# Naked Singularity Formation In $f({\mathcal{R}})$ Gravity

A. H. Ziaie<EMAIL_ADDRESS>Department of Physics, Shahid Beheshti
University, Evin, 19839 Tehran, Iran  K. Atazadeh<EMAIL_ADDRESS>Department of Physics, Azarbaijan University of Tarbiat Moallem, 53714-161
Tabriz, Iran  S. M. M. Rasouli<EMAIL_ADDRESS>Department of Physics,
Shahid Beheshti University, Evin, 19839 Tehran, Iran

###### Abstract

We study the gravitational collapse of a star with barotropic equation of
state $p=w\rho$ in the context of $f({\mathcal{R}})$ theories of gravity.
Utilizing the metric formalism, we rewrite the field equations as those of
Brans-Dicke theory with vanishing coupling parameter. By choosing the
functionality of Ricci scalar as $f({\mathcal{R}})=\alpha{\mathcal{R}}^{m}$,
we show that for an appropriate initial value of the energy density, if
$\alpha$ and $m$ satisfy certain conditions, the resulting singularity would
be naked, violating the cosmic censorship conjecture. These conditions are the
ratio of the mass function to the area radius of the collapsing ball,
negativity of the effective pressure, and the time behavior of the Kretschmann
scalar. Also, as long as parameter $\alpha$ obeys certain conditions, the
satisfaction of the weak energy condition is guaranteed by the collapsing
configuration.

##  I Introduction

Einstein’s General theory of relativity is the classical theory of one of the
four fundamental forces, gravity, which is the weakest but most dominant force
of nature governing phenomena at large scales, and is described by a
mathematically well-founded and elegant structure i.e., differential geometry
of curved spacetime. The Einstein’s field equations, a system of non-linear
partial differential equations, relate the geometric property of spacetime to
the four-momentum (energy density and linear momentum) of matter fields
leading to precise predictions that have received considerable experimental
confirmations with high accuracy such as solar system tests (see Solar  and
references therein). One of the most engrossing but open debates in general
relativity is that of the final fate of gravitational collapse with
possibility of the existence of spacetime singularities, the ultra-strong
gravity regions where the densities and spacetime curvatures blow up, leading
to a spacetime which is geodesically incomplete Hawking-Ellis  and the
structure of any classical theory of fields is vanquished. A star with a mass
many times than that of the Sun would undergo a continual gravitational
collapse due to its self-gravity without achieving an equilibrium state in
contrast to a neutron star or a white dwarf. Then, according to singularity
theorems established by Hawking and Penrose Hawking-Penrose  a singularity is
reached as the collapse endstate. Such a singularity may be a black hole
hidden from external observers by an event horizon or visible to the outside
Universe (naked singularity). In the latter collapse procedure, the
information on super-dense regions can be transported via suitable non-
spacelike trajectories to a distant observer. Although the occurrence of a
spacetime singularity as the final outcome of a collapse scenario is proved by
the singularity theorems, they do not specify the nature of such a
singularity. The cosmic censorship conjecture first articulated by Penrose CCC
states that a black hole is always formed in complete gravitational collapse
of reasonable matter fields, or a physically reasonable spacetime contains no
naked singularities. However, up to now many exact solutions of Einstein’s
field equations describing singularities, not hidden behind an event horizon
of spacetime, are known. A remarkable study is the one by Shapiro and
Teukolsky, who showed numerically that gravitational collapse of a spheroidal
dust may end in a naked singularity Shapiro . Also many exact solutions of
Einstein’s field equations with a variety of field-sources admitting naked
singularities have been surveyed. The examples studied so far include
gravitational collapse of a pressure-less matter Dust-Patil , radiation
Radiation-Patil , perfect fluids PF-Patil , imperfect fluids IPF-Patil  and
null strange quark fluids NSQF . Beside general theory of relativity, there
exist alternative theories of gravity explaining gravitational phenomena. Such
theories have been studied for a long time Alternative . From the theoretical
standpoint there has been many attempts to correct the Einstein-Hilbert action
in order to renormalize general relativity to build a quantum theory of
gravity or at least some effective action (including the low-energy limit of
string theories), or to quantize the scalar fields in curved spacetimes Callan
. From the observational point of view, the discovery of current acceleration
of the Universe using CMB Ia supernova CMB  suggests that such acceleration
may be explained within the framework of general relativity by assuming that
$76\%$ of energy content of the Universe is filled with a mysterious form of
$dark~{}energy$ with equation of state $p\sim-\rho$ (where $\rho$ and $p$ are
the energy density and pressure of the cosmic fluid, respectively). Another
possibility is to include a cosmological constant $\Lambda$ of a very small
magnitude in Einstein’s field equation, but encounters such difficulties as
the well-known cosmological constant problem and the coincidence problem.
Another alteration is $f({\mathcal{R}})$ theories of gravity FR  where the
Ricci scalar in the Einstein-Hilbert Lagrangian is replaced by a general
function of it, providing alternative gravitational models for dark energy
since the explanation of the cosmic acceleration comes back to the fact that
we do not understand gravity at large scales. Such theories can describe the
transition from deceleration to acceleration in the evolution of the Universe
Nojiri-Odintsov . Moreover, the coincidence problem may be solved simply in
such theories by the Universe expansion. Also some models of modified theories
of gravity are predicted by string/M-theory considerations MString . Recently,
it has been shown that the accelerated expansion of the Universe may be the
result of a modification to the Einstein-Hilbert action in the context of
higher order gravity theories Farhoudi . Our purpose here is to consider the
gravitational collapse of a star within the framework of $f({\mathcal{R}})$
theories of gravity, whose matter content obeys the barotropic equation of
state, $p=w\rho$. We investigate the conditions under which the resulting
singularity may be naked or not. In Section II we apply the metric formalism
to the action of $f({\mathcal{R}})$ gravity Metric-Formalism  and rewrite it
as that of the Brans-Dicke theory with $\omega_{BD}=0$. Choosing
$f({\mathcal{R}})=\alpha{\mathcal{R}}^{m}$ Capozzillo  in Section III and
fixing the corresponding potential, we find $m$ as a function of initial
energy density and $\alpha$. In Section IV we study the behavior of the
expansion parameter which is the key factor in examining the formation or
otherwise of trapped surfaces during the dynamical evolution of the collapse
scenario. In Section VI we examine the global features of the nakedness of the
resulting singularity by investigating the behavior of the Kretschmann scalar
as a function of time. In order to fully complete the model we utilize the
Vaidya metric to match the interior spacetime to that of the exterior one.

