# Collapsing scenario for the k-essence emergent generalised Vaidya spacetime
in the context of massive gravity’s rainbow

Saibal Ray<EMAIL_ADDRESS>Centre for Cosmology, Astrophysics and Space
Science (CCASS), GLA University, Mathura 281406, Uttar Pradesh, India  Arijit
Pandaa<EMAIL_ADDRESS>Department of Physics, Prabhat Kumar
College, Contai, Purba Medinipur-721404, India  Bivash Majumder
<EMAIL_ADDRESS>Department of Mathematics, Prabhat Kumar College,
Contai, Purba Medinipur-721404, India  Md. Rabiul Islam<EMAIL_ADDRESS>a Department of Physics, Raiganj University, Raiganj, Uttar Dinajpur-733 134,
West Bengal, India.  Goutam Mannab<EMAIL_ADDRESS>
b Corresponding author  Department of Physics, Prabhat Kumar College, Contai,
Purba Medinipur-721404, India

###### Abstract

In this paper, we study the collapsing scenario for the k-essence emergent
Vaidya spacetime in the context of massive gravity’s rainbow. For this study,
we consider that the background metric is Vaidya spacetime in massive
gravity’s rainbow. We show that the k-essence emergent gravity metric
resembles closely to the new type of generalized Vaidya massive gravity metric
with the rainbow deformations for null fluid collapse where we consider the
k-essence scalar field as a function solely of the advanced or the retarded
time. The k-essence emergent Vaidya massive gravity rainbow mass function is
also different. This new type k-essence emergent Vaidya massive gravity
rainbow metric has satisfied the required energy conditions. The existence of
the locally naked central singularity, the strength and the strongness of the
singularities for the rainbow deformations of the k-essence emergent Vaidya
massive gravity metric are the interesting outcomes of the present work.

##  I Introduction

Based on the nonlinear or doubly special relativity (DSR) [1, 2, 3], Magueijo
et al. [4] were the first to develop the rainbow gravity. Kimberly et al. [5]
discussed the details of this gravity, which is basically a deformed version
of the special relativity (SR). One of the major characteristics of this
theory is that, it was proposed to keep inertial frames relative. Secondly,
Planck energy can be made of an invariant scale [4, 6, 7] by this theory. Here
comes the proposal of a dual to non-linear realization of relativity in the
momentum space resulting, the spacetime invariant, an energy-dependent metric.
It is noteworthy that, there exist a variety of theories in literature where,
the modification of standard energy-momentum dispersion relation has been done
in the limit of the Planck scale, such as, string theory [8], loop quantum
gravity [9] and non-commutative geometry [10]. However, to assimilate the
curvature of the rainbow gravity [4] generalizes the DSR taking the energy-
dependency of spacetime into consideration. This type of consideration permits
the quanta of different energies to observe different classical geometries.
The argument for the allocation of same inertial frames is that, we want to
conserve the equivalence principle in a modified frame and that’s because all
the measurements of distance and time become dependent on the energy of the
quanta, which are to be engaged for the testing purposes.

The deformed dispersion relations, in the Planck length scale are,
$E^{2}f^{2}(E)-(\overrightarrow{p})^{2}g^{2}(E)=m^{2}$ [4]. However, the
standard linear Lorentz transformations do not ensure the invariance of these
relations. Whether the invariance of these have been proved under non-linear
Lorentz transformations by Kimberly et al. [5]. It has already been
acknowledged that, in place of single spacetime, dual to momentum space is
actually the energy-dependent family of the metric. A notable point is that
$E$ in the metric $g_{\mu\nu}(E)$ has nothing to do with the energy of the
spacetime, instead it is the scale at which the geometry of spacetime is
probed. Magueijo et. al. [4] argued that seeing a particle (or plane wave or
wave-packet) with energy $E$ by an observer means that particular particle
experiences the metric $g_{\mu\nu}(E)$. Moreover, if this particle is seen
with different $E^{\prime}(\neq E)$ by a different observer, then a different
metric will be assigned to the particle, i.e. $g_{\mu\nu}(E^{\prime})$. The
requirement of the covariance for a non-linear representation of the Lorentz
group in momentum space may demand, seeing a given particle being affected by
different metrics by different observers, as well as, assigning different
metrics to different particles moving in the same region at the same time by
same observer.

The modification of the equivalence principle have also been established by
Magueijo et al. [4], which states that, the spacetime is described by a one-
parameter family of metrics given in terms of a one-parameter family of
orthonormal frame fields as $g(E)=\eta^{\mu\nu}e_{\mu}(E)\otimes e_{\nu}(E)$,
where the frame fields depends on the energy as,
$e_{0}(E)=\frac{1}{f(E)}\tilde{e}_{0}$ and
$e_{i}(E)=\frac{1}{g(E)}\tilde{e}_{i}$ in the Planck length scale. Here the
tilde bearing terms refer to the energy-independent frame fields, that specify
the geometry probed by the low energy quanta, the limit being $El_{P}<<1$ with
$g_{\mu\nu}\simeq\eta_{\mu\nu}$, where $l_{P}$ is the Planck length.

However, an interesting point, which we would like to mention here that, if
the energy $E$ is just a non-dynamical parameter of the theory then, rescaling
can gauge it away easily. Unfortunately, the dependence on the coordinate is
dynamical and actually breaks the diffeomorphisms symmetry of the whole
metric. By the fact that, the local symmetry in gravity’s rainbow is not
Lorentz symmetry, because it depends on the deformation of the usual energy-
momentum dispersion relation and consequently breaks the Lorentz symmetry in
the UV limit also [11]. We should also remember that, the energy $E$ is an
implicit function of the coordinates whereas, rainbow functions are explicit
functions of the energy $\mathcal{E}$ which is defined as the ratio $E/E_{P}$,
$E_{P}$ being the Planck energy. That immediately concludes that, these are
also dynamical functions of the coordinates, restricting us to gauge these
away. The difficulty to find such an explicit dependence of rainbow functions
on energy for different systems also goes in vain because, these are implicit
dynamical functions of the coordinates, that means, gauging away is not going
to entertain us anyway. So we can expect that these functions may produce
physically different results from general relativity [12].

Ali et al. [13] have shown that the gravitational collapse can be explained in
the context of gravity’s rainbow. Heydarzade et. al. in [14, 15] have
discussed the energy dependent deformation and time dependent geometry of
massive gravity with the help of massive gravity’s rainbow formalism.
Vainshtein and the dRGT mechanisms have been used side by side for the energy
dependent massive gravity and additionally their works includes the analysis
of a ghost free theory of massive gravity’s rainbow with the help of the
radiating Vaidya solution. Phenomenologically, the massive gravity can also be
motivated from accelerated cosmic expansion [16, 17, 18, 19, 20, 21]. The
problems with the massive gravity can be resolved using the Vainshtein
mechanism [22, 23] easily. However, the Vainshtein mechanism can raise the
Boulware-Deser ghosts [24], which again can be resolved by using the dRGT
mechanism [25, 26, 27, 28, 29, 30, 31, 32].

Some related theories of rainbow gravity and their subsequent investigation in
charged dilatonic black holes, Gauss-Bonnet gravity, Lovelock gravity,
combination of Rastall and rainbow theories, $AdS_{4}$ dyonic black holes,
deformed Starobinsky model, thermodynamics of black holes, Galileon gravity,
horizon effect, violation of weak cosmic censorship and Branes are addressed
in [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 11, 43, 44, 45, 46, 47].

Either a black hole or a naked singularity may be produced from Gravitational
collapse [49, 50]. Also, it may provide mechanism to communicate with far-away
observers in the universe. The radiating Schwarzschild spacetime also known as
the Vaidya spacetime [51] can describe the geometry outside a radiating
spherically symmetric star. Particularly, the formation of naked singularities
can be produced from the solution of the null dust fluid with spherical
symmetry in gravitational collapse, as shown by Papapetrou [52]. Additionally,
cosmic censorship conjecture (CCC) [53] can be encountered in a different way
by this example. The causal trajectories joining the singularities in the
ingoing Vaidya situation have been studied by the authors of [54, 55]. They
also provide the whole story of constraints and classification the non-
spacelike geodesics that, make connection with the past naked singularity.
Then, this naked singularity was shown to be a strong curvature singularity in
a more effective sense.

The solution of Vaidya spacetime [56, 57, 58, 59, 60, 61] was generalised
including all the possible solutions of Einstein’s field equations and
combining Type-I and Type-II matter fields in the works of Husain [62] and
Wang and Wu [63]. the generalization of the Vaidya solution in the context of
the cosmic censorship hypothesis, which is basically the study of the
gravitational collapse for the generalized Vaidya spacetime has been done in
[64]. It was their findings that, to terminate the collapse the classes of
generalized Vaidya mass functions should brought in the scenario with a
locally naked central singularity. Along with they also calculated the
strength of these singularities. For the termination of the non-spacelike
geodesics directed towards the future at the singularity in the past they
studied the conditions on the mass function, developing a general mathematical
framework. The fact that, the final outcome of collapse can be determined in
terms of either a black hole or a naked singularity, for a given generalized
Vaidya mass function was also established. The gravitational collapse of
higher dimension, in the charged-Vaidya spacetime was studied by the authors
in [65]. Their works include that, singularities arising in a charged null
fluid in a higher dimensional spacetime are always naked, violating strong
cosmic censorship hypothesis (CCH). The holographic complexity and the
holographic entropy cone in AdS-Vaidya spacetime was explored in [66, 67].
Sharma et al. [68] revisited the Vaidya-Tikekar stellar model in the linear
regime. Manna et al. established the general connection between the k-essence
geometry and the Vaidya spacetime in [69, 70, 71]. More specifically, using a
generalized Vaidya-type metric [62, 63], they have worked on the gravitational
collapse in the k-essence emergent gravity in [71] in the context of cosmic
censorship hypothesis. Alongside, the existence of the locally naked central
singularity, the strength, and the strongness of the singularities for the
k-essence emergent Vaidya metric was established by them.

