\section{Introduction}
The cosmic censorship conjecture (CCC) articulated 
by Penrose 
\cite{rp}
is fundamental to many aspects of theory and astrophysical 
applications of black hole physics today. Despite many attempts
over past decades no theoretical proof or even any 
satisfactory mathematical formulation of CCC is available 
as of today in the case of dynamical gravitational collapse.
 In the mean time, many authors have studied 
mainly spherical gravitational collapse of a massive matter 
cloud within the framework of general relativity. As the 
nuclear fuel of a massive star exhausts, it loses its 
equilibrium and gravity becomes the central dominant force 
which lends the star to its perpetual collapse. 
The gravitational collapse studies then show that the 
collapse end state is either a black hole (BH) or a naked 
singularity (NS), depending on the nature of the initial 
data from which the collapse evolves, arising from a regular 
initial state to the final super dense state 
\cite{jg,jr1,r1,dj2,wl,op,ct,rw}.

The works such as above analyze physically relevant
general matter fields that include most of the known
physical forms of matter like dust, perfect fluids, and such 
others. Wide classes of solutions to Einstein equations 
are shown in these cases to exist where the collapse end state 
is a black hole or naked singularity depending on the nature of 
initial data and the collapse evolutions allowed by the 
Einstein equations subject to regularity and physical 
reasonability conditions. 

An equation of state for the matter is, however, not always 
assumed in many of these investigations. As such, the physical 
characteristics of the matter field constituting the star 
are described by an equation of state relating the density 
and pressures, and therefore it is important to know if 
the naked singularities would still form when a suitable 
equation of state is assumed for the collapsing cloud. 
But it is to be noted that such an equation of state is 
not precisely known for a collapsing massive star which 
will be assuming super dense states of matter closer to 
the final later stages of the collapse, where the physical 
region has ultra-high densities, energies and pressures. 
It is far from clear as of now what would be a physically 
realistic equation of state that would describe such an 
ultra-high density region of collapse, and also whether 
the equation of state would remain unchanged or it would 
actually evolve and change as the collapse develops.

Nevertheless it is important to understand the final fate
of a gravitational collapse when a suitable equation of state 
is assumed, and to know if the naked singularities still
form as collapse final states. One can thus choose the equation of 
state to be linear isentropic or polytropic to describe the 
collapse of a massive star, right after its departure from the 
equilibrium configuration where gravity was balanced by the 
nuclear reactions taking place at the core region of the 
star. 

The gravitational collapse of a perfect fluid with a linear 
equation of state is of interest from both theoretical as well
as numerical relativity perspectives. 
Our motivation also comes from certain other questions 
such as, what if the value of $k$
increases in the range $0 \leq k \leq 1$ when a naked
singularity appears as collapse final state for a given 
spacetime dimension and for a critical positive tangent
to the singularity curve as to be discussed below. 
In such a case will the singularity sustain its 
nature, or a formation of event horizon shall precede the 
formation of singularity to preserve the CCC? We believe
as answer to these and similar issues would be important 
to understand better the physical conditions for collapse
and the role of an equation of state towards providing 
a suitable mathematical and physical formulation 
of CCC.


We therefore consider here a linear isentropic equation 
of state, $p = k \rho,\; \; 0 \leq k \leq 1$ in the study of 
spherical gravitational collapse of a perfect fluid. 
Self-similar perfect fluid collapse models with a linear 
equation of state were considered through numerical simulations 
by Ori and Piran
\cite{op}
and analytically by Joshi and Dwivedi 
\cite{dj2}
to show how black holes and naked singularities develop
as collapse final states in this scenario. 
Further, Goswami and Joshi
\cite{rgj} 
studied the case of an isentropic perfect fluid 
with a linear equation of state in four dimensional spacetime 
without the self-similarity assumption, wherein they
showed that the occurrence of BH and NS evolving from regular
initial data depends on the choice of rest of the free 
functions available. It was also proved that in a general 
$N$-dimensional dust collapse, the occurrence of NS can be 
removed when one goes to a higher dimensional spacetime, 
thus restoring CCC 
\cite{rgpj-1}.
Dadhich {\it et al} 
\cite{DSD} 
studied spherically symmetric collapse of  
a fluid with a non-vanishing radial pressure in higher 
dimensional spacetimes with an equation of state $p_{r} =k \rho$ 
using the construction of the Joshi-Dwivedi root equation 
that governs the nature (BH versus NS) of the central singularity. 
Also other cases of collapse with pressure have been studied 
by various authors to understand the collapse
final fate
\cite{JhHaGiHa}.


Our aim here is to give an explicit class of collapse 
models for the perfect fluid case with a linear equation of state,
and to analyze the same for the collapse end states in terms 
of formation of black holes and naked singularities. We analyze 
here the marginally bound case for the sake of simplicity 
and transparency and the results show that BH and NS form 
for a wide range of values of the parameter $k$ in the 
equation of state $p = k \rho$.   


The plan of work is as follows: In section II, 
we study Einstein field equations for spherically symmetric 
metric. We obtain the function $M(r,v)$
as a class of solutions to the constraint equation that 
enters through dynamical equations for $p=k \rho$. 
The dynamics of collapse for occurrence of NS/BH 
phases due to the effect of regular initial data is studied 
in the subsection II.1. The 
implication of the function $M(r,v)$ on singularity curve is  
investigated, which corroborate the study of apparent horizon in section III. 
The occurrence of NS/BH phases is investigated for increasing 
value of the parameter $k$, providing illustration
of its behavior in subsections III.1 and III.2. 
The conclusions are specified in section IV.