##  II $f({\mathcal{R}})$ Field equations

We begin by the general action in modified theories of gravity given by

${\mathcal{A}}=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}f({\mathcal{R}})+{\mathcal{A}}_{matter}(g_{\mu\nu},\psi),$ (1)

where $\kappa=8\pi G$, G is the gravitational constant, g is the determinant
of the metric, ${\mathcal{R}}$ represents the Ricci scalar and $\psi$
collectively denotes the matter fields. Introducing an auxiliary field $\Psi$,
one can write the dynamically equivalent action as (See Faraoni  and
references therein)

${\mathcal{A}}=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}\left[f(\Psi)+f^{\prime}(\Psi)({\mathcal{R}}-\Psi)\right]+{\mathcal{A}}_{matter}(g_{\mu\nu},\psi),$
(2)

where variation with respect to $\Psi$ leads to the following equation as

$f^{\prime\prime}(\Psi)({\mathcal{R}}-\Psi)=0.$ (3)

If $f^{\prime\prime}(\Psi)\neq 0$, one can then recover action (1) by setting
$\Psi={\mathcal{R}}$. Redefining the field $\Psi$ by $\phi=f^{\prime}(\Psi)$
and setting

$V(\phi)=\Psi(\phi)\phi-f(\Psi(\phi)),$ (4)

action (2) will take the following form

${\mathcal{A}}=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}\left[\phi{\mathcal{R}}-V(\phi)\right]+{\mathcal{A}}_{matter}(g_{\mu\nu},\psi),$
(5)

which corresponds to the Jordan frame representation of the action of Brans-
Dicke theory with Brans-Dicke parameter $\omega_{BD}=0$. Brans-Dicke theory
with $\omega_{BD}=0$ is sometimes called massive dilaton gravity Wands  which
was originally suggested in O Hanlon  in order to generate a Yukawa term in
the Newtonian limit. Extremizing the action yields the following field
equations as (we set $\kappa=8\pi G=1$ in the rest of this paper)

$G_{\mu\nu}=T^{({\rm eff})}_{\mu\nu},$ (6)

and

$3\Box\phi+2V(\phi)-\phi\frac{dV(\phi)}{d\phi}=T^{m},$ (7)

where $T^{m}$ stands for the trace of $T^{m}_{\mu\nu}$ and the subscript “$m$”
refers to the matter fields (fields other than $\phi$) and we have defined the
effective stress-energy tensor as

$T^{({\rm eff})}_{\mu\nu}=\frac{1}{\phi}\left(T^{\rm
m}_{\mu\nu}+T^{\phi}_{\mu\nu}\right),$ (8)

with

$T^{\phi}_{\mu\nu}=(\nabla_{\mu}\nabla_{\nu}\phi-
g_{\mu\nu}\Box\phi)-\frac{1}{2}g_{\mu\nu}V(\phi),$ (9)

and

$T^{\mu m}_{\nu}={\rm diag}\left(\rho_{m},p_{m},p_{m},p_{m}\right),$ (10)

being the stress-energy tensors of the scalar field and a perfect fluid,
respectively.

##  III Gravitational collapse of a homogeneous cloud with
$f({\mathcal{R}})=\alpha{\mathcal{R}}^{m}$

Let us now build and investigate a homogeneous class of collapsing models in
$f({\mathcal{R}})$ gravity with $m\neq 0$, where the trapping of light is
avoided till the formation of singularity, allowing the singularity to be
visible to outside observers. In order to achieve our purpose we examine a
spherically symmetric homogeneous scalar field, $\Phi=\Phi(\tau)$ originating
from geometry. Since the interior spacetime is a dynamical one, we
parameterize its line element as follows

$ds^{2}=-d\tau^{2}+a^{2}(\tau)(dr^{2}+r^{2}d\Omega^{2}),$ (11)

where $\tau$ is the proper time of a free falling observer whose geodesic
trajectories are distinguished by the comoving radial coordinate $r$ and
$d\Omega^{2}$ is the standard line element on the unit $2$-sphere. It is worth
mentioning that here we assume that starting from the homogeneous initial
data, the collapsing configuration remains homogeneous till the singularity is
formed. But as the collapse proceeds there may be some inhomogeneities
occurring throughout the collapse scenario, the existence of which can be
investigated by perturbation theory, that is, imposing inhomogeneous
perturbations on the energy density, scale factor and BD scalar field and then
see whether the terms rising from inhomogeneity are dominant in the formation
of the singularity or not. Here we do not deal with such an issue but for more
details the reader may consult per  and references there in. Since the
presence of matter acting as a “seed” field prompts the collapse of the BD
scalar field, we have considered perfect fluid models obeying barotropic
equation of state as

$p_{m}=w\rho_{m}.$ (12)

Using the conservation equation for the matter
($\nabla^{\alpha}T^{m}_{\alpha\beta}=0$) together with the use of above
equation, one gets the following relations between $\rho_{m},~{}p_{m}$ and the
scale factor as

$\rho_{m}=\rho_{{}_{{}_{0}}m}a^{-3(1+w)};~{}~{}p_{m}=w\rho_{{}_{{}_{0}}m}a^{-3(1+w)},$
(13)

where $\rho_{{}_{{}_{0}}m}=\rho_{m}(a=1)$, is the initial value of energy
density of matter on the collapsing volume. Making use of equation (8) and
equation (11) one finds the following equations for the effective stress-
energy tensor as

$\rho_{{}_{({\rm eff})}}=-T^{\tau}\,_{\tau}{}^{({\rm
eff})}=\frac{1}{\phi}\left(\rho_{m}+\rho_{{}_{\phi}}\right)=\frac{1}{\phi}\left[\rho_{m}-3\frac{\dot{a}}{a}\dot{\phi}+\frac{V(\phi)}{2}\right],$
(14)

and

$p_{{}_{({\rm eff})}}=T^{r}\,_{r}{}^{({\rm
eff})}=T^{\theta}\,_{\theta}{}^{({\rm eff})}=T^{\varphi}\,_{\varphi}{}^{({\rm
eff})}=\frac{1}{\phi}\left(p_{{}_{m}}+p_{\phi}\right)=\frac{1}{\phi}\left[p_{m}+2\frac{\dot{a}}{a}\dot{\phi}+\ddot{\phi}-\frac{V(\phi)}{2}\right],$
(15)

with all other off-diagonal terms being zero and the radial and tangential
profiles of pressure are equal due to the homogeneity and isotropy.
Substituting the line element (11) into Einstein’s equation one gets the
interior solution as

$\rho_{{}_{({\rm
eff})}}=3\frac{\dot{a}^{2}}{a^{2}}=\frac{{\mathcal{M}}^{\prime}}{R^{2}R^{\prime}};~{}~{}~{}~{}p_{{}_{({\rm
eff})}}=-\left[\left(\frac{\dot{a}}{{a}}\right)^{2}+2\frac{\ddot{a}}{a}\right]=-\frac{\dot{{\mathcal{M}}}}{R^{2}\dot{R}},$
(16) $\dot{R}^{2}=\frac{{\mathcal{M}}}{R},$ (17)

where ${\mathcal{M}(\tau,r)}$ rises as a free function from the integration of
Einstein’s field equation which can be interpreted physically as the total
mass within the collapsing cloud at a coordinate radius $r$ with
${\mathcal{M}}\geq 0$, and $R(\tau,r)=ra(\tau)$ is the physical area radius
for the volume labeled by the comoving coordinate $r$. From equation (14) and
first part of equation (16), one can solve for the mass function