The k-essence model, particularly the scalar field model reveals that the role
of kinetic energy is much more dominating than the potential energy of the
field. The kinetic energy term does not depend explicitly on the field and
therefore can not be separated in the theoretical form of Lagrangian with non-
canonical kinetic terms theoretical form. Born and Infeld [72, 73, 74] first
proposed a theory with a non-canonical kinetic term which was related to the
infinite self-energy of the electron.

The general form of the Lagrangian for k-essence model can be written as:
$L=-V(\phi)F(X)$ where $\phi$ is k-essence scalar field,
$X=\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$ and shows the non-
dependency on $\phi$ to start with [75, 76, 77, 78, 79, 80, 81, 82, 83, 84,
85]. The k-essence theory with non-canonical kinetic terms and the
relativistic field theories with canonical kinetic terms differs in the non-
trivial dynamical solutions of the k-essence equation of motion (EOM). The
metric for the perturbations around these solutions are changed by solutions
of the EOM breaking the Lorentz invariance spontaneously. This allows the
perturbations to propagate in the so-called emergent or analogue curved
spacetime [75, 76, 77, 78, 79] with the metric, different from the
gravitational one. The emergent gravity metric $\tilde{G}_{\mu\nu}$ is not
conformally equivalent to the gravitational metric $g_{\mu\nu}$.

Based on the following articles [14, 15, 71], our work in this paper includes
the exploration of the collapsing scenario for the k-essence emergent Vaidya
spacetime in the context of massive gravity’s rainbow. To do this, we started
our work by considering the background metric as Vaidya spacetime in massive
gravity’s rainbow [14]. A notable thing to mention is that, the quantization
problem of the gravity is beyond our work for this article. Our consideration
for this work is that, the background metric is rainbow deformations of the
massive gravity Vaidya type. Doing this, we have discussed the collapsing
scenario on the basis of the k-essence geometry.

Let us now discuss some importance of the k-essence theory as follows: The
Coincidence problem has questioned most of the dark energy models (e.g.,
cosmological constant) in recent years by bringing some of the observational
data (Large-Scale Structure, searches of type Ia Supernovae and measurements
of Cosmic Microwave Background anisotropy [86]) in the scenario. The problem
is the fine tuning of the initial energy density which is of the orders of 100
or more smaller than the initial matter-energy density. The solution of the
abovementioned problem lies in the k-essence theory which is a nonlinear
kinetic energy of the scalar field [87]. An elaborated discussion about the
motivation of K-essence theory has been given in Ref. [88]. Not only in
cosmology [89, 90, 91, 92, 93] but also other fields of gravity has dealt with
K-essence theory [71, 94, 95]. All these aspects have motivated us to consider
the issues in the present work.

We embodied this paper as follows: The k-essence geometry based on the Dirac-
Born-Infeld action, rainbow theory of gravity and massive gravity have been
revisited in Sect. 2. In Sect. 3, we concentrate to build the rainbow
deformations of the k-essence emergent Vaidya massive gravity spacetime
considering the background metric as Vaidya massive gravity’s rainbow. A
scalar field with a restriction of being an arbitrary function of the advanced
or retarded Eddington-Finkelstein time has been considered where, the scalar
field is independent of the other variables of the four-dimensional spacetime.
The Ricci and Einstein tensors corresponding to our rainbow deformations of
the k-essence emergent Vaidya massive gravity metric has also been
constructed. This section also include the computation of the components of
the emergent energy-momentum tensor by direct substitution into an emergent
Einstein equation. This emergent tensor must obey the energy conditions of the
emergent geometry. In Sect. 4, we have developed the collapsing scenario for
the k-essence emergent Vaidya spacetime in massive gravity’s rainbow. In this
section, we have analyzed the structure of the central singularity to find out
the conditions on the k-essence emergent Vaidya massive gravity’s rainbow mass
function with the rainbow functions, existence of outgoing nonspacelike
geodesic and the strength of the singularities for the above k-essence
emergent geometry. The last Sect. 5 contains some conclusion as well as
discussion of our work.

##  II Brief review of the K-essence geometry, gravity’s rainbow theory and
massive gravity

###  II.1 The K-essence geometry

The action of the k-essence scalar field $\phi$, minimally coupled to the
background spacetime metric $g_{\mu\nu}$ is given by [75]–[79]

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

where $X={1\over 2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$.

The energy-momentum tensor can be written as

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

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

The equation of motion of the scalar field is

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

where

$\tilde{G}^{\mu\nu}\equiv
L_{X}g^{\mu\nu}+L_{XX}\nabla^{\mu}\phi\nabla^{\nu}\phi$ (4)

and $1+{2XL_{XX}\over L_{X}}>0$.

The conformal transformation gives $G^{\mu\nu}\equiv{c_{s}\over
L_{x}^{2}}\tilde{G}^{\mu\nu}$, with $c_{s}^{2}(X,\phi)\equiv{(1+2X{L_{XX}\over
L_{X}})^{-1}}$, the inverse metric of $G^{\mu\nu}$ takes the form

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

Again, applying a conformal transformation [80, 81]
$\bar{G}_{\mu\nu}\equiv{c_{s}\over L_{X}}G_{\mu\nu}$, we get

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

To make Eqs. (1)–(4) physically meaningful, we should have $L_{X}\neq 0$ for
$c_{s}^{2}$ to be positive definite. If $L$ does not directly depend on
$\phi$, then the EOM (3) reduces to

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

The Dirac-Born-Infeld (DBI) type Lagrangian [72, 73, 74, 80, 81, 82] has the
form

$L(X,\phi)=1-V(\phi)\sqrt{1-2X},$ (8)

with $V(\phi)=V=$constant and kinetic energy of $\phi>>V$, i.e
.$(\dot{\phi})^{2}>>V$. This ensures the domination of kinetic energy
dominates over the potential energy for the k-essence fields and gives us
$c_{s}^{2}(X,\phi)=1-2X$. For scalar fields
$\nabla_{\mu}\phi=\partial_{\mu}\phi$. Therefore, the effective emergent
metric (6) ends up as

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

The new Christoffel symbols and old Christoffel symbols are related to each
other [103, 80] as

$\bar{\Gamma}^{\alpha}_{\mu\nu}=\Gamma^{\alpha}_{\mu\nu}-\frac{1}{2(1-2X)}[\delta^{\alpha}_{\mu}\partial_{\nu}X+\delta^{\alpha}_{\nu}\partial_{\mu}X].$
(10)

Now, we can write the new geodesic equation for the k-essence theory in terms
of the new Christoffel connections $\bar{\Gamma}$ as

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

where $\lambda$ is an affine parameter.

###  II.2 Gravity’s rainbow theory

Inspired by the works of Magueijo et al. [4, 1, 2, 3] as well as Kimberly et
al. [5], the deformed energy-momentum dispersion relation can be provided as

$E^{2}\mathcal{F}^{2}(\mathcal{E})-p^{2}\mathcal{G}^{2}(\mathcal{E})=m^{2},$
(12)

where $\mathcal{E}=\frac{E}{E_{P}}$ is the energy ratio, which is obviously a
dimensionless quantity, $E$ and $p$ has been used to express the energy and
momentum (respectively) of the test particle and $E_{P}$ as the Planck energy.

The fact that, the energy of a test particle cannot exceed the Plank energy,
gives us the limit, $\mathcal{E}$ as $0<\mathcal{E}\leq 1$. Therefore, the
energy dependent rainbow functions $\mathcal{F}(\mathcal{E})$ and
$\mathcal{G}(\mathcal{E})$ will satisfy the below two conditions:

$\lim_{\mathcal{E}\rightarrow
0}\mathcal{F}(\mathcal{E})=1~{}\text{and}~{}\lim_{\mathcal{E}\rightarrow
0}\mathcal{G}(\mathcal{E})=1$ (13)

and the general relativity is recovered in the IR limit of the theory [96, 97,
98, 99, 100, 101, 102].

Again, the energy dependent contribution in the metric is given by

$g(\mathcal{E})=\eta^{\mu\nu}e_{\mu}(\mathcal{E})\otimes
e_{\nu}(\mathcal{E}),$ (14)

with the energy dependence of the frame fields as follows:
$e_{0}(\mathcal{E})=\frac{1}{f(\mathcal{E})}\tilde{e}_{0}$ and
$e_{i}(\mathcal{E})=\frac{1}{g(\mathcal{E})}\tilde{e}_{i}$ in the Planck
length scale. Here $\tilde{e}_{0}$ and $\tilde{e}_{i}$ are the energy
independent frame fields.

The choice of the rainbow functions can be of various type [104, 105, 14, 13]
such as, from the background motivation of loop quantum gravity and
$\kappa-$Minkowski noncommutative spacetime:

$\mathcal{F}(\mathcal{E})=1~{}\text{and}~{}~{}~{}\mathcal{G}(\mathcal{E})=\sqrt{1-\alpha\mathcal{E}^{q}},$
(15)

where we can take the rainbow functions with the constant velocity of light
[3] as

$\mathcal{F}(\mathcal{E})=\mathcal{G}(\mathcal{E})=\frac{1}{1-\alpha\mathcal{E}}.$
(16)

For the hard spectra from gamma-ray burster’s at cosmological distances, it is
also possible to choose rainbow functions [106] as

$\mathcal{F}(\mathcal{E})=\frac{e^{\alpha\mathcal{E}}-1}{\alpha\mathcal{E}}~{}\text{and}~{}\mathcal{G}(\mathcal{E})=1.$
(17)

Whatever be the choice, the main property of all these rainbow functions are
spacetime energy dependent.

###  II.3 Massive gravity

The 4-dimensional massive gravity action [14, 15, 107, 108] can be written as

$S=\int
d^{4}x\sqrt{-g}\Big{[}\mathcal{R}+\mathbb{M}^{2}\sum_{i}^{4}c_{i}\mathcal{U}_{i}(g,f)+\mathcal{L}_{m}\Big{]},$
(18)

$f$ being a fixed symmetric tensor (also known as the reference metric),
$c_{i}$ are constants, $\mathbb{M}$ are the massive gravity parameter and
$\mathcal{U}_{i}$ are symmetric polynomials of the eigenvalues of the $d\times
d$ matrix $\mathcal{K}^{\mu}_{\nu}=\sqrt{g^{\mu\alpha}f_{\alpha\nu}}$.