\section{Collapse with a linear equation of state}
The general spherically symmetric metric describing 
spacetime geometry of a collapsing cloud can be described in
comoving coordinates \ $(t, r, \theta, \phi)$ 
by the metric,
\begin{equation}
 ds^2 = - e^{2 \nu(t,r)}dt^2 +  e^{2 \psi(t,r)}dr^2 + R^2(t,r) d\Omega^2 \label {Dm01}
\end{equation}
where 
\[ d\Omega^2= d \theta^2 + \sin^2(\theta) d \phi^2 \] 
is the
metric on a two-sphere.
We aim here at the question, whether the choice of
an equation of state $ p = k \rho$, $k \in [0,1]$ contributes 
through the parameter $k$ in the development of BH/NS phases. 
The stress-energy tensor for the type I matter fields,
and in particular for perfect fluids, is in a diagonal form 
in a comoving coordinate system and is given by 
\cite{HE}, 
\[
 T^{t}_{t}  = -\rho, \; T^{r}_{r}  =
 T^{\theta}_{\theta} = T^{\phi}_{\phi}=\; p \;.
\] 
The quantities $\rho$ and $p$ are the energy
density and pressure respectively. We take the matter field to
satisfy the weak energy condition which implies ${{\rho} \ {\geq}}0$;
${\rho+p \ {\geq}} 0$. The linear equation of state for a 
perfect fluid is,
\begin{equation}
p(t,r) = k \ \rho(t,r) \ \text{where} \  k \in [0, 1]\; . \label{Des04}
\end{equation}
The Einstein field equations for the metric 
(\ref {Dm01}) are derived as 
\cite{rgj}
\begin{equation}
 \rho = \frac{{F}'}{ R^{2} R'} \;  \;
 = - \frac{1} {k} \frac{\dot{{F}}}{R^{2} \dot{R}} \label {Dsfe05} 
\end{equation}
\begin{equation}
 \nu \;' = - \frac{k}{k + 1} [ \ln(\rho)]' \label {Dsfe06} 
\end{equation}
\begin{equation}
 R' \dot{G} - 2 \dot{R} G \ {\nu}' = 0  \label {Dsfe07} 
\end{equation}
\begin{equation}
 G - H = 1 - \frac{{F}}{R} \; \; . \label {Dsfe08}
\end{equation}
Here $F = F(t, r)$ is an arbitrary function, and 
has an interpretation of the mass function for the cloud. 
It gives the total mass in a shell of comoving radius $r$ 
on any spacelike hypersurface $t = const.$ The energy conditions 
impose the restriction on $F$, namely that $F(t,r) \geq 0$. 
In order to preserve the regularity at the initial epoch, 
we have $F(t_{i}, 0) = 0$, that is, the mass function has to 
vanish at the center of the cloud. Since we are considering 
collapse, we have $\dot{R} < 0$, {\it i.e.} the physical radius 
$R$ of the collapsing cloud keeps decreasing in time and 
ultimately it reaches $R = 0$, which denotes the spacetime 
singularity where all the matter shells collapse to a zero 
physical radius. The functions $G$ and $H$ are defined as 
$G(t, r) = e^{-2 \psi} {R'}^2$ and $H(t, r) = e^{-2 \nu} \dot{R}^2$.


Let us define a new function  $v(t, r)$ by
$v(t, r)= R/r$,  
and now we use the scaling independence of the comoving
coordinate $r$ to write
\begin{equation}
R(t, r) = r \; v(t, r) \label{Df09}
\end{equation}
This gives $v(t_{i}, r) = 1, v(t_{s}(r), r) = 0$ and
the collapse condition is now written as $\dot{v} < 0$.
The time $t = t_{s}(r)$ corresponds to the shell-focusing 
singularity at $R = 0$, where all the matter shells collapse to a
vanishing physical radius. The introduction of the parameter 
$v$ as above allows us to distinguish the genuine spacetime 
singularity from the regular center at $r=0$, with $v = 1$ 
at the initial epoch including the center $r = 0$, and $v$ 
then decreases monotonically with time as collapse progresses 
to the value $v = 0$ at the singularity $R = 0$. 
At the regular center the mass function $F(t,r)$ behaves 
suitably so that the density remains finite
and regular there at all times till the occurrence 
of singular epoch. 
From the Einstein equation for density, we see that 
the mass function for the cloud can be written in general 
as,
\begin{equation}
{F}(t,r) = r^{3} M(r,v) \label {Dsm12}
\end{equation}
where $M(r,v)$ is regular and continuously twice
differentiable. Using equation (\ref {Dsm12}) in equations (\ref
{Dsfe05}), we obtain
\begin{equation}
\rho =  \frac{3 M + r [
M_{, r} + M_{, v}
 \ v']}{ v^{2} ( v + r v')}  
  =   - \frac{ M_{, v}} { k \ v^{2}}.  \label {Dsm13}
\end{equation}
Then as $ v \rightarrow 0 \; , \rho \rightarrow \infty $ and $p
\rightarrow \infty $, {\it  i.e.} both the density and pressure 
blow up at the singularity. 
We rearrange equation (\ref{Dsm13}) as follows,
\begin{equation}
 k r {M}_{,\ r} +[(k + 1) r v' + v]
{M}_{,\ v} = - 3 k M \label {Dsv14.1}
\end{equation}
The general solution of equation (\ref {Dsv14.1}) represents 
many classes of solutions but only
those classes are to be considered which satisfy
the energy conditions, which are regular and those which
give $\rho \rightarrow \infty$ as $v \rightarrow 0$. 
This means that the energy conditions and
equation of state $ p = k \rho$ isolate the class of functions
${M}(r,v)$ so that the mass function ${F}(t,r)$,
the metric function $\nu (t,r)$ and other concerned functions 
of $v$ evolve as the collapse begins according to the Type I
field equations. 