${\mathcal{M}}=\frac{R^{3}}{3\phi}(\rho_{{}_{\phi}}+\rho_{m}).$ (18)

Using the above equation together with equation (17) we arrive at a relation
between $\dot{a}$ and the effective energy density as follows

$\dot{a}^{2}=\frac{a^{2}}{3\phi}(\rho_{{}_{\phi}}+\rho_{m}).$ (19)

Since we are concerned with a continual collapsing scenario, the time
derivative of the scale factor should be negative $(\dot{a}<0)$ implying that
the physical area radius of the collapsing volume for constant value of $r$
decreases monotonically. The singularity arising as the final state of
collapse at $\tau=\tau_{s}$ is given by $a(\tau_{s})=0$. On the other hand
when the scale factor and physical area radius of all the collapsing shells
vanish, the collapsing cloud has reached a singularity. A point at which the
energy density and pressure blows up, the Kretschmann scalar
${\mathcal{K}}=R^{abcd}R_{abcd}$ diverges and the normal differentiability and
manifold structures break down. In order to solve the field equations we
proceed by substituting for $\rho_{{}_{({\rm eff})}}$ and $p_{{}_{({\rm
eff})}}$ from equation (16) into equations (14) and (15) and rewrite them as
follows

$3\frac{\dot{a}^{2}}{a^{2}}=\frac{1}{\alpha
m\Phi}\left[\rho_{{}_{{}_{0}}m}a^{-3(1+w)}-3\alpha
m\frac{\dot{a}}{a}\dot{\Phi}+\frac{V(\Phi)}{2}\right],$ (20)
$-\left[\left(\frac{\dot{a}}{{a}}\right)^{2}+2\frac{\ddot{a}}{a}\right]=\frac{1}{\alpha
m\Phi}\left[w\rho_{{}_{{}_{0}}m}a^{-3(1+w)}+\alpha m\ddot{\Phi}+2\alpha
m\frac{\dot{a}}{a}\dot{\Phi}-\frac{V(\Phi)}{2}\right],$ (21)

where for later convenience we have rescaled the scalar field as $\phi=\alpha
m\Phi$ ($\alpha$ and $m$ are real constants), and by the virtue of equation
(4) the associated potential to $f({\mathcal{R}})=\alpha{\mathcal{R}}^{m}$ can
be fixed as

$V(\Phi)=\alpha(m-1)\Phi^{\frac{m}{m-1}};~{}~{}~{}~{}m\neq 1.$ (22)

Since the scalar field must diverge at the singularity we examine its behavior
by taking the following ansatz for the scalar field

$\Phi(\tau)=a^{\delta}(\tau),$ (23)

where $\delta$ is a constant whose sign decides the divergence of the scalar
field. Substituting the first and second time derivatives of the scale factor
from equations (20) and (21) into equation (7) together with the use of
equation (22) one finds

$a^{-\left[\delta+3(1+w)\right]}\left\\{\frac{2\rho_{{}_{{}_{0}}m}(\delta+3w-1)}{3\alpha
m\delta(2+\delta)}\right\\}+a^{\left[\frac{\delta}{m-1}\right]}\left\\{\frac{4+\delta(2m-1)-2m}{6m\delta+3m\delta^{2}}\right\\}=0.$
(24)

Matching the powers of scale factor in equation (24) we arrive at the
following expression for $\delta$

$\displaystyle\delta=\frac{3(1+w)(1-m)}{m},$ (25)

whence by substituting the above equation into the pair of square brackets in
equation (24) one gets the following equation to be satisfied by $m$

$\displaystyle\alpha\Big{\\{}3(1+w)+m\left[-13-9w+m(8+6w)\right]\Big{\\}}+2\rho_{{}_{{}_{0}m}}\left[4m-3(1+w)\right]=0.$
(26)

Solving the above equation, we find $m$ as a function of $\alpha$ and
$\rho_{{}_{{}_{0}m}}$ for $w$ as

$\displaystyle
m_{\pm}=\frac{13\alpha-8\rho_{{}_{{}_{0}m}}\pm\left[73\alpha^{2}-16\alpha\rho_{{}_{{}_{0}m}}+64\rho_{{}_{{}_{0}m}}^{2}\right]^{\frac{1}{2}}}{16\alpha},$
(27)

for $w=0$,

$\displaystyle
m_{\pm}=-\frac{-5\alpha+4\rho_{{}_{{}_{0}m}}\pm\left[13\alpha^{2}-16\alpha\rho_{{}_{{}_{0}m}}+16\rho_{{}_{{}_{0}m}}^{2}\right]^{\frac{1}{2}}}{6\alpha},$
(28)

for $w=-\frac{1}{3}$,

$\displaystyle
m_{\pm}=-\frac{-7\alpha+8\rho_{{}_{{}_{0}m}}\pm\left[33\alpha^{2}-80\alpha\rho_{{}_{{}_{0}m}}+64\rho_{{}_{{}_{0}m}}^{2}\right]^{\frac{1}{2}}}{8\alpha},$
(29)

for $w=-\frac{2}{3}$, and

$\displaystyle
m_{\pm}=\frac{4\alpha-2\rho_{{}_{{}_{0}m}}\pm\sqrt{2}\left[3\alpha^{2}+2\alpha\rho_{{}_{{}_{0}m}}+2\rho_{{}_{{}_{0}m}}^{2}\right]^{\frac{1}{2}}}{5\alpha},$
(30)

for $w=\frac{1}{3}$, where $\alpha\neq 0$.

##  IV Time Behavior Of The Scale Factor and singular epoch

One would like to study time behavior of the scale factor as the collapse
evolves, considering matter fields. If at time $\tau=\tau^{*}$ (or
equivalently for some $a=a^{*}$) the collapse begins, then by integrating
equation (20) together with the use of equations (22) and (23) near the
singularity with respect to time one gets the time behavior of the scale
factor as

$a(\tau)=\left[a^{\ast\frac{1}{2}\left(\delta+3(1+w)\right)}-\frac{1}{2}\sqrt{\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}}\left(\delta+3(1+w)\right)(\tau-\tau_{\ast})\right]^{\frac{2}{\delta+3(1+w)}},$
(31)

and the corresponding singular epoch as

$\tau_{s}=2\sqrt{\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}}\frac{a^{\ast\frac{1}{2}\left(\delta+3(1+w)\right)}}{\delta+3(1+w)}+\tau^{\ast},$
(32)

where the time $\tau_{s}$ corresponds to a vanishing scale factor. Thus the
collapse reaches the singularity in a finite proper time. This result for the
scale factor completes the interior solution within the collapsing cloud,
providing us with the required construction.