The symmetric polynomials, mentioned above, can be written as

$\displaystyle\mathcal{U}_{1}$ $\displaystyle=$ $\displaystyle[\mathcal{K}],$
$\displaystyle\mathcal{U}_{2}$ $\displaystyle=$
$\displaystyle[\mathcal{K}]^{2}-[\mathcal{K}^{2}],$
$\displaystyle\mathcal{U}_{3}$ $\displaystyle=$
$\displaystyle[\mathcal{K}]^{3}-3[\mathcal{K}][\mathcal{K}^{2}]+2[\mathcal{K}^{3}],$
$\displaystyle\mathcal{U}_{4}$ $\displaystyle=$
$\displaystyle[\mathcal{K}]^{4}-6[\mathcal{K}^{2}][\mathcal{K}]^{2}+8[\mathcal{K}^{3}][\mathcal{K}]+3[\mathcal{K}^{2}]^{2}-6[\mathcal{K}^{4}],$

where $\mathcal{K}=\mathcal{K}^{\mu}_{\mu}$.

With the help of the variational principle the modified field equations for
the massive gravity can be obtained as

$\mathcal{R}_{\mu\nu}-\frac{1}{2}\mathcal{R}g_{\mu\nu}+\mathbb{M}^{2}\chi_{\mu\nu}=T_{\mu\nu}.$
(20)

Here, $\chi_{\mu\nu}$ denotes the massive term, which can be expressed as

$\displaystyle\chi_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{c_{1}}{2}\Big{(}\mathcal{U}_{1}g_{\mu\nu}-\mathcal{K}_{\mu\nu}\Big{)}-\frac{c_{2}}{2}\Big{(}\mathcal{U}_{2}g_{\mu\nu}-2\mathcal{U}_{1}\mathcal{K}_{\mu\nu}+2\mathcal{K}^{2}_{\mu\nu}\Big{)}$
(21) $\displaystyle-$
$\displaystyle\frac{c_{3}}{2}\Big{(}\mathcal{U}_{3}g_{\mu\nu}-3\mathcal{U}_{2}\mathcal{K}_{\mu\nu}+6\mathcal{U}_{1}\mathcal{K}^{2}_{\mu\nu}-6\mathcal{K}^{3}_{\mu\nu}\Big{)}$
$\displaystyle-$
$\displaystyle\frac{c_{4}}{2}\Big{(}\mathcal{U}_{4}g_{\mu\nu}-4\mathcal{U}_{3}\mathcal{K}_{\mu\nu}+12\mathcal{U}_{2}\mathcal{K}^{2}_{\mu\nu}-24\mathcal{U}_{1}\mathcal{K}^{3}_{\mu\nu}+24\mathcal{K}^{4}_{\mu\nu}\Big{)},$

considering $8\pi G=1$ in the geometrized units.

Heydarzade et al. [15] in their paper took a spatial reference metric in the
basis of $(v,r,\theta,\Phi)$ [14, 110, 111]

$f_{\mu\nu}=diag(0,0,c^{2}h_{ij}),$ (22)

in which $c^{2}$ is a positive constant and $h_{ij}$ is the two dimensional
Euclidean metric.

We can write the barotropic relation and the energy-momentum tensor
respectively as

$p=\kappa\rho,$ (23) $T_{\mu\nu}=T^{(n)}_{\mu\nu}+T^{(m)}_{\mu\nu},$ (24)

with

$\displaystyle T^{(n)}_{\mu\nu}$ $\displaystyle=$ $\displaystyle\sigma
l_{\mu}l_{\nu},$ $\displaystyle T^{(m)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle(\rho+p)(l_{\mu}n_{\nu}+l_{\nu}n_{\mu})+p~{}{g}_{\mu\nu}.$ (25)

The terms $T^{(n)}_{\mu\nu}$ and $T^{(m)}_{\mu\nu}$, in the above expression
are the energy-momentum tensor for the Vaidya null radiation and the energy-
momentum tensor of the perfect fluid, respectively. $\sigma$, $\rho$ and $p$
denoting the null radiation density, energy density and pressure of the
perfect fluid, respectively, and $l_{\mu}$, $n_{\mu}$ are the two null
vectors. These null vectors are defined as [15]: $l_{\mu}=(1,0,0,0)$,
$n_{\mu}=\Big{(}\frac{1}{2}(1-\frac{m(v,r)}{r}),-1,0,0\Big{)}$ with
$l_{\mu}l^{\mu}=n_{\mu}n^{\mu}=0$ and $l_{\mu}n^{\mu}=-1$.

Taking the above assumptions into consideration, Heydarzade et al. [15] have
constructed the following metric for the Vaidya spacetime in massive gravity:

$ds^{2}=-\Big{(}1-\frac{m(v,r)}{r}\Big{)}dv^{2}+2dv~{}dr+r^{2}d\Omega^{2},$
(26)

with $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\Phi^{2}$, and

$\displaystyle
m(v,r)=\frac{r^{1-2\kappa}}{1-2\kappa}f_{1}(v)+f_{2}(v)-\frac{1}{2}\mathbb{M}^{2}c_{1}cr^{2}-\mathbb{M}^{2}c_{2}c^{2}r,$
(27)

under the constraint $\kappa\neq\frac{1}{2}$, where $f_{1}(v)$ and $f_{2}(v)$
are arbitrary functions of $v$ and can be expressed as

$\displaystyle
f_{1}(v)=\rho(v,r)r^{2(1+\kappa)};~{}~{}~{}\sigma(v,r)=\frac{r^{-(1+2\kappa)}}{1-2\kappa}\dot{f}_{1}(v)+\frac{1}{r^{2}}\dot{f}_{2}(v),$
(28)

where a ’dot’ represents the derivative with respect to $v$. Here, $v$
represents the null coordinate corresponding to the Eddington advanced time
with $r$ decreasing towards the future along a ray related to $v=$ constant.

We can establish the following metric with the help of [15], considering the
rainbow deformations of the Vaidya spacetime in massive gravity [14]

$\displaystyle
ds^{2}=-\frac{1}{\mathcal{F}^{2}(\mathcal{E})}\Big{(}1-\frac{m(v,r)}{r}\Big{)}dv^{2}+\frac{2}{\mathcal{F}(\mathcal{E})\mathcal{G}(\mathcal{E})}dvdr+\frac{1}{\mathcal{G}^{2}(\mathcal{E})}r^{2}d\Omega^{2},$
(29)

with

$\displaystyle m(v,r)$ $\displaystyle=$
$\displaystyle\frac{r^{1-2\kappa}}{1-2\kappa}f_{1}(v)+f_{2}(v)-\frac{1}{2\mathcal{G}(\mathcal{E})}\mathbb{M}^{2}c_{1}cr^{2}-\mathbb{M}^{2}c_{2}c^{2}r,~{}~{}~{}(\kappa\neq\frac{1}{2})$
(30)

and

$\displaystyle
f_{1}(v)=\frac{\rho(v,r)}{\mathcal{G}^{2}(\mathcal{E})}r^{2(1+\kappa)};~{}~{}~{}~{}\sigma(v,r)=\mathcal{F}(\mathcal{E})\mathcal{G}(\mathcal{E})\Big{[}\frac{r^{-(1+2\kappa)}}{1-2\kappa}\dot{f_{1}(v)}+\frac{1}{r^{2}}\dot{f_{2}(v)}\Big{]}.$
(31)

As expected, the mass function, given in (30) is different from the mass
function of Heydarzade et al. [14]

##  III K-essence Vaidya spacetime in massive gravity’s rainbow

In this section, we will be busy to discuss the k-essence Vaidya geometry in
the context of the massive gravity’s rainbow. Manna et al. have established
the connection between the k-essence geometry and the Vaidya spacetime based
on the DBI type action in [69, 71].

So, let’s assume the background metric to be the Vaidya massive gravity’s
rainbow (29) with the definitions (30) and (31).

The k-essence emergent line element can be written from Eq. (9)

$dS^{2}=ds^{2}-\partial_{\mu}\phi\partial_{\nu}\phi dx^{\mu}dx^{\nu}.$ (32)

Assuming the k-essence scalar field $\phi(x)=\phi(v)$ only, the emergent line
element in the context of massive gravity’s rainbow (32) can be written as

$\displaystyle
dS^{2}=-\Big{[}\frac{1}{\mathcal{F}^{2}(\mathcal{E})}\Big{(}1-\frac{m(v,r)}{r}\Big{)}-\phi_{v}^{2}\Big{]}dv^{2}+\frac{2}{\mathcal{F}(\mathcal{E})\mathcal{G}(\mathcal{E})}dvdr+\frac{1}{\mathcal{G}^{2}(\mathcal{E})}r^{2}d\Omega^{2},$
(33)

which also can be represented as [71]:

$\displaystyle
dS^{2}=-\frac{1}{\mathcal{F}^{2}(\mathcal{E})}\Big{(}1-\frac{\mathcal{M}(v,r)}{r}\Big{)}dv^{2}+\frac{2}{\mathcal{F}(\mathcal{E})\mathcal{G}(\mathcal{E})}dvdr+\frac{1}{\mathcal{G}^{2}(\mathcal{E})}r^{2}d\Omega^{2}.~{}~{}~{}~{}$
(34)

Here, we define the k-essence emergent Vaidya massive gravity’s rainbow mass
function as

$\displaystyle\mathcal{M}(v,r)$
$\displaystyle=m(v,r)+r\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}$ (35)
$\displaystyle=\frac{r^{1-2\kappa}}{1-2\kappa}f_{1}(v)+f_{2}(v)-\frac{1}{2\mathcal{G}(\mathcal{E})}\mathbb{M}^{2}c_{1}cr^{2}-\mathbb{M}^{2}c_{2}c^{2}r+r\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2},~{}~{}~{}~{}(\kappa\neq\frac{1}{2}),$

where $\phi_{v}^{2}$ ($\phi_{v}=\frac{\partial\phi}{\partial v}$) is the
kinetic energy of the scalar field $\phi$, which should not be equal to zero.

