The generalized Vaidya spacetime with polytropic equation of state

Vitalii Vertogradov

Physics department, Herzen state Pedagogical University of Russia,

48 Moika Emb., Saint Petersburg 191186, Russia

SPb branch of SAO RAS, 65 Pulkovskoe Rd, Saint Petersburg 196140, Russia

<EMAIL_ADDRESS>
Abstract. The process of the gravitational collapse might lead not only to a
black hole but also to naked singularity formation. In this paper, we consider
the generalized Vaidya spacetime with polytropic and generalized polytropic
equations of state. We solve the Einstein and Einstein-Maxwell equations to
obtain the explicit form of a mass function. The process of the gravitational
collapse has been then considered. We have found out that the conditions of
the naked singularity formation don’t depend on the polytropic index.

Key words: Generalized Vaidya spacetime, Gravitational collapse, Polytropic
equation of state, Naked singularity, black hole.

## Introduction

Nowadays, we have strong experimental evidence of the existence of a black
hole through gravitational wave detection [1, 2] and the image of a black hole
shadow [3, 4, 5]. When a star is at the end of its lifecycle i.e. when all
fuel that supports its stability is exhausted, the star undergoes the
gravitational collapse. Depending on the mass this process can lead to white
dwarf, neutron stars, or black hole formations. The cosmic censorship
conjecture [CCC] states that any singularity must be covered with a horizon.
However, it has been shown that under realistic conditions, the result of
continuous gravitational collapse might be a naked singularity which leads to
the CCC violation [6, 7]. A thorough investigation of different models
revealed that under some physically relevant conditions, the naked singularity
might be the result of the gravitational collapse [8, 9, 10, 11, 12].
Moreover, the shadow that can cast a naked singularity can’t be distinguished
from a black hole one [13].

Oppenheimer and Snyder [14] built the model of the gravitational collapse with
the assumption of the pressureless homogeneous matter. Homogeneous dust
collapse always leads to a black hole formation. However, if one considers
inhomogeneous dust distribution then the result of such collapse might be a
naked singularity [15, 16]. We say that a naked singularity might form during
the gravitational collapse if the following conditions are held:

  * •

The time of the naked singularity formation is less than the time of the
apparent horizon formation;

  * •

there exists a family of non-spacelike, future-directed geodesics which
terminate in the central singularity in the past.

Vaidya spacetime [17], alternatively referred to as radiating Schwarzschild
black hole, is one of the earliest examples of CCC violation [18]. The Vaidya
spacetime is widely used in many astrophysical applications with strong
gravitational fields. In general relativity, this spacetime assumed added
importance with the completion of the junction conditions at the surface of
the star by Santos [19]. The pressure at the surface is non-zero, and the star
dissipates energy in the form of heat flux. This made it possible to study
dissipation and physical features associated with gravitational collapse, as
shown by Herrera et al. [20, 21, 22]. The backreaction of the accreting matter
leads to the vaidya spacetime in spherically-symmetric case [23]. The question
about the dynamical shadow formation in Vaidya and charged Vaidya spacetimes
are discussed in [24, 25, 26]. The horizon structure and entropy of this
solution are investigated for an empty background in [27, 28, 29], for Vaidya
surrounded by cosmological fields in [30, 31, 32]. Some recent studies of the
temperature properties inside the radiating star have been done in [33, 34,
35]. The Vaidya spacetime can be extended to include both null dust and null
string fluids leading to the generalized Vaidya spacetime [36]. A detailed
investigation of the properties of the generalized Vaidya spacetime can be
found in [37, 38, 39]. The generalized Vaidya spacetime has been used to
investigate gravitational collapse [40, 41, 42, 43, 44, 45, 46]. The conformal
symmetries and embedding and other properties are discussed in [47, 48, 49,
50, 51]. The generalized Vaidya spacetime has the off-diagonal term which may
lead to the negative energy states for a particle like in Kerr spacetime [52].
The absence of such particles has been proven in [53]. However, there is a
similar effect for the charged particles and a generalized Penrose process may
take place in static Reissner-Nordstrom [54] and dynamically charged Vaidya
[55] cases. The forces in Vaidya spacetime are discussed in [56, 57].
Recently, a new generalization of Vaidya spacetime has been found by the
gravitational decoupling method [58] which can describe the Vaidya black hole
distorted by dark matter.