To obtain such a class of solutions, 
 we consider here the ansatz, 
\begin{equation}
\frac{r v'} {v} = g(v) \label{CON01}
\end{equation}
due to which the equation (\ref {Dsv14.1}) has a general 
solution of the form,
\begin{eqnarray}
M(r,v) && = f( r e^{- Z(v)} ) e^{- 3 Z(v)} \label{SoD14} \\
\text{where} \; Z(v) && = \int_{1}^{v} \frac {k} {v[(k+1) g(v) + 1]} 
dv \label{SoD15.1}
\end{eqnarray}
and $f$ is an arbitrary function. Therefore, we can choose 
a physically relevant $M$ as given by, 
\begin{equation}
M(r,v) = \texttt{M}_{o} \; sech{( r e^{- Z(v)} )} \; e^{- 3 Z(v)} \label{SoD15} 
\end{equation}
as a solution of equation (\ref {Dsv14.1}), 
where $\texttt{M}_{o}$ is a positive constant and 
the choice of the function $g(v)$ should be such that 
integrand of equation (\ref{SoD15.1}) should be at least 
piecewise continuous on $[0,1]$ and 
 also we note that 
$g(1) =0$ and $\ g(v)= v'(0, t_{s})$  for 
$v \rightarrow 0$, $r \rightarrow 0$. Also at $r=0$, 
$g(v)=0$ i.e. $g(v)$ vanishes at the central shell
at all regular points of the spacetime.
The introduction of the equation
(\ref{CON01}) demands its compatibility with other 
field equations or their subsequent 
equation (\ref{RD19}), which is discussed in the Appendix. 
 
 
From equation (\ref{SoD15}), we can write
\begin{eqnarray}
M_{0}(v)=M(0,v)= \texttt{M}_{o} e^{-3Z}, \; 
M_{1}(v)= \frac{M_{,r}(0,v)} {1 !}=0 \; \nonumber \\
\text{and} \;  
M_{2}(v)= \frac{M_{,rr}(0,v)} {2 !}= - \frac{\texttt{M}_{o} e^{-5Z}} 
{2}. \hspace{0.5in}\nonumber
\end{eqnarray}
Here $M_{1}(v)=0$ is in accordance with the requirement that 
the energy density has no cusps at the center and that its first
derivative vanishes at the center. The density profile 
for this class of models then takes the form,
\begin{eqnarray}
\rho(r,v) = \frac{ \texttt{M}_{o} e^{- 3 Z(v)} sech{( r e^{- Z(v)} )}} 
{v^{3}[(k+1) g(v) + 1] } \hspace{0.8in} \nonumber \\
\times [3 - r e^{-Z(v)} \tanh(r e^{-Z(v)})].  \label{Sr16} 
\end{eqnarray}
We note that at the initial epoch, $g(1)=0,Z(1) =0$, 
and the regular density distribution takes the form
\begin{equation}
\rho_{o}(r) = \texttt{M}_{o}\;  sech{( r )} 
[3 - r  \tanh(r)] . \label {Sr17}
\end{equation}
Such a density profile is decreasing away from the 
center $r=0$, which is a physically reasonable feature for
the collapsing matter cloud. For example, at the epoch 
$v=1$, $\rho_{o}(r)=0$ at (an approximate) value 
$r= 3.014482$. Hence we would take such a value of $r$ 
as the boundary of the cloud where the energy density 
is zero, and where the interior is matched to a suitable
exterior metric. 

At $r=0$ we have $g(v)=0$ and therefore 
$e^{- 3 Z(v)}= v^{- 3k} $, and the energy density at the center 
takes the form,
\[\rho(0,v) = \frac{3 \; \texttt{M}_{o} } {v^{3(k+1)} }, 
\]
therefore $\rho$ satisfies the energy condition $\rho \geq 0$ 
at all the epochs.
The gradient of $\rho(r,v)$ is obtained as
\begin{eqnarray}
\rho_{,r}(r,v) = \frac{- {\texttt{M}}_{o} e^{-4Z} sech(re^{-Z})} 
{ v^{3}[(k+1)g(v)+1]} \times \hspace{1in} \nonumber \\
  \left\{4 \tanh(r e^{-Z}) - r e^{-Z} [2 \tanh^{2}
(r e^{- Z}) - 1] \right\}. \nonumber
\end{eqnarray}
At the initial epoch $v=1$, we get $\rho_{,r}=0$ at the center 
$r=0$. The radial coordinate $r$ has the range 
$0 \leq r \leq r_{b}$ which defines the interior of the 
compact collapsing object, and wherein the density profile 
is smooth and monotone decreasing away from the center of 
the cloud as one moves towards the boundary of 
the cloud.



\subsection {Dynamics of collapse and the BH/NS phases}
Based on the above, we would now like to determine the 
final states for such a perfect fluid collapsing cloud in terms
of the black holes and naked singularities. As we shall
point out, the key factor that determines this final
outcome is the geometry of the trapped surfaces. If the 
trapped surfaces developed early enough in collapse, well 
before the formation of the singularity, then a black hole 
is the collapse final state. On the other hand, when the 
trapped surface formation is delayed as the collapse evolves,
due to the internal dynamics of the collapsing cloud, 
in that case the singularity is no longer fully covered 
by an event horizon, and we can have light rays and 
material particles escaping away from an arbitrary vicinity 
of the same, thus giving rise to a naked singularity.   

To decide and understand this trapped surface geometry,
we need to determine the behavior of the singularity curve
in the spacetime, which describes how and when different 
collapsing shells arrive at the zero physical radius, 
thus forming the spacetime singularity.