For well-defined signature of the above metric (33) the values of
$\phi_{v}^{2}$ should lie between $0$ and $1$, i.e., $0<\phi_{v}^{2}<1$.
Since, in general, spherical symmetry would only require that
$\phi=\phi(v,r)$, the k-essence scalar field $\phi$ actually violates local
Lorentz invariance. The choice of $\phi(v,r)=\phi(v)$ additionally implies
that outside of this particular choice of frame, a spherically symmetric
$\phi$ is actually a function of both $v$ and $r$. Since, the dynamical
solutions of the k-essence equation of motion spontaneously break Lorentz
invariance and also change the metric for the perturbations around these
solutions, the k-essence theory allows us to take this type of Lorentz
violation. Also, notice that the choice of $\phi(v,r)~{}(=\phi(v))$ is equally
important in the context of cosmology and gravitation [69, 70, 71, 89, 88].

Using the following definitions

$\displaystyle\mathcal{M}_{v}=\dot{\mathcal{M}}(v,r)\equiv\frac{\partial\mathcal{M}(v,r)}{\partial
v},~{}~{}~{}\mathcal{M}_{r}=\mathcal{M}^{{}^{\prime}}(v,r)\equiv\frac{\partial\mathcal{M}(v,r)}{\partial
r},$ (36)

the non-zero components of the emergent Ricci tensors can be represented as

$\displaystyle\bar{R}^{v}_{v}=\bar{R}^{r}_{r}=\frac{\mathcal{G}^{2}(\mathcal{E})}{2r}\mathcal{M}_{rr}=\frac{\mathcal{G}^{2}(\mathcal{E})}{2r}m_{rr}~{};~{}\bar{R}^{\theta}_{\theta}=\bar{R}^{\Phi}_{\Phi}=\frac{\mathcal{G}^{2}(\mathcal{E})}{r^{2}}\mathcal{M}_{r}=\frac{\mathcal{G}^{2}(\mathcal{E})}{r^{2}}(m_{r}+\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}).$
(37)

Also the Ricci scalar is given by

$\displaystyle\bar{R}=\frac{\mathcal{G}^{2}(\mathcal{E})}{r}\mathcal{M}_{rr}+\frac{2\mathcal{G}^{2}(\mathcal{E})}{r^{2}}\mathcal{M}_{r}=\frac{\mathcal{G}^{2}(\mathcal{E})}{r}m_{rr}+\frac{2\mathcal{G}^{2}(\mathcal{E})}{r^{2}}(m_{r}+\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}).$
(38)

The non-vanishing components of the Einstein tensor are

$\displaystyle\mathbb{G}^{v}_{v}$ $\displaystyle=$
$\displaystyle\mathbb{G}^{r}_{r}=-\frac{\mathcal{G}^{2}(\mathcal{E})\mathcal{M}_{r}}{r^{2}}=-\frac{\mathcal{G}^{2}(\mathcal{E})}{r^{2}}\Big{(}m_{r}+\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}\Big{)}~{},~{}$
$\displaystyle\mathbb{G}^{r}_{v}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{G}^{2}(\mathcal{E})}{r^{2}}\mathcal{M}_{v}=\frac{\mathcal{G}^{2}(\mathcal{E})}{r^{2}}\Big{(}m_{v}+2r\mathcal{F}^{2}(\mathcal{E})\phi_{v}\phi_{vv}\Big{)}~{},~{}$
$\displaystyle\mathbb{G}^{\theta}_{\theta}$ $\displaystyle=$
$\displaystyle\mathbb{G}^{\Phi}_{\Phi}=-\frac{\mathcal{G}^{2}(\mathcal{E})}{2r}\mathcal{M}_{rr}=-\frac{\mathcal{G}^{2}(\mathcal{E})}{2r}m_{rr},$
(39)

where from Eq. (35), we have

$\displaystyle\mathcal{M}_{r}$ $\displaystyle=$ $\displaystyle
f_{1}(v)r^{-2\kappa}-\frac{1}{\mathcal{G}(\mathcal{E})}\mathbb{M}^{2}c_{1}cr-\mathbb{M}^{2}c_{2}c^{2}+\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}$
(40) $\displaystyle\mathcal{M}_{v}$ $\displaystyle=$
$\displaystyle\frac{r^{1-2\kappa}}{1-2\kappa}\dot{f_{1}(v)}+\dot{f_{2}(v)}+2r\mathcal{F}^{2}(\mathcal{E})\phi_{v}\phi_{vv}$
(41) $\displaystyle\mathcal{M}_{rr}$ $\displaystyle=$ $\displaystyle-2\kappa
r^{-(1+2\kappa)}f_{1}(v)-\frac{1}{\mathcal{G}(\mathcal{E})}\mathbb{M}^{2}c_{1}c.$
(42)

The k-essence emergent Einstein equation is

$\mathbb{G}^{\mu}_{\nu}={\cal T}^{\mu}_{\nu}.$ (43)

If we consider, $8\pi G=1$, then it immediately leads us to the components
${\cal T}^{\mu}_{\nu}$, which can be parametrized exactly as in refs. [71, 62,
63, 64] in terms of the components $\sigma,\rho$ and $p$ given by

$\displaystyle{\cal T}_{\mu\nu}={\cal T}_{\mu\nu}^{(n)}+{\cal
T}_{\mu\nu}^{(m)}=\left[\begin{array}[]{cccc}(\sigma/2+\rho)&\sigma/2&0&0\\\
\sigma/2&(\sigma/2-\rho)&0&0\\\ 0&0&p&0\\\ 0&0&0&p\end{array}\right],$ (48)

where ${\cal T}_{\mu\nu}^{(n)}=\sigma l_{\mu}l_{\nu}$; ${\cal
T}_{\mu\nu}^{(m)}=(\rho+p)(l_{\mu}n_{\nu}+l_{\nu}n_{\mu})+p\bar{G}_{\mu\nu}$
with $l_{\mu}$ and $n_{\mu}$ are two null vectors. The doubt in using the
perfect fluid energy-momentum tensor for the k-essence theory can be erased by
the form of the Lagrangian $L(X)=1-V\sqrt{1-2X}$, where $V$ is a constant, and
it does not depend explicitly on $\phi$. This class of models can be thought
as equivalent to perfect fluid models with zero vorticity and the pressure
(Lagrangian) can be expressed through the energy density only [79].

Therefore, the three independent components are expressed as

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{\mathcal{G}^{2}(\mathcal{E})}{r^{2}}\mathcal{M}_{v}=\frac{\mathcal{G}^{2}(\mathcal{E})}{r^{2}}\Big{(}m_{v}+2r\mathcal{F}^{2}(\mathcal{E})\phi_{v}\phi_{vv}\Big{)},$
(49) $\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{\mathcal{G}^{2}(\mathcal{E})}{r^{2}}\mathcal{M}_{r}=\frac{\mathcal{G}^{2}(\mathcal{E})}{r^{2}}\Big{(}m_{r}+\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}\Big{)},$
(50) $\displaystyle\Sigma$ $\displaystyle=$
$\displaystyle-\frac{\mathcal{G}^{2}(\mathcal{E})}{2r}\mathcal{M}_{rr}=-\frac{\mathcal{G}^{2}(\mathcal{E})}{2r}m_{rr}.$
(51)

The energy conditions for the combination of Type-I and Type-II matter fields
energy momentum tensor ${\cal T}_{\mu\nu}$, which are defined [109, 71] as
following:

(a) The weak and strong energy conditions:

$\sigma\geq 0~{},~{}\rho\geq 0~{},~{}p\geq 0~{}~{}~{}(\sigma\neq 0).$ (52)

(b) The dominant energy conditions:

$\sigma\geq 0~{},~{}\rho\geq p\geq 0~{}~{}~{}~{}~{}(\sigma\neq 0).$ (53)

So, following the above energy conditions (52) and (53), imposed on ${\cal
T}_{\mu\nu}$ will be constrained in $m(v,r)$ and $\phi(v)$ and their
derivatives. Thus

$\displaystyle\sigma>0\Rightarrow
m_{v}+2r\phi_{v}\phi_{vv}>0\Rightarrow\frac{r^{1-2\kappa}}{1-2\kappa}\dot{f_{1}(v)}+\dot{f_{2}(v)}+2r\mathcal{F}^{2}(\mathcal{E})\phi_{v}\phi_{vv}>0,$
(54) $\displaystyle\rho>0\Rightarrow m_{r}+\phi_{v}^{2}>0\Rightarrow
f_{1}(v)r^{-2\kappa}-\frac{1}{\mathcal{G}(\mathcal{E})}\mathbb{M}^{2}c_{1}cr-\mathbb{M}^{2}c_{2}c^{2}+\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}>0~{}~{}$
(55) $\displaystyle p>0\Rightarrow m_{rr}<0\Rightarrow 2\kappa
f_{1}(v)r^{-(1+2\kappa)}+\mathbb{M}^{2}c_{1}c>0.~{}~{}~{}$ (56)

##  IV Collapsing scenario for the k-essence Vaidya spacetime in massive
gravity’s rainbow

Now, let’s cultivate the collapsing scenario of the k-essence emergent Vaidya
spacetime in the context of the massive gravity’s rainbow. At the beginning,
we define $K^{\mu}$ as the tangent to the non-spacelike geodesics with
$K^{\mu}=\frac{dx^{\mu}}{d\lambda}$ where $\lambda$ is the affine parameter.
The geodesic equation takes the form of [64, 71]

$\bar{G}_{\mu\nu}K^{\mu}K^{\nu}=\beta,$ (57)

where $\beta$ is a constant. Here $\beta=0$ and $\beta<0$ describes the null
geodesics and timelike geodesics respectively.