The thorough investigation of the gravitational collapse of the generalized
Vaidya spacetime revealed the possibility of the naked singularity formation
[40] for general matter distribution. It is well-known that the presence of an
equation of state introduces a differential relation for a mass function.
There has been considered a barotropic [37] and Hagedorn [8] equations of
state. An equation of state is a constitutive relation that provides the link
between some state quantities describing the system. Typically, for a
collapsing matter an equation of state is provided once the pressure can be
expressed as a function of the energy density $P=P(\rho)$. The equations of
state that are usually considered for perfect fluids at equilibrium are two. A
barotropic relation $P=\alpha\rho$ and polytropic one $P=\alpha\rho^{\gamma}$.
The study of the collapse of a perfect fluid with the polytropic equation of
state is more complicated. In both cases, solving the differential equation
for the mass function $M(v,r)$ might prove to be unattainable.

The polytropic equation of state has been used in describing the physical
properties of stellar objects and anisotropic relativistic stars [59, 60, 61,
62, 63]. Here, we have decided to consider the simple model of the
gravitational collapse with this equation of state. Motivated by the fact that
the Einstein equations for generalized Vaidya spacetime are linear in mass, we
might try to find the analytical solution by connecting pressure $P$ and
energy density $\rho$ via a polytropic equation of state. The solution of the
Einstein field equations in this case can be expressed in combination.
Moreover, it turns out that the Einstein-Maxwell field equations can be solved
analytically for this case. When we find the explicit expression for the mass
function, then one can try to consider the question about the gravitational
collapse of generalized Vaidya spacetime with the polytropic equation of state
to find out how the polytropic index affects the end state of such process.

The polytropic equation of state has been used in describing the physical
properties of stellar objects and anisotropic relativistic stars [59, 60, 61,
62, 63]. Here, we have decided to consider the simple model of the
gravitational collapse with this equation of state. Motivated by the fact that
the Einstein equations for generalized Vaidya spacetime are linear in mass, we
might try to find the analytical solution by connecting pressure $P$ and
energy density $\rho$ via a polytropic equation of state. The solution of the
Einstein field equations in this case can be expressed in combination.
Moreover, it turns out that the Einstein-Maxwell field equations can be solved
analytically for this case. When we find the explicit expression for the mass
function, then one can try to consider the question about the gravitational
collapse of generalized Vaidya spacetime with the polytropic equation of state
to find out how the polytropic index affects the end state of such process.

$M^{\prime}\equiv\frac{\partial M}{\partial r}\,,\dot{M}\equiv\frac{\partial
M}{\partial v}\,.$ (1.1)

## Generalized Vaidya solution with polytropic equation of state

The Vaidya spacetime can be extended to include both null dust and null string
fluids leading to the generalized Vaidya spacetime [36]. It has the form

$ds^{2}=-\left(1-\frac{2M(v,r)}{r}\right)dv^{2}+2\varepsilon
dvdr+r^{2}d\Omega^{2}\,.$ (2.2)

This paper is organized as follows: in sec 2. we consider the exact solution
of the Einstein field equations for the generalized Vaidya spacetime with the
polytropic equation of state and naked singularity formation discussed. In
sec. 3 we consider the system of the Einstein-Maxwell equations and find the
analytical solution. Sec. 4 is the conclusion.