##  V The Conditions

We are now in a position to investigate the nature of singularity as the
endstate of a collapsing scenario. The singularity is called locally naked if
it is only visible to an observer being in the neighborhood of it (such a
singularity is necessarily covered by a spacetime event horizon) and is called
globally naked if there exists a family of future directed non-spacelike
geodesics reaching to the outside observers in the spacetime and terminating
at the past in the singularity. To investigate the nature of spacetime
singularity arising from the collapse procedure, we examine here whether such
a singularity could be naked, or necessarily covered within a spacetime event
horizon and if so under what conditions. Since $\alpha$ can be regarded as a
free parameter, we seek for the appropriate values of it satisfying the
following conditions

  * •

The ratio ${\mathcal{M}}/R$ stays less than unity till the singular epoch
which means that the singularity has been formed earlier than the formation of
the apparent horizon or the formation of trapped surfaces have been failed
till the singular epoch.

  * •

$\delta<0$ during the gravitational collapse scenario accompanied by
divergence of the scalar field.

  * •

For physical reason, weak energy condition, stated as $\rho_{{}_{({\rm
eff})}}\geq 0$ and $\rho_{{}_{({\rm eff})}}+p_{{}_{({\rm eff})}}\geq 0$ must
be satisfied during the dynamical evolution of the system.

  * •

The effective energy density and effective pressure blow up in the vicinity of
the singularity, the latter being negative during the evolution of the
collapse process, since the absence of trapped surfaces is accompanied by a
negative pressure. In fact the negativeness of the effective pressure implies
that $\dot{{\mathcal{M}}}<0$, that is, the mass contained in the collapsing
volume with comoving coordinate $r$ keeps decreasing leading to an outward
energy flux during the gravitational collapse scenario.

  * •

Kretschmann scalar diverges at the singular time and then converges to zero at
late times.

In order to determine whether the singularity is naked or not, one needs to
investigate the formation of trapped surfaces during the collapse procedure.
These surfaces are defined as compact two-dimensional (smooth) spacelike
surfaces such that both families of ingoing and outgoing null geodesics
orthogonal to them necessarily converge or the expansion parameter $\Theta$ of
the outgoing future-directed null geodesics is everywhere negative Frolov-
Malec . Consider a congruence of outgoing radial null geodesics having the
tangent vector $(V^{\tau},V^{r},0,0)$, where
$V^{\tau}=d\tau/dk~{}and~{}V^{r}=dr/dk$ and $k$ is an affine parameter along
the geodesics. For the spacetime metric (11), the geodesic expansion parameter
which is defined as the covariant divergence of the vector field $V^{\nu}$ is
given by Singh

$\Theta=\nabla_{\nu}V^{\nu}=\frac{2}{r}\left[1-\sqrt{\frac{{\mathcal{M}}}{R}}\right]V^{r}.$
(33)

If the null geodesics terminate at the singularity in the past with a definite
tangent, then at the singularity we have $\Theta>0$. If such family of curves
do not exist and the event horizon forms earlier than the singularity, a black
hole is formed. Utilizing equation (33), we now attempt to study the formation
of trapped surfaces during the dynamical evolution of the gravitational
collapse procedure. We show that physically, the formation of a black hole or
a naked singularity as the final state for the dynamical evolution is governed
by the rate of collapse and the presence of scalar field. It is seen that for
a specified range of variation of $\alpha$, the cosmic censorship conjecture
may be violated for all cases of matter considered below. In the following
subsections we consider first the four conditions mentioned in the beginning
of this section and postpone the last one to the next section. We begin by
calculating the ratio ${\mathcal{M}}/R$ in the general case which is
considered for the four cases of matter,
$w=\left\\{0,-\frac{1}{3},-\frac{2}{3},\frac{1}{3}\right\\}$ corresponding to
dust, cosmic strings, domain walls and radiation, respectively. By the virtue
of equation (17) we have

$\frac{{\mathcal{M}}}{R}=r^{2}\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta+3w+1)}.$ (34)

The weak energy condition which states that the energy density as measured by
any local observer must be non-negative can be written for any timelike vector
$V^{\mu}$ as follows

$T_{\mu\nu}V^{\mu}V^{\nu}\geq 0,$ (35)

whereby one gets the following conditions for the effective energy density
$(\rho_{{}_{({\rm eff})}}\geq 0)$

$\rho_{{}_{({\rm eff})}}=\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+3(w+1))}\geq 0,$ (36)

and the sum of effective energy density and pressure ($\rho_{{}_{({\rm
eff})}}+p_{{}_{({\rm eff})}}\geq 0$) as

$\displaystyle\rho_{{}_{({\rm eff})}}+p_{{}_{({\rm
eff})}}=\left(1+w+\frac{\delta}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-\left(\delta+3(w+1)\right)}\geq 0.$ (37)

Finally for the rate of change of mass function with respect to time one has

$\displaystyle\dot{{\mathcal{M}}}=r^{3}(\delta+3w)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta+3w+1)}\mid\dot{a}\mid,$ (38)

where the minus sign has been absorbed into $\dot{a}$. At the initial epoch
where $a(\tau^{*})=1$ there should not be any trapping of light, then by
assuming that $r=r_{b}$ is the boundary of the collapsing volume one may
easily see that for a suitable initial value of energy density the ratio
${\mathcal{M}}/R$ is less than unity at the initial time. This fact is in
accordance with the regularity conditions stating that if gravitationally
collapsing massive stars are to be modeled, then the energy density, pressure,
and other physical quantities must be finite and regular at the initial
spacelike hyper-surface from which the collapse commences. For the case of
homogeneous-density collapse the resulting singularity coincides with the
curves $R(\tau_{s},0)=0$ or $R(\tau_{s},r\neq 0)=0$, corresponding to a
central or non-central singularity, respectively. In the next subsections we
first consider the simpler case of non-central singularity and investigate
formation or otherwise of trapped surfaces for different values of $w$.

###  V.1 Dust ($w=0$)

For this case of matter we have the following relations (we set
$\rho_{{}_{{}_{0}m}}=1$ in the rest of this paper)

$\displaystyle\left\\{\begin{array}[]{ccccccc}\frac{{\mathcal{M}}}{R}=r^{2}\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta+1)},&&&&&&\\\ &&&&&&\\\ \rho_{{}_{({\rm
eff})}}+p_{{}_{({\rm
eff})}}=\left(1+\frac{\delta}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+3)},&&&&&&\left\\{\begin{array}[]{c}\rho_{{}_{({\rm
eff})}}=\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+3)},\\\ \\\ \\\ \\\
\dot{{\mathcal{M}}}=r^{3}\delta\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta+1)}\mid\dot{a}\mid.\\\ \end{array}\right.\\\
&&&&&&\\\ p_{{}_{({\rm
eff})}}=\frac{\delta}{3}\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+3)},&&&&&&\\\ \end{array}\right.$ (49)