Expressing the k-essence emergent geodesic equation (11) in terms of $K^{\mu}$
we get

$\frac{dK^{\alpha}}{d\lambda}+\bar{\Gamma}^{\alpha}_{\mu\nu}K^{\mu}K^{\nu}=0.$
(58)

Now, using Eqs. (34), (57) and (58), we get the geodesic equations [71, 54,
64] with the forms:

$\displaystyle\frac{dK^{v}}{d\lambda}+\frac{\mathcal{G}(\mathcal{E})}{2\mathcal{F}(\mathcal{E})}\Big{[}\frac{\mathcal{M}}{r^{2}}-\frac{\mathcal{M}_{r}}{r}\Big{]}(K^{v})^{2}-r\frac{\mathcal{F}(\mathcal{E})}{\mathcal{G}(\mathcal{E})}\Big{[}(K^{\theta})^{2}+\sin^{2}\theta(K^{\Phi})^{2}\Big{]}=0$
(59) $\displaystyle\Rightarrow$
$\displaystyle\frac{dK^{v}}{d\lambda}+\frac{\mathcal{G}(\mathcal{E})}{2\mathcal{F}(\mathcal{E})}\Big{[}\frac{m}{r^{2}}-\frac{m_{r}}{r}\Big{]}(K^{v})^{2}-\frac{\mathcal{F}(\mathcal{E})}{\mathcal{G}(\mathcal{E})}\frac{l^{2}}{r^{3}}=0,~{}$

$\displaystyle\frac{dK^{r}}{d\lambda}+\frac{\mathcal{G}(\mathcal{E})}{2\mathcal{F}(\mathcal{E})}\frac{\mathcal{M}_{v}}{r}(K^{v})^{2}-\frac{\mathcal{G}^{2}(\mathcal{E})}{2}\Big{[}\frac{\mathcal{M}}{r^{2}}-\frac{\mathcal{M}_{r}}{r}\Big{]}\times\Big{[}-\frac{1}{\mathcal{F}^{2}(\mathcal{E})}\Big{(}1-\frac{\mathcal{M}}{r}\Big{)}(K^{v})^{2}+\frac{K^{r}K^{v}}{\mathcal{F}(\mathcal{E})\mathcal{G}(\mathcal{E})}\Big{]}$
(60) $\displaystyle-$
$\displaystyle\Big{(}1-\frac{\mathcal{M}}{r}\Big{)}r\Big{[}(K^{\theta})^{2}+\sin^{2}\theta(K^{\Phi})^{2}\Big{]}=0$
$\displaystyle\Rightarrow$
$\displaystyle\frac{dK^{r}}{d\lambda}+\frac{\mathcal{G}(\mathcal{E})}{2\mathcal{F}(\mathcal{E})}\Big{(}\frac{m_{v}}{r}+2\mathcal{F}^{2}(\mathcal{E})\phi_{v}\phi_{vv}\Big{)}(K^{v})^{2}-\frac{l^{2}}{r^{3}}\Big{(}1-\frac{m}{r}-\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}\Big{)}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\beta\mathcal{G}^{2}(\mathcal{E})\Big{(}\frac{m}{r^{2}}-\frac{m_{r}}{r}\Big{)}=0~{},$

$\displaystyle\frac{dK^{\theta}}{d\lambda}+\frac{2}{r}K^{\theta}K^{r}-\sin\theta~{}\cos\theta(K^{\Phi})^{2}=0,$
(61)

$\displaystyle\frac{d}{d\lambda}(r^{2}\sin^{2}\theta K^{\Phi})=0,$ (62)

where we use the relation [54, 71]

$\displaystyle
K^{\Phi}=\frac{l\cos\delta}{r^{2}\sin^{2}\theta}~{}~{},~{}~{}K^{\theta}=\frac{l}{r^{2}}\sin\delta\cos\Phi,$
(63)

where $l$ and $\delta$ are constant of integration. which represents the
impact parameter and the isotropy parameter respectively, satisfying the
relation, $\sin\Phi~{}\tan\delta=\cot\theta$.

The definition of $K^{v}$ [54, 64] gives us

$\displaystyle K^{v}=\frac{P(v,r)}{r}$ (64)

and the relation $\bar{G}_{\mu\nu}K^{\mu}K^{\nu}=\beta$ provides

$\displaystyle K^{r}$ $\displaystyle=$
$\displaystyle\frac{P}{2r}\frac{\mathcal{G}(\mathcal{E})}{\mathcal{F}(\mathcal{E})}\Big{(}1-\frac{m}{r}-\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}\Big{)}-\frac{\mathcal{F}(\mathcal{E})}{\mathcal{G}(\mathcal{E})}\frac{l^{2}}{2Pr}+\mathcal{F}(\mathcal{E})\mathcal{G}(\mathcal{E})\frac{\beta
r}{2P},$ (65)

where $P(v,r)$ is an arbitrary function.

Next, on differentiation the equation (64) with respect to $\lambda$ we get

$\displaystyle\frac{dP}{d\lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{r}\Big{(}r^{2}\frac{dK^{v}}{d\lambda}+P\frac{dr}{d\lambda}\Big{)}$
(66) $\displaystyle=$
$\displaystyle\frac{\mathcal{G}(\mathcal{E})}{\mathcal{F}(\mathcal{E})}\frac{P^{2}}{2r^{2}}\Big{(}1-\frac{2m}{r}+m_{r}\Big{)}+\frac{\mathcal{F}(\mathcal{E})}{\mathcal{G}(\mathcal{E})}\frac{l^{2}}{2r^{2}}+\mathcal{F}(\mathcal{E})\mathcal{G}(\mathcal{E})\frac{\beta}{2}-\frac{\mathcal{G}(\mathcal{E})}{\mathcal{F}(\mathcal{E})}\frac{P^{2}}{2r^{2}}\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}.$

After this we would now examine the destiny of the collapse, whether it ends
with a black hole or a naked singularity for the given k-essence emergent
Vaidya massive gravity rainbow mass function (35). If there exist some
families of future directed non-spacelike trajectories which terminates in the
past at the singularity, which are reaching faraway observers in spacetime,
then a naked singularity forming as the final state of the collapse. On the
flip side, if no such families exist and an event horizon forms sufficiently
early to cover the singularity then we have a black hole.

The radial null geodesic $(l=0,~{}\beta=0)$ can be achieved using equations
(64) and (65) as

$\displaystyle\frac{dv}{dr}$
$\displaystyle=\frac{2r\mathcal{F}(\mathcal{E})}{\mathcal{G}(\mathcal{E})\Big{(}r-m(v,r)-r\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}\Big{)}}\equiv\frac{2\mathcal{F}(\mathcal{E})}{\mathcal{G}(\mathcal{E})\Big{(}1-\frac{\mathcal{M}(v,r)}{r}\Big{)}}.$
(67)

For a suitable choice of rainbow functions it has a singularity at
$r=0,~{}v=0$, provided $\phi_{v}^{2}\neq 0$.

###  IV.1 Structure of the Central singularity for the k-essence emergent
Vaidya spacetime in massive gravity rainbow

In general, the above Eq. (67) can be written as [64, 71]

$\frac{dv}{dr}=\frac{M(v,r)}{N(v,r)},$ (68)

with the singular points $r=0,~{}v=0$, where both the above functions $M(v,r)$
and $N(v,r)$ vanish.

As discussed in [64], we can also have the characteristic equation for the
existence and uniqueness of the above form of differential equation (68) in
the vicinity of the singularity as

$\chi^{2}-(A+D)\chi+AD-BC=0,$ (69)

where $A=M_{v}(0,0)~{},~{}B=M_{r}(0,0)~{},~{}C=N_{v}(0,0)~{},~{}D=N_{r}(0,0)$
and $AD-BC\neq 0$.

The roots of the above Eq. (69) are

$\chi=\frac{1}{2}\Big{[}(A+D)\pm\sqrt{(A-D)^{2}+4BC}\Big{]}.$ (70)

The singularity can be of two type, First one is node, if $(A-D)^{2}+4BC\geq
0$ and $BC>0$. and second one is a center or focus.

Comparing Eqs. (67) and (68), we have

$\displaystyle M(v,r)$ $\displaystyle=$ $\displaystyle
2r\mathcal{F}(\mathcal{E}),$ $\displaystyle N(v,r)$ $\displaystyle=$
$\displaystyle\mathcal{G}(\mathcal{E})\Big{(}r-\mathcal{M}(v,r)\Big{)}=\mathcal{G}(\mathcal{E})\Big{(}r-m(v,r)-r\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}\Big{)}.$
(71)

At the central singularity ($v=0,r=0$), we define

$\displaystyle\mathcal{M}_{0}=\lim\limits_{v\to 0,r\to
0}\mathcal{M}(v,r)~{};~{}\mathcal{M}_{v0}=\lim\limits_{v\to 0,r\to
0}\frac{\partial}{\partial v}\mathcal{M}(v,r)~{};~{}$
$\displaystyle\mathcal{M}_{r0}=\lim\limits_{v\to 0,r\to
0}\frac{\partial}{\partial
r}\mathcal{M}(v,r)~{};~{}\phi_{v0}=\lim\limits_{v\to 0,r\to
0}\frac{\partial}{\partial v}\phi(v)~{};~{}$ $\displaystyle
m_{v0}=\lim\limits_{v\to 0,r\to 0}\frac{\partial}{\partial
v}m(v,r)~{};~{}m_{r0}=\lim\limits_{v\to 0,r\to 0}\frac{\partial}{\partial
r}m(v,r).$ (72)

Using Eqs. (71) and (72), we get
$A=0,~{}B=2\mathcal{F}(\mathcal{E}),~{}C=-\mathcal{G}(\mathcal{E})m_{v0}~{}and~{}D=\mathcal{G}(\mathcal{E})\Big{(}1-m_{r0}-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}\Big{)}$.
Hence, roots of the characteristic equation (69) can be written as:

$\displaystyle\chi=\frac{1}{2}\Bigg{[}\mathcal{G}(\mathcal{E})\Big{(}1-m_{r0}-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}\Big{)}\pm\sqrt{\mathcal{G}^{2}(\mathcal{E})\Big{(}1-m_{r0}-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}\Big{)}^{2}-8\mathcal{F}(\mathcal{E})\mathcal{G}(\mathcal{E})m_{v0}}\Bigg{]}.$
(73)

The required conditions for the singular point at $r=0,~{}v=0$ to be a node

$\displaystyle\mathcal{G}(\mathcal{E})\Big{(}1-m_{r0}-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}\Big{)}^{2}\geq
8\mathcal{F}(\mathcal{E})m_{v0},~{}m_{v0}>0~{}and~{}\phi_{v0}^{2}>0.$ (74)

Therefore, to satisfy the condition (74) the k-essence emergent Vaidya massive
gravity rainbow mass function $\mathcal{M}(v,r)$ and the rainbow functions
($\mathcal{F}(\mathcal{E})~{}and~{}\mathcal{G}(\mathcal{E})$)can be chosen in
a certain way and then the singularity at the origin $(v=0,r=0)$ will be a
node resulting the outgoing non-spacelike geodesics come out of the
singularity with a definite value of the tangent.