To determine this, 
firstly on integrating equation (\ref {Dsfe06}), we obtain 
the general metric function,
\begin{equation}
\nu(r,t) = - l\; \ln(\rho) + a(t). \label {Dcv14}
\end{equation}
where $ l = \frac{k} {k + 1} $. We can set $a(t) =0$. 
Let $A(r, v)$ be a suitably differentiable function 
defined by $A(r, v)_{,\ v} = {\nu \;'} / {R'}$, and on using 
equations (\ref{CON01}) and (\ref {Dcv14}), it takes the form
\begin{eqnarray}
A(r,v)_{\; , \; v} = && - \frac{l \; {\rho}'} { {\rho} R'}  \nonumber \\
= && \frac{l e^{-Z} \{4 \; T - re^{-Z} [2 \; T^{2}  -1] \} }
 {v[1+g(v)]\;  [3 - r e^{-Z} \; T ]}, 
\label {Decv15}
\end{eqnarray}
\begin{equation}
\left[A(r,v)_{\; , \; v} \right]_{v=1} 
= \frac{l  \{4 \; T(r,1) - r [2 \; T^{2}(r,1) -1] \} }
 { \; [3 - r T(r,1)]}  \label {Decv15.1}
\end{equation}
where $T(r,v) = \tanh(r e^{-Z})$ and $T(r,1) = \tanh(r)$.

Then on integrating equation (\ref{Decv15}), we obtain
\begin{eqnarray}
A(r,v)  
=\int_{v}^{1}  \frac{l e^{-Z} \left\{4 \; T - re^{-Z} [2 \; T^{2}  -1] \right\} }
 {v[1+g(v)]\;  [3 - r e^{-Z} \; T ]} dv 
\label {Decv16}
\end{eqnarray}
We note that if we explicitly find $A(r,v)$, we then have an 
exact solution of the Einstein field equations for the perfect
fluid collapse case. As is known this is of course a complex 
problem, and in the present scenario, the unknown nature of 
the function $g(v)$ forbid such a possibility of 
writing down the exact metric for the full spacetime. 
However, we can still write the metric in the neighborhood 
of the center of the cloud as we point out below. 

Our aim below is to find the BH/NS phases for collapse 
for a given initial data in terms of the function $M(r,v)$,
and we write $v' = v g(v)/r \equiv w(r,v)$. 
The mass function $M(r,v)$ satisfies the regularity
condition $M_{,r}(0,v)=0$ and we have $A \approx r$
closer to the center $r=0$. The function $A$ can be expressed 
as an expansion and it takes the form,
\begin{equation}
A(r, v) = r A_{1}(v)+ r^{2} A_{2}(v) + r^3 A_{3}(v) + 
{\mathcal{O}}(r^{4}), \label{ARDef1}
\end{equation}
with the first terms $A_{i}(v)$ given by
\begin{eqnarray}
A_{1}(v)= && \frac{A_{,r}(0,v)} {1!} =  
\frac{5 \;l} {3} \int_{v}^{1} \frac{e^{- 2 Z(v)}} {v[1 + g(v)]} dv,  \nonumber \\
A_{2}(v)= && \frac{A_{,rr}(0,v)} {2!} = 0,   \nonumber \\
A_{3}(v) = && \frac{A_{,rrr}(0,v)} {3!} = 
\frac{- 5l}{9} 
\int_{v}^{1} \frac{e^{- 4 Z(v)}} {v[1 + g(v)]} dv  \label{As17}
\end{eqnarray}


Also, the use of definition of $A(r,v)$ in equation (\ref
{Dsfe07}) yields
\begin{equation}
G(t,r) = \textbf{d}(r) \; e^{2rA} \label {Dg18}
\end{equation}
where $ \textbf{d}(r) $ is another arbitrary continuously
differentiable function of $r$. Following comparison with the dust
collapse models, we can write
\begin{equation}
\textbf{d}(r) = 1 + r^2 \; \textbf{b}(r)  \label {Db17}
\end{equation}
where the function $\textbf{b}(r)$ basically characterizes 
the energy distribution for the collapsing shells.
For the sake of simplicity and transparency of consideration
and for a better understanding of the model, 
we consider here only the `marginally bound case',
which corresponds to $\textbf{b}(r)=0$.


As we see now, having supplied an explicit 
function $M(r,v)$ which is a solution to the first two 
Einstein equations, that determines $\rho(r,v)$,
and therefore the metric function $\nu(r,v)$ is fully 
determined as above. The function $A(r,v)$ is 
also determined as above and is now a known function, 
and therefore $G(r,v)$ is also determined. Finally, the 
metric function $R$ is determined as we shall mention 
below.  We can write the metric in the 
neighborhood of the center 
$r=0$ of the cloud as,
\begin{equation}
 ds^2 = - \rho^{-2\; l}dt^2 +  \frac{R'^{2}}{ e^{2rA}} dr^2 
+ R^2(t,r) d\Omega^2 \label {Dms01}
\end{equation}