If the interval for $\alpha$ is $-3.9<\alpha<-0.1$ and by picking out $m_{-}$
in equation (27), the corresponding parameter $\delta_{-}$ is always negative,
its absolute value varies between $1<\mid\delta_{-}\mid<3$ and its minimum and
maximum values are $\delta_{-}=-2.72484$ and $\delta_{-}=-1.00569$,
respectively. It is seen that for such values of $\alpha$ and $\delta_{-}$,
the ratio ${\mathcal{M}}/R$ stays less than unity and then the expansion
parameter is always positive up to the singularity, that is the singularity is
formed earlier than the formation of apparent horizon which is the boundary of
trapped surfaces. The negativeness of effective pressure for the allowed
values of $\alpha$ ensures that $\dot{{\mathcal{M}}}$ is negative as collapse
proceeds which means that the mass contained in the collapsing volume keeps
waning. Then there exists an outward energy flux which may be visible to
outside observers (for the case of globally naked singularity) since the
trapped surfaces do not form early enough to cover the singularity. In
addition, the validity of the weak energy condition can be easily checked by
the virtue of expressions obtained for $\rho_{{}_{({\rm eff})}}$ and
$\rho_{{}_{({\rm eff})}}+p_{{}_{({\rm eff})}}$.

###  V.2 Cosmic Strings ($w=-\frac{1}{3}$)

Cosmic strings are the result of hypothetical 1-dimensional (spatially)
topological defects which may have been constructed during a symmetry breaking
phase transition at the early Universe. The possibility of their existence was
first considered by Tom Kibble in 1976 TKible . A fluid of cosmic strings may
have an effective equation of state,
$p_{{}_{{}_{m}}}=-\frac{1}{3}\rho_{{}_{{}_{m}}}$, so one has the following
relations for this type of matter fluid as

$\displaystyle\left\\{\begin{array}[]{ccccccc}\frac{{\mathcal{M}}}{R}=r^{2}\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-\delta},&&&&&&\\\ &&&&&&\\\ \rho_{{}_{({\rm
eff})}}+p_{{}_{({\rm
eff})}}=\left(\frac{\delta+2}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+2)},&&&&&&\left\\{\begin{array}[]{c}\rho_{{}_{({\rm
eff})}}=\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+2)},\\\ \\\ \\\ \\\
\dot{{\mathcal{M}}}=r^{3}(\delta-1)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-\delta}\mid\dot{a}\mid.\\\ \end{array}\right.\\\
&&&&&&\\\ p_{{}_{({\rm
eff})}}=\left(\frac{\delta-1}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+2)},&&&&&&\\\ \end{array}\right.$ (60)

Choosing $m_{+}$ in equation (28), it can be seen that the valid range of
change for $\alpha$ is $-100<\alpha<-0.1$ upon which the absolute value of
parameter $\delta_{+}$ is restricted to vary between $0<\mid\delta_{+}\mid<2$
(in order to prevent the ratio ${\mathcal{M}}/R$ to be infinity,
$\delta_{+}=-1$ has to be excluded) and its minimum and maximum values are
$\delta_{+}=-1.86224$ and $\delta_{+}=-0.61558$, respectively. It should be
noted that one can choose the lower limit of $\alpha$ to be much less than
$-100$, but such a choice does not affect considerably the value of
$\delta_{+}$ and its magnitude remains close to $-0.6$. Taking these values
into account, the ratio ${\mathcal{M}}/R$ stays finite till the singular epoch
and causes the expansion parameter to be positive up to the singularity, and
if no trapped surfaces exist initially then none would form until the epoch
$a(\tau_{s})=0$ which is consistent with the fact that there exist families of
outgoing radial null geodesics emerging from the singularity. Also the weak
energy condition is satisfied ($\mid\delta_{+}\mid<2$) and the effective
pressure is negative (since $\delta_{+}<0$), consistent with the fact that
time derivative of mass function is negative i.e., the mass incorporated in
the region where the collapse procedure evolves keeps falling off. As a
result, there exists an outward energy flux during the collapse scenario which
may be visible to external Universe.

###  V.3 Domain Walls ($w=-\frac{2}{3}$)

Domain walls are two-dimensional objects that form when a discrete symmetry is
spontaneously broken at a phase transitionWeinberg . It has been noticed that
there exist a link between domain-walls and cosmologies such as brane
cosmologyArkani . The effective equation of state for a fluid of domain walls
may be $p_{{}_{{}_{m}}}=-\frac{2}{3}\rho_{{}_{{}_{m}}}$ and the mentioned
conditions for this case can be written as

$\displaystyle\left\\{\begin{array}[]{ccccccc}\frac{{\mathcal{M}}}{R}=r^{2}\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta-1)},&&&&&&\\\ &&&&&&\\\ \rho_{{}_{({\rm
eff})}}+p_{{}_{({\rm
eff})}}=\left(\frac{\delta+1}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+1)},&&&&&&\left\\{\begin{array}[]{c}\rho_{{}_{({\rm
eff})}}=\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+1)},\\\ \\\ \\\ \\\
\dot{{\mathcal{M}}}=r^{3}(\delta-2)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta-1)}\mid\dot{a}\mid.\\\ \end{array}\right.\\\
&&&&&&\\\ p_{{}_{({\rm
eff})}}=\left(\frac{\delta-2}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+1)},&&&&&&\\\ \end{array}\right.$ (71)

As long as $\alpha$ varies in the range $-100<\alpha<-0.1$ (lower limit of
$\alpha$ can be chosen much less than $-100$ but such a choice has no
noticeable influence on the maximum value of $\delta$), by choosing $m_{+}$ in
equation (29), the parameter $\delta_{+}$ is negative, its absolute value is
less than unity and its minimum and maximum values are $\delta_{+}=-0.953501$
and $\delta_{+}=-0.379572$, respectively. One then may easily see that at
initial epoch ($a(\tau^{*})=1$), the regularity condition (there should be no
trapped surfaces at the initial hyper-surface from which the collapse
commences) is satisfied and the ratio of mass function to physical area radius
of the collapsing volume is less than unity during the collapse procedure
denoting that the expansion parameter being positive up to the singularity. In
this case the collapse to a naked singularity may take place, where the
trapped surfaces do not form early enough or are avoided in the spacetime.
Also the mass contained in the collapsing ball reduces as the time advances
due to the fact that the effective pressure stays negative till the singular
epoch. It is obvious that for such values of $\alpha$ and $\delta_{+}$ the
weak energy condition is satisfied during the collapse scenario.