The null geodesic equation can be linearized near the central singularity
using the limits (72) as

$\frac{dv}{dr}=\frac{2r\mathcal{F}(\mathcal{E})}{\mathcal{G}(\mathcal{E})\Big{[}(1-m_{r0}-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2})r-m_{v0}v\Big{]}},$
(75)

which is also singular at $v=0,~{}r=0$ provided $\phi_{v}^{2}\neq 0$. From
(35), we get

$m_{r0}=-\mathbb{M}^{2}c_{2}c^{2};~{}m_{v0}=\frac{\partial f_{2}(v)}{\partial
v}\Big{|}_{0}\equiv\dot{f_{20}(v)}.$ (76)

###  IV.2 Existence of outgoing non-spacelike geodesics for the k-essence
emergent Vaidya spacetime in massive gravity rainbow

The summary upto now is that, the outgoing radial null geodesics have to end
up in the past at a singularity, particularly, in the central physical
singularity located at $v=0,~{}r=0$. With the help of this remark, we will now
explore the existence of naked singularity (NS) in the k-essence emergent
Vaidya spacetime in massive gravity rainbow. For a locally naked singularity,
such null geodesics exist. The possibility of this singularity to be either a
naked singularity or a black hole (BH) implies that if the singularity is not
a naked singularity, then the formation of a black hole is undoubtable. So,
the analysis of the radial null geodesics emerging from the singularity can
help us to understand the nature of such a singularity, which can be
established by a catastrophic gravitational collapse.

It is well-known in general relativity that, such a singularity which comes
out from a gravitational collapse, is always a black hole because of the
cosmic censorship in general relativity. Therefore, there will be an event
horizon by which the singularity is always covered in general relativity. But
to speak about the generalized case, it is possible for inhomogeneous dust
cloud to form a naked singularity through a collapse [112]. Also it is noted
that some interesting results have been obtained for fluids whose equations of
state (EOS) is not exactly similar to a dust cloud [113]. Thus it becomes
evident to generalize the cosmic censorship in general relativity [114].

Let us now consider that, the k-essence emergent Vaidya massive gravity mass
function $\mathcal{M}(v,r)$ with acceptable choices of rainbow functions
satisfies all the physical energy conditions (52) and (53) with the
constraints (54), (55) and (56). The partial derivatives of the mass function
which are continuous in the entire k-essence emergent Vaidya massive gravity’s
rainbow spacetime (35) also exist obeying the conditions (74) at the central
singularity. Following refs. [64, 65, 14, 15, 71], we would now determine the
nature (a black hole or a naked singularity) of the collapsing solutions. To
do this, we consider the function $X$ behave as, $X=\frac{v}{r}$ having
limiting value at the central singularity as follows:

$X_{0}=\lim\limits_{v\to 0,r\to 0}X=\lim\limits_{v\to 0,r\to 0}\frac{v}{r}.$
(77)

Using Eqs. (67), (77) and L’Hospital’s rule, we get

$\displaystyle X_{0}$ $\displaystyle=\lim\limits_{v\to 0,r\to
0}\frac{v}{r}=\lim\limits_{v\to 0,r\to 0}\frac{dv}{dr}=\lim\limits_{v\to
0,r\to
0}\frac{2\mathcal{F}(\mathcal{E})}{\mathcal{G}(\mathcal{E})\Big{(}1-\frac{\mathcal{M}(v,r)}{r}\Big{)}}.$
(78)

Again using (35), the above Eq. (78) can be written as

$\displaystyle\frac{2}{X_{0}}=\lim\limits_{v\to 0,r\to
0}\frac{\mathcal{G}(\mathcal{E})}{\mathcal{F}(\mathcal{E})}\Big{[}1-\frac{r^{-2\kappa}}{1-2\kappa}f_{1}(v)-\frac{f_{2}(v)}{r}+\frac{1}{2\mathcal{G}(\mathcal{E})}\mathbb{M}^{2}cc_{1}r+\mathbb{M}^{2}c^{2}c_{2}$
$\displaystyle-\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}\Big{]},$ (79)

under the constraint $\kappa\neq\frac{1}{2}$.

Now, considering $f_{1}(v)=\alpha v^{2\kappa}$ and $f_{2}(v)=\beta v$, the
algebraic equation of $X_{0}$ can be expressed as

$\displaystyle\frac{\alpha}{1-2\kappa}X_{0}^{1+2\kappa}+\beta
X_{0}^{2}-\Big{(}1+\mathbb{M}^{2}c^{2}c_{2}-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}\Big{)}X_{0}+2\frac{\mathcal{F}(\mathcal{E})}{\mathcal{G}(\mathcal{E})}=0,~{}~{}~{}~{}~{}~{}~{}~{}$
(80)

where $\alpha$ and $\beta$ are constants.

As mentioned before, the Eq. (80) is not similar to Eq. (3.7) of Heydarzade et
al. [15] due to the presence of the terms $\phi_{v0}^{2}$ and rainbow
functions. But we choose the same functions of $f_{1}(v)$ and $f_{2}(v)$ as
[15] along with the same characteristics. In the early universe ($\kappa\geq
0$), $f_{1}(v)$ grows with $v$, whereas in the late universe ($\kappa<0$),
$f_{1}(v)$ decays with time. On the other hand $f_{2}(v)$ is a linear function
of $v$. These choices of the arbitrary functions $f_{1}(v)$ and $f_{2}(v)$ are
self-similar in nature and this comes from the definition of $X_{0}$ given in
Eq. (78). For the same reasons, showed by Heydarzade et al. [15], we also
consider self-similar cases.

The formation of a black hole can be assured in this system by achieving a
non-positive solution of the above Eq. (80). However, the positive roots of
this equation can produce a naked singularity. With the specific choice of the
rainbow functions, we will examine the effects of the mass term and kinetic
energy of the k-essence scalar field ($\phi_{v0}^{2}$) on the formation of
naked singularity and observe that the effect of a mass term and
$\phi_{v0}^{2}$ on the formation of naked singularities. For this purpose, we
choose a specific rainbow function [115, 106, 14] as

$\mathcal{F}(\mathcal{E})=1,~{}\mathcal{G}(\mathcal{E})=\sqrt{1-\eta\Big{(}\frac{E_{1}}{E_{p}}\Big{)}}.$
(81)

In the above expressions, we have denoted Planck energy by $E_{p}$, given by
$E_{p}=1/\sqrt{G}=1.221\times 10^{19}$GeV, where G denotes gravitational
constant and $E_{1}=1.42\times 10^{-13}$ [115, 106, 14]. In [106, 14], the
authors have estimated the value of $\eta$ as, $\eta\approx 1$. So, in our
study we use $\eta=1$. It is very cumbersome to find exact solutions for
$X_{0}$, except for some particular values. Therefore, for some preferable
values we find the exact solutions as:

Case-I: $\kappa=0$: The zero value of $\kappa$ represents the pressure-less
dust regime of the universe. For this regime, the Eq. (80) reduces to

$\alpha X_{0}+\beta
X_{0}^{2}-\Big{(}1+\mathbb{M}^{2}c^{2}c_{2}-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}\Big{)}X_{0}+2\frac{\mathcal{F}(\mathcal{E})}{\mathcal{G}(\mathcal{E})}=0,$
(82)

and the solutions become

$\displaystyle X_{0\pm}$
$\displaystyle=\frac{\Big{(}U-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}-\alpha\Big{)}}{2\beta}\pm\frac{\sqrt{(U-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}-\alpha)^{2}-8\beta\mathcal{F}(\mathcal{E})/\mathcal{G}(\mathcal{E})}}{2\beta}~{}~{}~{}$
(83)

with $U=1+\mathbb{M}^{2}c_{2}c^{2}$.

For the demand of these two solutions to be real and finite, we must have
$\beta\neq 0$ and
$\mathcal{G}(\mathcal{E})(U+\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}-\alpha)^{2}\geq
8\beta\mathcal{F}(\mathcal{E})$. In our case the values of $\phi_{v0}^{2}$ are
$0<\phi_{v0}^{2}<1$, i.e., positive and the rainbow functions (81) are also
positive.

To make the solution $X_{0+}$ positive, we have the conditions for:

(i)
$\beta>0:~{}~{}(U-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}-\alpha)\geq-\sqrt{(U-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}-\alpha)^{2}-8\beta\mathcal{F}(\mathcal{E})/\mathcal{G}(\mathcal{E})}\\\
~{}~{}\Rightarrow~{}\beta\geq 0$.

But if the solutions are real and finite $\beta\neq 0$, then the solution
$X_{0+}$ is positive for any positive $\beta$.

(ii)
$\beta<0:~{}~{}~{}(U-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}-\alpha)\leq-\sqrt{(U-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}-\alpha)^{2}-8\beta\mathcal{F}(\mathcal{E})/\mathcal{G}(\mathcal{E})}\\\
~{}~{}\Rightarrow~{}~{}\beta\leq 0$.

As $\beta\neq 0$, the solution $X_{0+}$ has positive values for any negative
$\beta$. Therefore, in conclusion we can say that, for any non-zero real
values of $\beta$, the solution $X_{0+}$ is positive and this represents a
naked singularity.