In order to determine the nature of the singularity
curve $t_s(r)$ which corresponds to the physical singularity 
at $R=0$ or $v=0$, 
the field equation (\ref{Dsfe08}) can be written in the form
\begin{equation}
\dot{R}^2 = e^{2 \nu} \left[\frac{F}{R} + G -1 \right] \label{RD18}
\end{equation}
Using equations (\ref{Dsm12}) and (\ref{Dg18}), this 
can be expressed as,
\begin{equation}
\dot{v} = - e^{\nu} \sqrt{\frac{M(r,v)}{v} 
+\frac{ e^{2rA} -1}{r^2} } \label{RD19}
\end{equation}
where the negative sign is chosen since for collapse
we must have $ \dot{v} < 0 $.
Integrating the above equation, we have
\begin{equation}
 t(r,v) = t_{i} + \int_{v}^{1}  \frac{ e^{- \nu} dv}
 { \sqrt{\frac{M(r,v)}{v} +\frac{ e^{2rA} -1}{r^2} }} \; \; .
\hspace{.4in} \label {Dsi21}
\end{equation}
In above equation, the variable $r$ is treated as a constant.
Regularity ensures that, in general, $t(r, v)$ is 
at least $C^{2}$ near the singularity and therefore can be 
expanded around the center as,
\begin{equation}
t(r,v) = t(0,v) + r \chi_{1}(v) + \frac{r^2}{2 !} 
\chi_{2}(v)+{\mathcal{O}}(r^3) \label {Dt22}
\end{equation}
where 
\[ \chi_{1}(v) = \frac{dt}{dr} \Big{|}_{r=0}, \; \; 
\chi_{2}(v) =  \frac{d^{2}t}{dr^{2}} \Big{|}_{r=0} \]
and using equations (\ref{Dcv14}) and (\ref{ARDef1}), we have
 \begin{eqnarray}
t(0,v) = t_{i} + (3 \texttt{M}_{o})^{l} 
  \int_{v}^{1}  \frac{ e^{-3\;l Z} \; dv}
 { v^{3\;l} \sqrt{\frac{M(0,v)}{v}  + 2 A_{1}(v) } }. \nonumber
\end{eqnarray}
The time for other collapsing shells to arrive at 
the singularity can be expressed by
\begin{equation}
 t_{s}(r)=t(r,0) = t_{i} + \int_{0}^{1}  \frac{ e^{- \nu} dv}
  {\sqrt{{\frac{M(r,v)}{v}} +\frac{ e^{2rA} -1}{r^2} }}.
\label {Dt26}
\end{equation}
Since the energy density has no
cusps at the center, that means $M_{1} = 0$. The singularity 
curve then takes the form,
\begin{equation}
t_{s}(r) = t_{0} + r \chi_{1}(0) + \frac{r^{2}}{2 !} 
\chi_{2}(0) + {\mathcal{O}}(r^{3}) ,
\end{equation}
where $t_{0} = t(0, 0)$ is the time at which the central 
shell becomes singular, and which is obtained  as 
\begin {eqnarray}
 t(0,0)= t_{i} + 3^{l} (\texttt{M}_{o})^{l-3/2} \times 
\hspace{1.5in} \nonumber \\
\left\{\frac{2 \; \texttt{M}_{o}} {3(1-k)}  
 - \frac{5}{3(k+1)}\left[ \frac{1}{5-k} - \frac{1}{5+3k}\right] 
\right\} . \nonumber
\end{eqnarray}
The time taken by the central shell to reach the 
singularity should be positive and finite, and hence we have 
the {\it model realistic condition} (MRC), namely that,
\begin{equation}
\left\{\frac{2 \; \texttt{M}_{o}} {3(1-k)}  
 - \frac{5}{3(k+1)}\left[ \frac{1}{5-k} - \frac{1}{5+3k}\right] 
\right\} > 0 \label{MRC01}
\end{equation}
and it must be finite for any 
$k \in [0,1)$. \\
\begin{figure}[t]
\begin{center}
\resizebox{7.0cm}{7.0cm}
{\includegraphics{MRC8.5-fig01.eps}}
\caption{The model realistic choice of $\texttt{M}_{o}$ 
is shown above. The allowed values for $\texttt{M}_{o}$ have 
an absolute maximum (and also a local maximum) around $k=0.4$. 
This maximum value of $\texttt{M}_{o}$ is a necessary
bound for the models for all $k \in [0,1)$.}
\end{center}
\end{figure}
Noting that $\nu \;'(0,v)=0$, 
we have
 \begin{equation}
 \chi_{1}(v)= - \frac{1}{2} \int_{v}^{1} 
\frac{2 e^{-\nu} A_{2}} {\left[\frac{M(0,v)} {v} 
+ 2 A_{1} \right]^{3/2}} dv \label{CHI1}
\end{equation}
and since $A_{2}(v)=0$,  we have $\chi_{1}(0)= 0$ for all $k$.
Also, at $r=0$, $g(v)=0$ and therefore $Z= k \ln(v)$ 
and $e^{-3Z}=v^{-3k}$, so we have
\[A_{1}(v)= \frac{5}{6(k+1)} [v^{-2k} -1], A_{3}(v)
= \frac{5}{36(k+1)}(1-v^{-4k}) \]
Since $\chi_{1}(0)=0$, using expressions for $M_{0}(v)$, 
$A_{1}(v)$ and $A_{3}(v)$, we compute $\chi_{2}(0)$ for an 
analysis of the nature of the central singularity
in terms of its visibility or otherwise. We get, 
\begin{widetext}
\begin{eqnarray}
 \chi_{2}(v) = - (3 \texttt{M}_{o})^{l}  
  \int_{v}^{1}  
  \left\{ \frac{5 l}{3} \frac{v^{-5k} } 
{ \left[\frac{M_{0}(v)} {v} + 2 A_{1}  \right]^{1/2}} 
   +  \frac{ [\frac{ M_{2}}{v} + 2 A_{3} + 2 A_{1}^{2}] \; v^{-3k}} 
{\left[\frac{M_{0}(v)} {v} + 2 A_{1}  \right]^{3/2}}
   \right\} dv  \label{ch201}
 \end{eqnarray}
It then follows that,
\begin{eqnarray}
  \chi_{2}(0)= - \frac{(3 \texttt{M}_{o})^{l}}{ \texttt{M}_{o}^{5/2}}  
  \left\{ \frac{- \texttt{M}_{o}^2}{3(k+1)} 
 +\frac{2 \; l \; \texttt{M}_{o} } {9 (1-k)} \left[ \frac{ c_{1}} 
{k+1} - \frac{10 \; l} {5-k} \right]  + \frac{800 \; c_{2} \; l^{2} }
{9(k+1) } \right\} \label{ch202}
\end{eqnarray}
where $c_{1}=\left[\frac{31 k^{2} + 136 k +25}{(5+3k)(5-k)}  \right]$ 
and $c_{2}=\frac{ 3 k^2 - 20 k +7}{(7+k)(7-3k)(7+5k)(7+9k)}$  
are functions of $k$.
\end{widetext}