###  V.4 Radiation ($w=\frac{1}{3}$)

Finally, for this type of matter we have following relations for the said
conditions

$\displaystyle\left\\{\begin{array}[]{ccccccc}\frac{{\mathcal{M}}}{R}=r^{2}\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta+2)},&&&&&&\\\ &&&&&&\\\ \rho_{{}_{({\rm
eff})}}+p_{{}_{({\rm
eff})}}=\left(\frac{\delta+4}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+4)},&&&&&&\left\\{\begin{array}[]{c}\rho_{{}_{({\rm
eff})}}=\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+4)},\\\ \\\ \\\ \\\
\dot{{\mathcal{M}}}=r^{3}(\delta+1)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta+2)}\mid\dot{a}\mid.\\\ \end{array}\right.\\\
&&&&&&\\\ p_{{}_{({\rm
eff})}}=\left(\frac{\delta+1}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+4)},&&&&&&\\\ \end{array}\right.$ (82)

The suitable range of variation of $\alpha$ for $m_{-}$ in equation (30) is
$-0.66<\alpha<-0.01$ which makes the corresponding values of $\delta_{-}$ to
vary in the range $2<\mid\delta_{-}\mid<4$ with $\delta_{-}=-2.01007$ and
$\delta_{-}=-3.95037$ are the maximum and minimum values of this parameter
respectively. For such values of $\alpha$ and $\delta_{-}$ the ratio
${\mathcal{M}}/R$ stays less than unity as the collapse procedure ends. Then
the expansion parameter remains positive up to the singularity denoting that
the apparent horizon is failed to form. On the other hand, weak energy
condition is satisfied for these values of $\alpha$ and $\delta_{-}$ and the
effective pressure is negative till the singular epoch. It is worth noticing
that if one chooses the parameter $\alpha$ and the power of Ricci scalar in
equations (27)-(30) other than the ones determined in the above subsections,
weak energy condition together with the conditions on effective pressure and
the ratio ${\mathcal{M}}/R$ would be violated. The central singularity
occurring at $R=0$, $r=0$ can be naked if we have any future-directed outgoing
null geodesics terminating in the past at the singularity. In order to examine
the possibility of existence of such families let us introduce a new variable
$y=r^{\gamma}$ with $\gamma>1$ defined in such away that the ratio
$R^{\prime}/r^{\gamma-1}$ is a unique finite quantity in the limit
$r\rightarrow 0$. Now consider the outgoing radial null geodesic equation
which is given by

$\displaystyle\frac{d\tau}{dr}=a(\tau).$ (83)

In terms of variables $y=r^{\gamma}$ and $R$ the above equation reads

$\displaystyle\frac{dR}{dy}=\frac{1}{\gamma
r^{\gamma-1}}\left[\dot{R}\frac{d\tau}{dr}+R^{\prime}\right],$ (84)

whence using equation (17) we have

$\displaystyle\frac{dR}{dy}=\frac{R^{\prime}}{\gamma
y^{\frac{\gamma-1}{\gamma}}}\left[1-\sqrt{\frac{{\mathcal{M}}}{R}}\right].$
(85)

If there exist outgoing radial null geodesics in the past at the central
singularity which occurs at $\tau=\tau_{s}$, then along such geodesics we have
$R\rightarrow 0$ as $r\rightarrow 0$, or in terms of the variables $y$ and
$R$, the point $y=0$, $R=0$ is a singularity of the above first order
differential equation. For such a congruence of geodesics $dR/dy$ must be
positive up to the singularity. Then as long as $\alpha$,
$\rho_{{}_{{}_{0}m}}$, and $m$ satisfy the values which was determined
previously, trapped surfaces would fail to form as the collapse evolves.

##  VI nakedness of the singularity

The final fate of a continual gravitational collapse of a matter cloud ends in
either a black hole or a naked singularity where in the former there exists an
event horizon of spacetime developing earlier than the formation of the
singularity to cover it. Thus the regions of extreme curvatures and densities
are concealed from the outside observers. The event horizon or surface of a
black hole is defined as the boundary of the spacetime that is causally
connected to future null infinity Revisiting EH Finders , or in other word the
boundary between events from which light rays emitted inside this boundary
surface can not escape to future infinity while those emitted outside in a
suitable direction can. Both event and apparent horizons coincide in
stationary spacetimes, however this case is not generally true in dynamical
ones. Although the existence of an apparent horizon predicates the existence
of spacetime event horizon, the converse is not always true and the event
horizon may veil the singularity even if apparent horizon does not emerge on a
spatial slice. So far we have discussed the conditions under which the
collapse scenario ends in a locally naked singularity, that is, the
singularity is visible to an observer being in the neighborhood of it. In this
case, the trajectories coming out of the singularity do not actually come out
to a distant observer but fall back into the singularity again at a later time
without going out of the boundary of the star. Thus the locally naked
singularity could still be covered by the event horizon and only strong
version of cosmic censorship conjecture is violated but the week form of it is
intact TPSinghPSJ . However if the event horizon is delayed to form or the
singularity forms early enough before the formation of event horizon, then it
would be visible to external observers and thus a globally naked singularity
would born as the endstate of collapse rather than a black hole. Therefore in
such a situation curvature invariants namely the Kretschmann scalar should
increase near the singularity, diverge at the singular epoch and then converge
to zero at late times Kretschmann . In previous section we showed that for
suitable values of $\alpha$, trapping of light can be avoided which means that
the apparent horizon is failed to form. But since the failure of formation of
an apparent horizon does not necessarily bode the absence of an event horizon,
we investigate the nakedness of the singularity in spherically symmetric
collapse of a fluid by considering the behavior of Kretschmann scalar with
respect to time. For the line element (11) this quantity is given by

${\cal K}\equiv
R^{abcd}R_{abcd}=\frac{12}{a^{4}}\left[a^{2}\ddot{a}^{2}+\dot{a}^{4}\right].$
(86)

By the virtue of equation (31) one can easily obtain this quantity as a
function of time for
$w=\left\\{0,-\frac{1}{3},-\frac{2}{3},\frac{1}{3}\right\\}$ and the results
are sketched in Figures 1-4. It is seen that Kretschmann scalar diverges at
singular time and then tends to zero at late times. In order to better
understand the situation one should resort to critical behavior at the black
hole threshold. Such a behavior was discovered in gravitational collapse of a
spherically symmetric massless scalar field 14H , axisymmetric gravitational
waves 15H , spherical system of a radiation fluid 16H  and spherical system of
a perfect fluid obeying the equation of state $p=w\rho$ 17H . Consider a type
II critical collapse in which the black hole mass is scaled as
$M_{BH}\propto\mid p-p_{*}\mid^{\gamma}$, where $p$ parameterizes a family of
initial data sets evolving through Einstein’s equations, $p_{*}$ is a critical
value and $\gamma$ is a positive constant which is called a critical exponent
Kretschmann ; Harada-LivingCarsten . For a sufficiently large value of $p$ the
collapse procedure develops to a black hole and for a sufficiently small one
it evolves to a dispersion Harada-LivingCarsten . The boundary between these
two regimes is the black hole threshold. Now consider the limit from
supercritical collapse $(p>p_{*})$ to a critical collapse, i.e., $p\rightarrow
p_{*}$. In such a limit, the black hole mass tends to zero and the maximum
value of curvature diverges just outside the event horizon. Since we have
arbitrarily strong curvature outside the event horizon by fine-tuning, the
black hole threshold can be regarded as a globally naked singularity Harada-
LivingCarsten . Let us now consider the geometry of the exterior spacetime. In
order to complete the model we need to match the interior spacetime of the
dynamical collapse to a suitable exterior geometry. The Schwarzschild solution
is a useful model to describe the spacetime outside stars but the spacetime
outside such a star may be filled with radiated energy from the star in the
form of electromagnetic radiation. The Schwarzschild solution does not
describe this properly as it deals with an empty spacetime given by
$T_{ab}=0$. The spacetime outside a spherically symmetric star surrounded by
radiation emitted from the star is described by the Vaidya metricVaidya  which
can be given in the following form as