For positivity of the solution $X_{0-}$, we have, for $\beta>0$,
$(U-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}-\alpha)\geq\sqrt{(U-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}-\alpha)^{2}-8\beta\mathcal{F}(\mathcal{E})/\mathcal{G}(\mathcal{E})}~{}\Rightarrow~{}\beta\geq
0$.
For the above said reason as explained above, here also $\beta>0$, which
implies that the solution is positive. Again for $\beta<0$, we can get the
positive solution following the same reason. Hence, we can conclude that, this
solution also represents a naked singularity. Finally, for $\kappa=0$, we will
get naked singularity as the destiny of the collapse. Here, we would like to
mention that, the conditions for the positive solutions of Eq. (80) are
similar type as Heydarzade et al. [15].

Case-II: $\kappa=1$: This condition represents the early stiff fluid [116] era
of our universe. In this scenario Eq. (80) reduces to

$-\alpha X_{0}^{3}+\beta
X_{0}^{2}-(U-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2})X_{0}+2\frac{\mathcal{F}(\mathcal{E})}{\mathcal{G}(\mathcal{E})}=0.$
(84)

If we consider $\alpha$, $\beta$,
$U-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}$, $\mathcal{F}(\mathcal{E})$ and
$\mathcal{G}(\mathcal{E})$, all to be positive, then by the well-known theorem
of Descartes, also known as Descartes’ Rule of Sign, the number of variations
of sign is 3. Which means the number of positive roots of Eq. (84) is either 1
or 3. The minimum number of positive root being 1, we can announce that the
naked singularity forms for $\kappa=1$.

It is noteworthy, for the case $\kappa>\frac{1}{2}$, the energy conditions
(50-52) are violated apparently in our case and these energy conditions can be
made satisfied easily if we redefine the mass function $\mathcal{M}(v,r)$.

Case-III: $\kappa=-1/2$: This represents the dark energy dominated
accelerating universe. For $\kappa=-1/2$, Eq. (80) is reduces to

$\frac{\alpha}{2}+\beta
X_{0}^{2}-(1+\mathbb{M}^{2}c_{2}c^{2}-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2})X_{0}+2\frac{\mathcal{F}(\mathcal{E})}{\mathcal{G}(\mathcal{E})}=0,$
(85)

which bears the following solutions

$\displaystyle
X_{0\pm}=\frac{1}{2\beta}\Big{[}(U-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2})\pm\sqrt{(U-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2})^{2}-\frac{2\beta}{\mathcal{G}(\mathcal{E})}\Big{(}4\mathcal{F}(\mathcal{E})+\alpha\mathcal{G}(\mathcal{E})\Big{)}}\Big{]}.~{}~{}~{}~{}$
(86)

In this case, to get the real and finite solutions, we must have $\beta\neq 0$
and
$(U-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2})^{2}\geq\frac{2\beta}{\mathcal{G}(\mathcal{E})}(4\mathcal{F}(\mathcal{E})+\alpha\mathcal{G}(\mathcal{E}))$.
Following the same procedure of case-I, the positiveness of these solutions
(86), we can show for $\beta>0$, we get the condition
$\alpha\frac{\mathcal{G}(\mathcal{E})}{\mathcal{F}(\mathcal{E})}\geq-4$ and
for $\beta<0$,
$\alpha\frac{\mathcal{G}(\mathcal{E})}{\mathcal{F}(\mathcal{E})}\leq-4$. Due
to the presence of the rainbow functions, the solutions we get here are
different from Heydarzade et al. [15].

###  IV.3 Numerical solution of $X_{0}$

Using numerical methods, we will try to find a numerical solutions of $X_{0}$
by considering the specific rainbow functions (81) in this subsection. The
dynamics of collapse lies in the knowledge of $X_{0}$ for the whole
cosmologically meaningful region $\kappa<1$, i.e., from early to late
universe, not only at discrete points of $\kappa$. So, we decide to visualize
these solutions obtaining various contours for $\kappa-X_{0}$ for different
numerical values of the involved parameters.

Figure 1: Figs. (a) and (b) show the variation of $X_{0}$ with $\kappa$ for
different values of $\mathbb{M}$ and $\alpha$ respectively (in Fig. (a) $M$
are equal to $\mathbb{M}$) with $\phi_{v0}^{2}=0.7$, $E_{p}=1.221\times
10^{19}$ and $E_{1}=1.42\times 10^{-13}$. In Fig. (a) the initial conditions
are $\alpha=0.5$, $\beta=2$, $c=2$ and $c_{2}=2$ whereas in Fig. (b) the
initial conditions are $\mathbb{M}=2$, $\beta=2$, $c=2$ and $c_{2}=2$.

Figure 2: Figs. (a) and (b) show the variation of $X_{0}$ with $\kappa$ for
different values of $c$ and $c_{2}$ respectively with $\phi_{v0}^{2}=0.7$,
$E_{p}=1.221\times 10^{19}$ and $E_{1}=1.42\times 10^{-13}$. In Fig. (a) the
initial conditions are $\mathbb{M}=2$, $\alpha=0.5$, $\beta=2$ and $c_{2}=2$
whereas in Fig. (b) the initial conditions are $\mathbb{M}=2$, $\alpha=0.5$,
$\beta=2$ and $c=2$. Figure 3: Variation of $X_{0}$ with $\kappa$ for
different values of $\phi_{v0}^{2}$ with $E_{p}=1.221\times 10^{19}$ and
$E_{1}=1.42\times 10^{-13}$ and the initial conditions are $\alpha=0.5$,
$\beta=2.0$, $c=2.0$, $c2=2.0$ and $\mathbb{M}=2.0$

Looking at the above figures, we get to know that, the trajectories run across
the positive range of $X_{0}$, which is the indication of formation of naked
singularity (NS). Fig. 1(a) reveals the dependence of $X_{0}$ on the equation
of state (23) parameter ($\kappa$) for particular values of the massive
gravity parameter $\mathbb{M}(=2,3,5,7)$ with the fixed $\phi_{v0}^{2}$. Also,
from this figure, we see that with increasing $\mathbb{M}$ the tendency of
formation of NS decreases. Therefore, we can say that the addition of graviton
mass to this system with the specific choice of rainbow functions (81) deforms
the dynamics of the system. In Fig. 1(b), we obtain the $\kappa-X_{0}$
trajectories by varying the values of $\alpha(=0.5,5,10,15)$. In this figure,
we see that, with greater values of $\alpha$, the tendency to form NS is
greater.

In Fig. 2(a), we observe how $c$ affects the collapsing scenario. It is clear
that, for the greater value of $c(=2,3,5,10)$, the tendency to form NS is
lesser. In Fig. 2(b), we can observe the effect of $c_{2}$ on the collapsing
system. Like $c$, here also we see that an increase in $c_{2}$ decreases the
possibility of NS.

In Fig. 3, we have showed the effect of kinetic energy ($\phi_{v0}^{2}$) of
the k-essence scalar field on the collapsing scenario. It has been observed
that, all the values of $\phi_{v0}^{2}$ predict the same trajectories of the
system and form the NS of the collapsing system. Here we consider restriction
of $\phi_{v0}^{2}$ not to exceed unity.

From all the above observations and their corresponding discussions of the
$\kappa-X_{0}$ trajectories makes it clear that the trajectories reside in the
positive $X_{0}$ region, and assures the formation of naked singularity from
the collapse of this system.

###  IV.4 Strength of singularity for the k-essence emergent Vaidya spacetime
in massive gravity rainbow

Taking the recommendation of [71, 15, 65, 117, 118, 119], we can conclude
that, a singularity $(r=v=\lambda=0)$ would be strong if the following
condition is satisfied

$\lim\limits_{\lambda\to 0}\lambda^{2}\psi=\lim\limits_{\lambda\to
0}\lambda^{2}\bar{R}_{\mu\nu}K^{\mu}K^{\nu}>0,$ (87)

where $\bar{R}_{\mu\nu}$ is the Ricci tensor and
$\psi=\bar{R}_{\mu\nu}K^{\mu}K^{\nu}$ is defined as a scalar of the k-essence
emergent Vaidya massive gravity rainbow spacetime. We would like to mention
that, this scalar $\psi$ is not the k-essence scalar field and is not coupled
with the background gravitational metric $g_{\mu\nu}$. Stepping into the
footsteps of [71, 64], with the conditions of (74), it can be shown that

$\lim\limits_{\lambda\to
0}\lambda^{2}\psi=\mathcal{G}^{2}(\mathcal{E})\Big{(}\mathcal{M}_{v0}\Big{)}\frac{1}{4}X_{0}^{2}>0.$
(88)

So far we have seen that, if this condition is satisfied for some real and
positive roots of $X_{0}$, the naked singularity of the k-essence emergent
Vaidya massive gravity rainbow spacetime is strong. On the other hand, if
there is no positive real root of $X_{o}$, we can conclude that there is no
outgoing future directed null geodesics from the singularity exist, i.e., the
collapse ends in a black hole.

Now from the definition of $f_{1}(v)$ and $f_{2}(v)$ and from (35), we have

$\mathcal{M}_{v}=\frac{2\kappa\alpha}{1-2\kappa}\Big{(}\frac{v}{r}\Big{)}^{2\kappa-1}+\beta+2r\mathcal{F}^{2}(\mathcal{E})\phi_{v}\phi_{vv}.$
(89)

Thus, we have

$\mathcal{M}_{v0}=\frac{2\kappa\alpha}{1-2\kappa}X_{0}^{2\kappa-1}+\beta\equiv
m_{v0}.$ (90)

In the above three cases, i.e., $\kappa=0$, $\kappa=1$ and $\kappa=-1/2$, we
have achieved the positive roots of $X_{0}$ for the respective conditions.
Also, from Eq. (90), we observe the following situations for the above three
cases:

(i) $\kappa=0$, $\mathcal{M}_{v0}=\beta$ which is positive for any positive
value of $\beta$,

(ii) $\kappa=1$, $\mathcal{M}_{v0}=\alpha X_{0}+\beta$, positivity of
$\mathcal{M}_{v0}$ implies that $\alpha X_{0}+\beta>0$ and

(iii) $\kappa=-1/2$, $\mathcal{M}_{v0}=\beta-\frac{\alpha}{2X_{0}^{2}}$, here
the positive value of $\mathcal{M}_{v0}$ implies that $2\beta
X_{0}^{2}>\alpha$.