 
Since the term $M_{0}(v)/v + 2 A_{1}(v) > 0$, and with the 
suitable continuity of the functions involved in the integrand of 
equation (\ref{ch201}), the sign of the term $\chi_{2}(v)$ totally 
depends on the behavior of the term $M_{2}/v + 2 A_{3}$. 
Therefore the positive or negative sign of $\chi_{2}(0)$, for 
various values of $k \in [0,1)$, depends on the behavior of the 
functions $M_{2}(v)$, and $A_{3}(v)$.
We also note that the positive constant $\texttt{M}_{o}$ 
appearing in the mass function $F(t,r)$ is related with density 
by the expression $ 3 \texttt{M}_{o}= \rho(0,1)$. So, 
$\texttt{M}_{o}$ in fact represents one-third of density 
of the star at the center taken on a cross section of an 
initial surface $v=1$ (at the beginning of the collapse at 
the instant $t=t_{i}$).
\begin{figure}[ht]
\begin{center}
\resizebox{7.0cm}{7.0cm}
{\includegraphics{Chi8.5-fig02.eps}}
\caption{For a fixed value of the central density 
corresponding to $\texttt{M}_{o}=1$, the quantity 
$\chi_{2}(0)$ takes positive values for $k=0$ and also 
for other small values of $k$. Then it crosses the $k$-axis 
and takes negative values as $k \rightarrow 1$. 
 }
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\resizebox{7.0cm}{7.0cm}
{\includegraphics{Chi8.5-fig03.eps}}
\caption{The nature of $\chi_{2}(0)$ is illustrated through the above
graphs. We note that if the model begins with a higher value 
of $\texttt{M}_{o}$, higher values of $k$ would satisfy the condition 
$\chi_{2}(0) > 0$. Thus, it is the initial central density that 
can specify and would have ramification on the nature of 
singularity in terms of its visibility or otherwise.}
\end{center}
\end{figure}
\section{The apparent horizon}
The significance of the term $\chi_2(0)$ towards deciding
the visibility or otherwise of the final spacetime 
singularity is as follows. As seen by the equation for $t_s(r)$ 
above for the singularity curve, this term and its sign basically 
decides the behavior of the singularity curve away from the 
center. In particular, when the singularity curve is increasing, 
it is shown that the apparent horizon which begins also at the 
central singularity at $r=0$, is increasing necessarily, and 
therefore it follows that there is no formation of trapped 
surfaces as the gravitational collapse of the matter cloud evolves, 
all the way till the singularity formation epoch. In such a
case, as has been shown earlier
\cite{dj2},
the singularity is necessarily locally naked and a family
of future directed null (and also timelike) geodesics come out
which terminate in the past at the singularity. Such a singularity
can also be globally visible depending on the nature of the
mass function away from the center
\cite{dj2}.    
On the other hand, when the singularity curve is
constant ($\chi_2$ and other higher order terms are all vanishing),
or would be decreasing, then a black hole will necessarily
form as the collapse final state.

 
Basically, a naked singularity could occur when a comoving 
observer (at a fixed radius $r$) does not encounter any 
trapped surfaces until the time of singularity formation. 
On the other hand, for a black hole the trapped surfaces
form before the singularity. Thus for a black hole to
form we require,
\begin{equation}\label{bh}
t_{\rm ah}(r)\leq t_0~\mbox{for}~r>0\,,~\mbox{near}~r=0\,.
\end{equation}
where $t_0$ is the epoch at which the central shell
hits the singularity.
In the general case this condition is violated when
$\chi_2>0$ as seen from the equation for $t_s(r)$ above. 
The apparent horizon curve that initiates at the singularity 
$r=0$ at the epoch $t_0$, then increases with
increasing $r$, moving to the future and we have 
$t_{\rm ah}>t_0$ for $r>0$ near the center. The behavior of 
the outgoing families of null geodesics has been analyzed in detail 
in these cases, and it is known that the geodesics terminate 
at the singularity in the past, which results in a naked singularity. 
In such cases then the extreme strong gravity regions can 
communicate with external observers. 

As these aspects are already discussed earlier 
as cited above, we therefore mainly focus here on determining 
the behavior of the quantity $\chi_2$ for the perfect fluid 
collapse with an equation of state $p = k \rho$, and we 
examine the impact of changing the values of the parameter 
$k$ on the visibility or otherwise of the singularity. 



\subsection{Dust collapse}
We note that for the well-studied case of the 
dust collapse, we have $p=0$ at $k=0$ and the MRC reduces 
to the form \ $\texttt{M}_{o} > 0$. We  have 
\begin{eqnarray}
\chi_{1}(0)= 0 \; \text{and} \; \chi_{2}^{k=0}(0) =  
\frac{1}{3 \; \texttt{M}_{o}^{1/2} } > 0. \hspace{0.5in}   
       \label{CHI203}
\end{eqnarray}
In this case a naked singularity exists for the given 
initial data of mass function.



\subsection {The Radiation collapse and other values of $k$}
The radiation collapse can be considered through 
the case $k=1/3$ for which the
MRC takes the form 
 $\texttt{M}_{o}> 5/84$ and we have
 $\chi_{1}(0)= 0 $
 and
 \begin{eqnarray}
 \chi_{2}^{k=1/3}(0) = -
 \frac{(3)^{1/4}} {\texttt{M}_{o}^{9/4}}
  \left[- \frac{1}{4}  \; \texttt{M}_{o}^{2} + \frac{121}{1008} \texttt{M}_{o}
    + \frac{5}{6864} \right] \hspace{0.2in} \label{CHI1302}
 \end{eqnarray}
Since the bracketed term has varying sign, 
we find that $\chi_{2}(0) < 0$ for small positive values 
of $\texttt{M}_{o}$, and for higher values of $\texttt{M}_{o}$, $\chi_{2}(0)$ 
changes its sign.
  