$ds^{2}_{out}=-\left[1-\frac{2M(r_{v},v)}{r_{v}}\right]dv^{2}-2dvdr_{v}+r_{v}^{2}d\Omega^{2},$
(87)

where $v$ is the retarded null coordinate, $r_{v}$ and $M(r_{v},v)$ are the
Vaidya radius and Vaidya mass, respectively. In what follows, we use the
Isreal-Darmois junction conditions to match the interior spacetime to a Vaidya
exterior geometry at the boundary hyper-surface $\Sigma$ given by $r=r_{b}$.
The spacetime metric just inside $\Sigma$ is given by

$ds^{2}_{in}=-d\tau^{2}+a^{2}(\tau)\left[dr^{2}+r_{b}^{2}d\Omega^{2}\right],$
(88)

whereby matching the area radius at the boundary one gets

$R(r_{b},\tau)=r_{v}(v).$ (89)

One then gets the interior and exterior metrics on the hyper-surface $\Sigma$
as follows

$ds^{2}_{\Sigma in}=-d\tau^{2}+a^{2}(\tau)r_{b}^{2}d\Omega^{2},$ (90)
$ds^{2}_{\Sigma
out}=-\left[1-\frac{2M(r_{v},v)}{r_{v}}+2\frac{dr_{v}}{dv}\right]dv^{2}+r_{v}^{2}d\Omega^{2},$
(91)

where matching the first fundamental form gives

$\left[\frac{dv}{d\tau}\right]_{\Sigma}=\frac{1}{\left[1-\frac{2M(r_{v},v)}{r_{v}}+2\frac{dr_{v}}{dv}\right]^{\frac{1}{2}}},~{}~{}~{}(r_{v})_{\Sigma}=r_{b}a(\tau).$
(92)

In order to match the second fundamental form (extrinsic curvature) for
interior and exterior spacetimes we need to find the unit vector field normal
to the hyper-surface $\Sigma$. We then proceed by taking into account the fact
that any spacetime metric can be written locally in the following form as

$ds^{2}=-\left(\alpha^{2}-\beta_{i}\beta^{i}\right)d\tau^{2}-2\beta_{i}dx^{i}d\tau+h_{ij}dx^{i}dx^{j},$
(93)

where $\alpha$, $\beta^{i}$, and $h_{ij}$ are the lapse function, shift
vector, and induced metric, respectively and $i$, $j$ run in $\\{1,2,3\\}$.
Comparing the above equation with equations (87) and (88) together with using
the following normalization condition for $n^{v}$ and $n^{r_{v}}$

$n^{v}n_{v}+n^{r_{v}}n_{r_{v}}=1,$ (94)

one gets the normal vector fields for the interior and exterior spacetimes as

$n^{a}_{in}=[0,a(\tau)^{-1},0,0],$ (95)
$n^{v}=-\frac{1}{\left[1-\frac{2M(r_{v},v)}{r_{v}}+2\frac{dr_{v}}{dv}\right]^{\frac{1}{2}}},~{}~{}~{}~{}~{}~{}~{}n^{r_{v}}=\frac{1-\frac{2M(r_{v},v)}{r_{v}}+\frac{dr_{v}}{dv}}{\left[1-\frac{2M(r_{v},v)}{r_{v}}+2\frac{dr_{v}}{dv}\right]^{\frac{1}{2}}}.$
(96)

The extrinsic curvature of the hyper-surface $\Sigma$ is defined as the Lie
derivative of the metric tensor with respect to the normal vector n, given by
the following relation as

$K_{ab}=\frac{1}{2}{\cal
L}_{\textbf{n}}g_{ab}=\frac{1}{2}\left[g_{ab,c}n^{c}+g_{cb}n^{c}_{,a}+g_{ac}n^{c}_{,b}\right],$
(97)

whereby the nonzero $\theta$ components of the extrinsic curvature read

$K^{in}_{\theta\theta}=r_{b}a(\tau),~{}~{}~{}~{}~{}~{}K^{out}_{\theta\theta}=r_{v}\frac{1-\frac{2M(r_{v},v)}{r_{v}}+\frac{dr_{v}}{dv}}{\left[1-\frac{2M(r_{v},v)}{r_{v}}+2\frac{dr_{v}}{dv}\right]^{\frac{1}{2}}}.$
(98)

Setting $\left[K^{in}_{\theta\theta}-K^{out}_{\theta\theta}\right]_{\Sigma}=0$
on the hyper-surface $\Sigma$, together by using equations (17) and (92) one
gets the following relation between the mass function and Vaidya mass on the
boundary as

${\cal M}(\tau,r_{b})=2M(r_{v},v).$ (99)

From equation (99) and equation (18) it is seen that the behavior of Vaidya
mass is decided by the allowed values of $\rho_{{}_{{}_{0}}m}$, $\alpha$ and
$m$ which prompt the gravitational collapse scenario to end in the formation
of a naked singularity. In order to get a relation describing the rate of
change of Vaidya mass with respect to $r_{v}$ one has to match the $\tau$
component of the extrinsic curvature on the hyper-surface $\Sigma$. Setting
$\left[K^{in}_{\tau\tau}-K^{out}_{\tau\tau}\right]=0$ we have

$M(r_{v},v)_{,r_{v}}=\frac{{\mathcal{M}}}{2r_{v}}+r_{b}^{2}a\ddot{a}.$ (100)

Now it can be seen that at the singular time, $\tau=\tau_{s}$ the ratio
$2M(r_{v},v)/r_{v}$ tends to zero. Thus the exterior spacetime at the singular
epoch reads

$ds^{2}=-dv^{2}-2dvdr_{v}+r_{v}^{2}d\Omega^{2},$ (101)

which describes a Minkowski spacetime in retarded null coordinates. Hence, the
exterior generalized Vaidya metric at singular time can be smoothly extended
to the Minkowski spacetime as the collapse completesPSJ . The occurrence of a
naked singularity as the final fate of a collapse scenario depends on the
existence of families of non-spacelike trajectories reaching faraway observers
and terminating in the past at the singularity. In order to show this we begin
by equation (92) and after using equation (99) we get

$\left[\frac{dv}{d\tau}\right]_{\Sigma}=\frac{1-r_{b}\dot{a}}{1-\frac{{\mathcal{M}}(\tau,r_{b})}{r_{v}}}.$
(102)

It is seen that imposing the null condition on the Vaidya metric leads to the
same relation as the above. This means that null geodesics can come out from
the singularity and reach distant observers before it evaporates into the free
space. On the other hand, since for the allowed values of $\alpha$,
$\rho_{{}_{{}_{0}m}}$ and $m$ formation of trapped surfaces in spacetime is
avoided and from another side the singularity emerges outside of the event
horizon, such a congruence of trajectories can be detected by the outside
observer.