Throughout the paper, we will use the geometrized system of units in which
$G=1=c$, also we use the signature $-+++$. We also introduce the following
notations:

Where $M(v,r)$ is the mass function depending on both the areal coordinate $r$
and Eddington time $v$. $\varepsilon=\pm 1$ depending on ingoing or outgoing
radiation. We are interested in gravitational collapse and put
$\varepsilon=+1$. $v$ is Eddington advanced ($\varepsilon=+1$) or retarded
($\varepsilon=-1$) time. $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$
is the metric on unit two-sphere. When the mass function depends only on the
time $v$ then the metric (2.2) reduces to the usual Vaidya spacetime with
energy-momentum tensor representing the null dust

$T^{(ND)}_{ik}=\mu(v,r)l_{i}l_{k}\,.$ (2.3)

Here $l_{i}$ is the nul vector ($l^{i}l_{i}=0$) and $\mu$ is the energy
dencity of the null dust

$\mu=\frac{2\dot{M}}{r^{2}}\,.$ (2.4)

The generalized Vaidya spacetime (2.2) is supported with the energy-momentum
tensor which represents the combination of the null dust (2.3) and the null
strings

$T^{(NS)}_{ik}=(\rho+P)(l_{i}n_{k}+l_{k}n_{i})+Pg_{ik}\,.$ (2.5)

Where $n^{i}$ is another null vector with properties

$n^{i}n_{i}=0\quad,\quad n^{i}l_{i}=-1\,.$ (2.6)

$\rho$ and $P$ are the energy density and pressure respectivelly

$\begin{split}\rho=\frac{2M^{\prime}}{r^{2}}\,,\\\
P=-\frac{M^{\prime\prime}}{r}\,.\end{split}$ (2.7)

The null vectors $l^{i}$ and $l^{i}$ have the following form

$\begin{split}l_{i}=\delta^{0}_{i}\,,\\\
n_{i}=\frac{1}{2}\left(1-\frac{2M}{r}\right)\delta^{0}_{i}-\delta^{1}_{i}\,.\end{split}$
(2.8)

So the total energy-momentum tensor for the generalized Vaidya spacetime is
given by(2.2):

$T_{ik}^{total}=T_{ik}^{(ND)}+T_{ik}^{(NS)}\,.$ (2.9)

They consider the model of the gravitational collapse to find out if this
process can lead to the naked singularity formation. We demand our model to be
physically relevant i.e. the total energy-momentum tensor must satisfy the
energy conditions [64]. For generalized Vaidya spacetime they read [40]:

  * •

Weak and strong energy conditions.

$\mu>0\,,P\geq 0\,,\rho\geq 0\,,\mu\neq 0\,.$ (2.10)

  * •

Dominant energy condition.

$\mu>0\,,\rho\geq P\geq 0\,,\mu\neq 0\,.$ (2.11)

The equation of the state, which is usually considered for perfect fluids at
equilibrium, is polytropic one

$P=\alpha\rho^{\gamma}\,.$ (2.12)

Where $\alpha$ is constant and another constant $\gamma$ is usually written as

$\gamma=1+\frac{1}{n}\,.$ (2.13)

Here $n$ is a polytropic index of the system. If $\gamma=1$ one comes to the
linear equations of the state. This solution is known as Husain solution [37].
Thus, in our consideration, we assume that $\gamma\neq 1$. It can be also
shown that $n\leq 5$ otherwise the matter cloud has no boundary at
equilibrium.

The Einstein equations for the generalized Vaidya spacetime (2.2) with the
matter distribution of the form (2.9) with the polytropic equation of the
state (2.12) lead to the linear equation

$M^{\prime\prime}+2^{\gamma}\alpha\frac{(M^{\prime})^{\gamma}}{r^{2\gamma-1}}=0\,.$
(2.14)

In order to solve this equation we introduce a new function $W=M^{\prime}$
then (2.14) becomes

$\begin{split}W^{\prime}+2^{\gamma}\alpha\frac{W^{\gamma}}{r^{2\gamma-1}}=0\,,\rightarrow\\\
W(v,r)=\left(D(v)-2^{\gamma-1}\alpha
r^{2-2\gamma}\right)^{\frac{1}{1-\gamma}}\,.\end{split}$ (2.15)

Where $D(v)$ is an integration function of the time $v$. From (2.15) one can
easily obtain the mass function in the form

$M(v,r)=\int Wdr+C(v)\,.$ (2.16)

Where $C(v)$ is another function of integration. To consider the gravitational
collapse, one should evaluate the integral $\int Wdr$ in (2.16). We consider
separately three cases:

  * •

$D(v)\equiv 0$;

  * •

$D(v)\ll 1$;

  * •

The case of the most general $D(v)$.

