarios, the
mass parameters exhibit a non-constant nature, since they are implicitly
dependent on time due to the presence of the kinetic energy term
($\phi_{v}^{2}$) associated with the K-essence scalar field. When we plot the
mass function with $\phi_{v}^{2}$ for different $\omega$ we get that in each
case the mass function reaches to zero at a particular value of
$\phi_{v}^{2}$, which is $0.75$ (3(a) $\&$ 4(a)). Now according to the
observation, we know that the present value of dark energy is about $0.74$.
That means the K-essence term $\phi_{v}^{2}$ can be considered as the dark
energy component of the universe [104, 105, 106, 107, 108]. The mass function
vanishes at the point $0.75$. This is a clear indication that the mass has
been converted into energy at this point indicating that the spacetime we are
now is Minkowski. We should remember the fact that we started with a
generalized Vaidya spacetime in the context of K-essence geometry.
Surprisingly it is seen that this phenomenon occurs within the context of flat
spacetime in the current era of our universe, specifically for a certain value
of $\phi_{v}^{2}$. Again the condition that makes the spacetime (19) physical
is that $\phi_{v}^{2}$ can have a value between $0$ and $1$. This enables us
to consider it as an exponential function of $v/v_{0}$
($\phi_{v}^{2}=e^{-v/v_{0}}\theta(v)$), which provides us with the scope to
study the variation of mass parameter with respect to the normalized time
coordinate $v/v_{0}$ (Figs. 3(b) and 4(b)). The zero point of mass function
arises at around $0.287682$ in the $v/v_{0}$ axis intending the similar
behavior of the mass function as achieved with $\phi_{v}^{2}$. In these
figures, we see the mass function sometimes becomes positive and sometimes
negative depending upon the value of $\omega$ we choose. Two tables (1) and
(2) have been arranged to study the behavior of the mass function in these two
cases namely, Cases 3 and 4. In conclusion, it is important to reiterate that
the presence of a positive and negative mass function in this study may
potentially signify the existence of a gravitational dipole across various
stages of the universe. Each curve on these graphs encompasses both positive
and negative values of the mass function, either before or succeeding the zero
value. The potential existence of this gravitational dipole might perhaps
account for the phenomenon of early inflation and the subsequent accelerated
expansion of the cosmos.

It is noteworthy to emphasize that the K-essence theory possesses the
capability to be utilized in a complete way, enabling the concurrent
investigation of both dark energies [92, 88, 104, 105, 106, 146] and a purely
gravitational perspective [109, 110, 112, 113]. The findings supporting the
existence of dark energy, a Minkowski universe for a fixed point, and the
gravitational dipole in our study agree well with the postulated theories.
Additionally, our research provides evidence for the existence of a global
naked singularity by examining gravitational collapse just from a
gravitational perspective, without considering the dark energy density
represented by $\phi_{v}^{2}$. Therefore, it can be concluded that the
K-essence theory possesses the potential to serve as both a dark energy
framework and a simply gravitational theory simultaneously, facilitating the
examination of diverse phenomena within the realm of cosmology.

## Acknowledgement

A.P. and G.M. acknowledges the DSTB, Government of West Bengal, India for
financial support through the Grants No.: 322(Sanc.)/ST/P/S&T/16G-3/2018 dated
06.03.2019. S.R. is thankful to the Inter-University Centre for Astronomy and
Astrophysics (IUCAA), Pune, India for providing Visiting Associateship under
which a part of this work was carried out who also gratefully acknowledges the
facilities under ICARD, Pune at CCASS, GLA University, Mathura. 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).

Data Availability Statement: The manuscript does not contain any connected
data.

Keywords: K-essence geometry, Vaidya spacetime, $f(\bar{R},\bar{T})$ gravity,
gravitational collapse, dark energy.

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