In general, to understand the nature of $\chi_{2}(0)$ 
in the interval $k \in [0,1)$,
we can obtain expressions for $\chi_{2}(0)$ for 
different values of $k$ in order to see how it
changes with a change in the parameter values $k$.
The graph of $\chi_{2}(0)$ for various specific $k$
values is given in Fig.3.
\begin{figure}[ht]
\begin{center}
\resizebox{8.70cm}{8.0cm}
{\includegraphics{I-Chi8.5-fig04.eps}}
\caption{An implicit graph of $\chi_{2}(0)$ specifies the 
regions for formation of naked singularity and black hole,
as determined by the variation in parameters 
$\texttt{M}_{o}$ and $k$. }
\end{center}
\end{figure}


The equation of apparent horizon in a
spherically symmetric spacetime is given by 
\ $F/R =1$. Use of equations (\ref{Dsm12}) and (\ref{SoD15}) 
give us the expression
\[ r^{2} \texttt{M}_{o} \ sech(r e^{-Z(v)}) e^{-3 Z(v)} = v \]
It is clear that as $v \rightarrow 0$, we must have 
$r \rightarrow 0$ on the apparent horizon. Therefore, the only 
singularity that can eventually be visible is that at the 
center of the collapsing cloud. Certainly, it is not possible 
to satisfy $1 - F/R \ > 0$, {\it  i.e.}
\[1- \frac{ r^{2} \texttt{M}_{o} sech(r e^{-Z(v)}) e^{-3 Z(v)} } 
{v} < 0 \; \text{as}\; v \rightarrow 0 \; \text{for}\; r >0\]
near the singularity and away from center, that is 
with $r > 0$. It follows that the region surrounding the 
singularity cannot be timelike, and therefore any singularity 
that might eventually form near the center with $r > 0$ 
must not be visible.


\section{Conclusions}
The investigation of gravitational collapse with a linear 
equation of state has revealed role of the parameter $k$ in deciding 
the nature of central singularity for the given initial data 
set of the mass function $M(r,v)$. We examined here the marginally
bound case with $b(r)=0$. 

In particular, we gave here an explicit class of mass 
functions that satisfies the Einstein equations and the metric 
obtained represents a class of solutions for a unique 
choice of functions $g(v)$, representing the exact solution 
for a perfect fluid sphere collapse with an isentropic 
equation of state. 

The occurrence of a locally naked singularity as collapse 
end state is shown for these collapse models for  
a wide variety of values of $k \in [0,1)$, and also formation of 
black hole is seen for a range of $k$ values for this 
class of mass functions which are an appropriate choice of 
$M(r,v)$ that are solutions to the Einstein field equations. 

We thus see, through an explicit demonstration, that
both black holes and naked singularities arise naturally 
as collapse final states for perfect fluid collapse
with a linear isentropic equation of state. The class of
solutions we gave here have several intriguing properties
some of which we discussed here. 
It is thus seen that the choice of $k \in [0,1)$ 
influences and contributes to decide in the formation of 
NS/BH phases of gravitational collapse. The model 
realistic condition discussed here emphasizes the role 
of pressure in the gravitational collapse due to which 
collapse could stop momentarily, and a bounce may take place. 
In other cases the cloud would collapse continually 
without a bounce. 

We note that the present model can be extended for a 
more general case with $b(r) \neq 0$, and the results on the 
same shall be reported elsewhere.  

{\bf{Acknowledgement}}: \\
We like to thank Daniele Malafarina for his valuable comments. 
Sanjay Sarwe acknowledges the facilities extended by IUCAA, Pune, 
while part of this work was being completed.

\begin {references}
\bibitem{rp} {R. Penrose},  Riv.  Nuovo Cimento Soc. Ital. Fis. {\bf 1}, 252
 (1969); `Singularities in Cosmology' in {\it General Relativity, 
an Einstein Centenary
 Volume}, edited by S. W. Hawking and W. Israel (Cambridge
 University Press, Cambridge, England, 1979).
\bibitem{jg} {P. S. Joshi and R. Goswami}, 
 {Physical Review D {\bf 69}, 064027 (2004)}
\bibitem{jr1} {R. Goswami and P. S. Joshi}, 
{Class. Quantum Grav.} {\bf 19}, 5229-5234 (2002).
\bibitem{r1}  {C. J. S. Clarke},
Class. Quantum Grav.  {\bf 10}, 1375 (1994); 
{R. M. Wald},  gr-qc/9710068; 
{S. Jhingan and G. Magli}, gr-qc/9902041; 
{T. P. Singh},
 {J. Astrophys. and Astron.} {\bf 20}, 221 (1999).
\bibitem{op} {A. Ori and T. Piran}, 
 {Phys. Rev. Lett.} {\bf 59}, 2137 (1987).
\bibitem{dj2} {P. S. Joshi and I. H. Dwivedi},
 {Commun. Math. Phys.},  {\bf 146}, 333 (1992);
  Lett. Math Phys. {\bf 27}, 235 (1993);
 Commun. Math. Phys. {\bf 166} 117-128 (1994);
{Class. Quantum Grav.} \textbf{16}, 41-59 (1999).
\bibitem{wl} {B. Waugh and K. Lake},
 {Phys. Rev. D} {\bf 40}, 2137 (1989);
{Phys. Rev. D} {\bf 42}, 1068 (1990).
\bibitem{ct} {M. E. Cahill and A. H. Taub}, 
   {Commun. Math. Phys.} {\bf 21}, 1 (1971). 
\bibitem{rw} {J. F. V. Rocha, A. Wang and N. O.~Santos}, 
  {Phys. Lett. A} {\bf 255}, 213 (1999).
\bibitem{rgj} P. S. Joshi and R. Goswami,
{Class. Quantum Grav} {\bf 21}, 3645 (2004).  
\bibitem{rgpj-1} {R. Goswami and P. S. Joshi},
{Phys. Rev. D} {\bf{69}}, 104002 (2004).
\bibitem{DSD} {N. Dadhich, S. G. Ghosh and D. W. Deshkar}
 Int. J. Mod. Phys. A {\bf 20}, 1495-1502 (2005).
 \bibitem{JhHaGiHa} S. Jhingan and G. Magli, {Phys. Rev. D} {\bf {61} }, 124006 (2000);
  T. Harada, {Phys. Rev. D} {\bf 58}, 104015 (1998);
  R. Giamb\`{o}, F. Giannoni, G. Magli and P. Piccione, {Gen. Rel. Grav.} {\bf 36}, 1279 (2004);  T. Harada, K. Nakao and H. Iguchi, {Class. Quantum Grav.} \textbf{16}, 2785 (1999).
 \bibitem{HE} {S.W. Hawking and G.F.R. Ellis},{\it The Large Scale Structure of 
Space-time} (Cambridge University Press, Cambridge, 1973).
\end {references}