With the above conditions and the specific rainbow functions (81), we see that
$\lim\limits_{\lambda\to 0}\lambda^{2}\psi>0$. Therefore, it can be admitted
that, the naked singularity is strong with the above conditions.

###  IV.5 Apparent horizon for the emergent spacetime and physical
consequences

####  IV.5.1 Apparent horizon

The authors in [69, 70], have described the dynamical horizon (DH) on the
basis of [120, 121, 122, 123, 124] in the context of k-essence geometry
considering Schwarzschild metric as the background metric. Whereas, in [64,
71], the authors have described the apparent horizon (AH) for the generalized
Vaidya type geometry. The AH is nothing but the boundary of the trapped
surface region in the given spacetime (34).The casual behaviour of the trapped
surfaces developed in the spacetime during the collapse evolution decides the
occurrence of a naked singularity or black hole.

We should remember that, not only the AHs are not invariant properties of a
spacetime, but also are distinct from event horizons (EHs). Within an AH,
light does not move away from it. This is in contrast with the EH, where in a
dynamical spacetime outgoing light rays exterior to an AH (but still interior
to the EH) can exist. Specifically, an AH is a local notion of the boundary of
a spacetime, whereas an EH is a global notion of a black hole.

####  IV.5.2 Physical consequences

The AH for the k-essence emergent Vaidya spacetime in massive gravity rainbow
(34) can be represented as

$\displaystyle\frac{\mathcal{M}(v,r)}{r}=1\Rightarrow\frac{m(v,r)}{r}+\mathcal{F}^{2}(\mathcal{E})\phi_{v}^{2}=1,$
(91)

with $\mathcal{F}(\mathcal{E})\neq 0$.

Considering refs. [64, 71], the slope of the AH at the central singularity
($r\rightarrow 0,v\rightarrow 0$) can be written as

$\displaystyle\big{(}\frac{dv}{dr}\big{)}_{AH}$
$\displaystyle=\frac{1-m_{r0}-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}}{m_{v0}}=\frac{1+\mathbb{M}^{2}c_{2}c^{2}-\mathcal{F}^{2}(\mathcal{E})\phi_{v0}^{2}}{\dot{f}_{20}(v)},$
(92)

where we have used Eq. (76).

Now, the sufficient conditions for the existence of a locally naked
singularity for the collapsing k-essence emergent Vaidya spacetime in massive
gravity’s rainbow has been attained. The emergent mass function
$\mathcal{M}(v,r)$ obeys all the energy conditions and constraints mentioned
in (54), (55) and (56). Also, like [71], there exist an open set of parameter
values for which, the singularity is locally naked for the case of k-essence
emergent Vaidya spacetime in massive gravity’s rainbow.

Not to forget about the k-essence emergent Vaidya spacetime in massive gravity
rainbow metric may exchange radiation with the surrounding but the mass
function (35) can not totally evaporate due the presence of massive gravity
terms and the kinetic part of the k-essence scalar field, which is function of
$v$. As an example, in [70], the authors have shown the decreasing nature of
black hole mass $m(v,r)$ in the K-essence emergent Schwarzschild Vaidya
spacetime but it does not completely vanish, by using DH equation as
$\phi_{v}^{2}\rightarrow 0^{+}$.

##  V Discussion and Conclusion

This paper shows the construction of a theory of the k-essence Vaidya
spacetime in massive gravity’s rainbow. We have also analyzed the energy
dependent deformation of the massive gravity in our work. We have used the
Vainshtein mechanism and the dRGT mechanism in the construction of massive
gravity where, deformation by the rainbow functions has been considered. This
paper also includs the analysis of the gravitational collapse of null fluids
in the radiating k-essence Vaidya black hole solution of massive gravity’s
rainbow based on the Dirac-Born-Infeld Lagrangian. The above said exploration
has been presented in the context of cosmic censorship hypothesis with the
k-essence emergent Vaidya massive gravity rainbow mass function. The Einstein
tensor and the energy conditions for the combination of Type-I and Type-II
matter fields energy-momentum tensor for the k-essence emergent Vaidya massive
gravity rainbow spacetime has also been studied thoroughly. We have shed some
light on the classification of non-spacelike geodesic for the k-essence
emergent Vaidya massive gravity rainbow spacetime, that connects the naked
singularity in the past, which is a strong curvature singularity in a stronger
sense. The central singularity is a node, which allows outgoing non-spacelike
geodesics to come out of the singularity pursuing a definite value of the
tangent. This positivity of the tangent implies the violation of the strong
cosmic censorship conjecture, i.e., the singularity is locally naked, though
not necessarily the weak one.

If somehow, the value of tangent does not seizes positive value, the central
singularity deviates from being a naked one because, in that case, there are
no outgoing future-directed null geodesics coming out from the singularity,
i.e., the collapse will always lead to a black hole. With these results we
have also studied the effects of the graviton mass, kinetic energy of the
k-essence scalar field and rainbow deformation for a time-dependent system.

We have obtained the contours for the tangent $X_{0}$ against the barometric
parameter $\kappa$ for different values of other parameters, namely
$\mathbb{M}$, $\alpha$, $\beta$, $\phi_{v0}^{2}$, etc. Various regimes of the
fluid content of the universe such as the radiation ($\kappa>0$), pressure-
less dust ($\kappa=0$), dark energy ($\kappa<0$) has been plotted. From those
figures it is clear that the negative region contains no trajectories for
specific choice of the rainbow functions. This outcomes make us believe the
existence of black holes cannot be possible as the end state of collapse in
the context of massive gravity rainbow with the considered parameters.

It should be noted that here we have studied the structure of gravitational
collapse of a non-rotating type spacetime. The violation of CCC allows us to
achieve a naked singularity which by definition is not bounded by any EH. On
the other hand, there have been considered several models when the non-
spacelike geodesics originate in the central singularity. However, this is not
the prove of the nakedness of the singularity. In [125], for the rotating case
there should be a consideration of ergosphere to describe the motion of the
non-spacelike geodesics. The authors also have studied in detail the geodesics
in the ergosphere of rotating black hole which have negative energies. An
ergoshpere is basically a region located outside a rotating black hole’s outer
EH. Particularly in non-extremal Kerr spacetime ($a<M$ case), a non-spacelike
geodesic for particles having negative, zero and positive energy can originate
at the central singularity, when the collapse is a black hole. In [126] the
authors have studied the dynamics of the high energy particles which have
negative and positive energies in the vicinity of black holes. They also have
studied the time of movement of particles with negative energy in the
ergosphere. Vertogradov in his paper [127] has examined the nature of the
geodesics for particles with negative energy in Kerr’s metric. Also, we would
like to mention that in our article we have not discussed about the formation
of ergosphere. We may try to explore this type of formation in near future.

In the paper we have found very much similarity between the k-essence emergent
gravity metric and the generalized Vaidya massive gravity metric with the
rainbow deformations. Obviously, this may establish another motivation or
foundation for gravity’s rainbow theory. The result continues to include that,
the rainbow deformations of the k-essence emergent Vaidya massive gravity
metric also satisfies the required energy conditions. Therefore, from totally
physical background, the results are viable and it demands for some
observational clues on the possibility to observe the naked singularity. The
crucial issue then becomes: How experimentally the naked singularity can be
tested? Let us elaborate this point a bit based on the available literature as
follows:

(i) The authors of [128], have studied the optical appearance of a
geometrically thin and optically thick accretion disk around the strongly
naked Janis-Newman-Winicour (JNW) singularity in their paper. Their solution
represents an example of a compact object spacetime, which possesses no photon
sphere. They [128] also have established a general observational signature for
distinguishable compact objects possessing no photon sphere from black holes.

(ii) In [129, 130], the authors have studied the distinction between black
holes and naked singularities using the images of their accretion disks. They
have considered a simplified model of spherical accretion onto the central
object and studied shadow and images of Joshi-Malafarina-Narayan (JMN) [131,
132] for naked singularities.

(iii) Bhattacharya et al. [133] have studied a new class of naked
singularities and their observational signatures on the basis of suitable
junction conditions matched with an expanding FLRW metric with a generic
contracting solution of Einstein’s equations, on a spacelike hypersurface.
They have also studied some observational aspects of the resulting two-
parameter family of static naked singularity backgrounds.

(iv) Kovacs et al. [134] have noted the properties of the accretion disk and
observationally different black holes from naked singularities. They have
considered a rotating solution of the Einstein-massless scalar field equations
for the naked singularity, which reduces to the Kerr solution when the scalar
charge tends to 1. They also have showed their interest to study the motion of
the particles in the gravitational potential of this solution. Depending on
the values of the mass, scalar charge, and the spin parameter, respectively,
there are two types of disks that could exist around naked singularities. The
first type are marginally stable orbits, locates outside the naked
singularity, while the second type are reachable for the particles and in
direct contact with the singularity.

(v) Ortiz et al. [135] have spent their time to study the observational
differences between black holes and naked singularities through the redshift
function. They said that, the photons propagating from past to future null
infinity through the center of the cloud obtains the frequency shift, which
can be measured by distant stationary observers and that can help to
discriminate naked singularities from black holes. They have showed that,
according to their model, it is possible to differentiate the formation of a
naked singularity from the formation of a black hole based on the detection of
photons traversing a collapsing object.

(vi) Vertogradov [136, 137] have proposed a method to study the formation of
the eternal naked singularity in the case of gravitational collapse of
generalized Vaidya spacetime. Also, in [138, 139], the authors have studied
the naked singularity, using the gravitational lensing.

We would like to end our discussion by mentioning that, we have analytically
discussed the formation of naked singularity in the k-essence Vaidya spacetime
in massive gravity’s rainbow. But from the above observational discussions of
the existence of naked singularity, one may conclude that this type of
singularity can also be possible to exist in reality, however which needs
observational assurance and evidences.

Data availability: Our manuscript has no associated data.

## Acknowledgement

The authors would like to thank the referee for illuminating suggestions to
improve the paper. SR 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.

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