Figure 1: The behavior of Kretschmann scalar (in units of $s^{-4}$) as a
function of proper time for $w=\frac{1}{3}$ (upper-left figure) and different
values of $\alpha$ and $\delta_{-}$. $\alpha=-0.65$, $m_{-}=2.02575$ and
$\delta_{-}=-2.02542$, Solid curve, $\alpha=-0.15$, $m_{-}=5.97355$ and
$\delta_{-}=-3.33038$, Dotted curve and $\alpha=-0.05$, $m_{-}=6.6128$ and
$\delta_{-}=-3.75922$, Dashed curve. For the initial energy density, scale
factor, and proper time we have adopted the values $\rho_{{}_{0m}}=1$,
$a^{\ast}=1$, and $\tau^{\ast}=0$, respectively. The corresponding singular
epoches are, $\tau_{s}=2.49674$ for Solid curve, $\tau_{s}=9.44081$ for Dotted
curve, and $\tau_{s}=27.8945$ for Dashed curve. Figure 2: The behavior of
Kretschmann scalar (in units of $s^{-4}$) as a function of proper time for
$w=0$ (upper-right figure) and different values of $\alpha$ and $\delta_{-}$.
$\alpha=-3.5$, $m_{-}=1.52406$ and $\delta_{-}=-1.03175$, Solid curve,
$\alpha=-2.5$, $m_{-}=1.60424$ and $\delta_{-}=-1.12996$, Dotted curve and
$\alpha=-1.5$, $m_{-}=1.8076$ and $\delta_{-}=-1.34034$, Dashed curve. For the
initial energy density, scale factor, and proper time we have adopted the
values $\rho_{{}_{0m}}=1$, $a^{\ast}=1$, and $\tau^{\ast}=0$, respectively.
The corresponding singular epoches are, $\tau_{s}=2.5083$ for Solid curve,
$\tau_{s}=2.70354$ for Dotted curve, and $\tau_{s}=3.19309$ for Dashed curve.
Figure 3: The behavior of Kretschmann scalar (in units of $s^{-4}$) as a
function of proper time for $w=-\frac{1}{3}$ (lower-left figure) and different
values of $\alpha$ and $\delta_{+}$. $\alpha=-90$, $m_{+}=1.44581$ and
$\delta_{+}=-0.616689$, Solid curve, $\alpha=-30$, $m_{+}=1.46909$ and
$\delta_{+}=-0.638609$, Dotted curve and $\alpha=-10$, $m_{+}=1.54031$ and
$\delta_{+}=-0.701562$, Dashed curve. For the initial energy density, scale
factor, and proper time we have adopted the values $\rho_{{}_{0m}}=1$,
$a^{\ast}=1$, and $\tau^{\ast}=0$, respectively. The corresponding singular
epoches are, $\tau_{s}=4.04176$ for Solid curve, $\tau_{s}=4.13327$ for Dotted
curve, and $\tau_{s}=4.38509$ for Dashed curve. Figure 4: The behavior of
Kretschmann scalar (in units of $s^{-4}$) as a function of proper time for
$w=-\frac{2}{3}$ (lower-right figure) and different values of $\alpha$ and
$\delta_{+}$. $\alpha=-100$, $m_{+}=1.61179$ and $\delta_{+}=-0.379572$, Solid
curve, $\alpha=-20$, $m_{+}=1.68699$ and $\delta_{+}=-0.407227$, Dotted curve
and $\alpha=-2$, $m_{+}=2.55425$ and $\delta_{+}=-0.608495$, Dashed curve. For
the initial energy density, scale factor, and proper time we have adopted the
values $\rho_{{}_{0m}}=1$, $a^{\ast}=1$, and $\tau^{\ast}=0$, respectively.
The corresponding singular epoches are, $\tau_{s}=10.2643$ for Solid curve,
$\tau_{s}=10.787$ for Dotted curve, and $\tau_{s}=16.808$ for Dashed curve.

##  VII Conclusion and outlook

One of the physical motivations for discussing naked singularities is that
these objects provide a useful laboratory for quantum gravity, since in such
ultra-strong gravity regions the length and time scales are comparable to the
Planck length and time. In other words, quantum effects occurring in such
super-dense regimes are no longer covered by the spacetime event horizon and
the chance to observe such effects in the universe is provided. An example is
quantum particle creation due to the formation of a naked singularity, which
has been studied in the literature QPC . During the past twenty years, cosmic
censorship conjecture has been extensively investigated in spherical models of
gravitational collapse of physically reasonable matter. The simplest of those
which has been scrutinized in detail and has been shown that both black holes
and naked singularities form from generic initial conditions is gravitational
collapse of a dust fluid. Dwivedi and Joshi in Dwivedi-Joshi  and Waugh and
Lake in Waugh-Lake  showed that a naked strong curvature singularity can be
formed in an inhomogeneous dust collapse and a self-similar one, respectively.
Also some examples of naked singularity formation in gravitational collapse of
a scalar field is given in Christodoulou . In this work we have studied the
process of gravitational collapse of a star where the matter fluid obeys the
barotropic equation of state $p=w\rho$, in the context of $f({\mathcal{R}})$
theories of gravity. Making use of metric formalism we wrote the action of
$f({\mathcal{R}})$ gravity as the Brans-Dicke one with vanishing coupling
parameter. Having solved the resulting field equations by taking the ansatz
(23) for scalar field we arrived at the expressions (27)-(30) for the exponent
of Ricci scalar as a function of $\alpha$ and initial energy density. In
Section V we imposed five conditions on the effective energy density and
pressure, the ratio ${\mathcal{M}}/R$, parameter $\delta$, time behavior of
the mass function, and Kretschmann scalar, the validity of which depends on
determining appropriate values of $\alpha$. As long as these conditions are
fulfilled the resulting singularity can be globally naked, i.e., ultra-dense
regions are no longer covered by a spacetime event horizon and physical
effects are allowed to be shared by the external Universe. It is worth
mentioning that there are future finite-time singularities in the dark energy
universe coming from modified gravity as well as in other dark energy
theories. Considering $f(\cal{R})$ gravity models that satisfy cosmological
viability conditions (chameleon mechanism), it is possible to show that
finite-time singularities emerge in several cases. Such singularities can be
classified according to the values of the scale factor $a(t)$, the density
$\rho$, and the pressure $p$ capo .

##  VIII Acknowledgments

The authors would like to express their sincere thanks to H. R. Sepangi for
useful discussions.

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