\section*{Appendix-\\
Compatibility condition}
The introduction of the equation
(\ref{CON01}) demands its compatibility with other field equations or their subsequent 
equation (\ref{RD19}). So, we consider $v' = v g(v)/r = W(r,v)$ and $\dot{v} = U(r,v)$. The condition of compatibility for non-linear partial differential equations of order one yields,
\begin{equation}
W U_{,v} -  U W_{,v} = - U_{,r}. \label{AC-01}
\end{equation}
where $U(r,v) = - e^{\nu(r,v)} \sqrt{D(r,v)} $ and 
\begin{equation}
 D(r,v) = \frac{M(r,v)}{v} + \frac{e^{2rA(r,v)}-1} {r^2} \label{AC-02}
\end{equation}
Equation (\ref{AC-01}) takes the form
\begin{eqnarray}
-\frac{e^{\nu}}{ 2 r \sqrt{D}} \Big[ 2D \left( v g(v) \; \nu_{,v} -g(v) 
- v \frac{dg}{dv}
+ r \; \nu_{,r} \right) \nonumber \\
+ v g(v) D_{,v} + r D_{,r}\Big] =0 \hspace{0.6in} \label{AC-03}
\end{eqnarray}
Using equation (\ref{AC-02}), and values of $\nu_{,r}$ , $\nu_{,v}$ ( and subsequently
 $\rho$, $\rho_{,v}$ and $Z_{,v}$ ) above equation becomes
\begin{eqnarray}
\frac{r^2 v^2 (1+g(v)) D}{Y}\; \frac{dg}{dv}  - r^3 v^2 A_{,v} D -\frac{r^3 M_{,r}}{2} \hspace{0.36in} \nonumber \\
+ \; r^2 v  g(v) D
\Big\{ \frac{l k X[ X[1- \tanh^{2}(X)] + \tanh(X) ]} {Y[3-X \tanh(X)]} \nonumber \\
+(1-3l) + lv \frac{M_{,v}}{M} - rv A_{,v} \Big\}  \nonumber \\
 +\; \frac{rg(v)}{2} [rM-2v^2 e^{2rA} A_{,v} -rvM_{,v}]  \hspace{0.72in} \nonumber \\
 - \; rve^{2rA} (A + rA_{,r}) + v(e^{2rA}-1)=0. \hspace{0.54in}  \label{AC-04}
\end{eqnarray}
where $X= re^{-Z(v)}$ and $Y=[1+(k+1)g(v)]$. Since $Z(v)$ and $A_{,r}$
are integration terms, therefore, above equation is an integro-differential equation.
 The dependent function $g(v)$ can be obtained as its solution in terms of $v$'s and integration constant. In turn, $v'$ will have a regulated form consistent with the field equations. So, $v'$ and $\dot{v}$ given by equations (\ref{CON01}) and
(\ref{RD19}) respectively, together determine $v$, and hence $R(t,r)$
 to obtain the requisite exact solution of the Einstein's field equations. \\
 
 In the case of dust model, for $k=0$, equation (\ref{AC-04}) gives
 \begin{equation}
 v \frac{dg}{dv} + \frac{3}{2} g(v)= -\frac{ r \; \tanh(r)} {2}  \label{AC-05}
 \end{equation}
 The solution of above equation with treating $r$-terms as constant and using initial condition $g(1)=0$ is obtained as
 \begin{equation}
 g(v)= \frac{r \tanh(r) }{3} \left[\frac{1}{v^{3/2}} -1 \right] \label{AC-06}
 \end{equation}
 and therefore, 
 \begin{equation}
 v'= v \frac{ \tanh(r) }{3} \left[\frac{1}{v^{3/2}} -1 \right]. \label{AC-07}
 \end{equation}
 On integrating it with respect to $r$, we have
 \begin{equation}
 v^{3/2}(t,r) = 1 -\sqrt{sech(r)} \; S(t)^{-1/2} \label{AC-08}
 \end{equation}
 where $S(t)$ is due to a constant of integration, and it can be defined aptly to satisfy
 the requisite boundary conditions. As expected above equation satisfies the equation 
 (\ref{RD19}) for $k=0$,
 \[ \dot{v}= - \sqrt{ \frac{M_{o} sech(r) } {v} } \;\; \text{with} \; \;
 \dot{S}(t) = - 3 \sqrt{M_{o}} S(t)^{3/2}.\]
 Thus, condition of compatibility has established that definition of $v'$ 
 given in equation (\ref{CON01}) is consistent with field equations. 
 Its use, determines the form of $g(v)$ and therefore role of $v'$ is further
 become more evident in the analysis of existence of naked singularity. 

\end {document}
    