###  § 2.1 The case $D(v)\equiv 0$

We begin our consideration with the most simple case when an integration
constant $D(v)$ is zero. In the case, the function $W(v,r)$ reads

$W(v,r)=-\frac{\alpha^{\xi}r^{2}}{2}\,.$ (2.17)

Where $\xi=\frac{1}{1-\gamma}$. Then the integration in (2.16) gives

$\int Wdr=-\frac{\alpha^{\xi}r^{3}}{6}\,.$ (2.18)

The mass function is then given by

$M(v,r)=C(v)-\frac{\alpha^{\xi}r^{3}}{6}\,,$ (2.19)

and the spacetime (2.2) takes the following form

$ds^{2}=-\left(1-\frac{2C(v)}{r}+\frac{\alpha^{\xi}r^{2}}{6}\right)dv^{2}+2dvdr+r^{2}d\Omega^{2}\,.$
(2.20)

This spacetime reminds the Vaidya-de Sitter spacetime [65], where $C(v)$ is
associated with Vaidya mass function $M(v)$ and $\frac{\alpha^{\xi}}{2}$ is
interpreted as cosmological constant $\Lambda$. Note, that depending on the
sign $\alpha$ we obtain either Vaidya-de Sitter spacetime $\alpha<0$ or
Vaidya-anti-de Sitter spacetime $\alpha>0$. As we mentioned in the
introduction, the naked singularity is the result of the gravitational
collapse if the following conditions are held:

  * •

The time of the singularity formation is less than the time of the apparent
horizon formation;

  * •

there is a family of non-spacelike, future-directed geodesics which terminate
in the central singularity in the past.

The apparent horizon equation for the spacetime (2.20) has the form

$1-\frac{2C(v)}{r}+\frac{\alpha^{\xi}r^{2}}{6}=0\,.$ (2.21)

At the time of the naked singularity formation $v=0$, the apparent horizon is
absent if the limit

$\lim\limits_{v\rightarrow 0,r\rightarrow 0}\frac{2C(v)}{r}=X_{1}\,,$ (2.22)

exists and finite. For considering the existence of a family of non-spacelike,
future-directed geodesic, we consider a family of radial null geodesics

$\frac{dv}{dr}=\frac{2}{1-\frac{2C(v)}{r}+\frac{\alpha^{\xi}r^{2}}{6}}\,.$
(2.23)

Here such geodesics terminate in the central singularity in the past and
future-directed if the limit

$\lim\limits_{v\rightarrow 0,r\rightarrow 0}\frac{dv}{dr}=X_{0}>0\,,$ (2.24)

finite. Note, that for the gravitational collapse case, $X_{0}=0$ doesn’t suit
us because $v=const.$ is the ingoing null geodesic and we are interested only
in the outgoing one. Let’s consider the function $C(v)$ in the linear form
i.e.

$C(v)=\mu v\quad,\quad\mu>0\,.$ (2.25)

Then, applying the limit (2.24), the geodesic equation (2.23) reads

$\begin{split}X_{0}=\frac{2}{1-2\mu X_{0}}\rightarrow\\\ 2\mu
X_{0}^{2}-X_{0}+2=0\rightarrow\\\
X_{0}^{\pm}=\frac{1}{4\mu}\left(1\pm\sqrt{1-16\mu}\right)\,.\end{split}$
(2.26)

And we obtain a well-known result that the naked singularity is the result of
the gravitational collapse of $\mu\leq\frac{1}{16}$ obtained in the paper
[22]. The gravitational collapse of Vaidya-de Sitter spacetime has been also
considered in [66]. Note, that the outcome of the gravitational collapse
doesn’t depend on the sign of $\alpha$.

###  § 2.2 The case $D(v)\ll 1$

In the previous subsection, we have found out that the second integration
function $C(v)$ we can associate with a black hole mass $M(v)$. Here we will
find the influence of the first integration function $D(v)$ on the
gravitational collapse. For this purpose, let’s assume that $D(v)\ll 1$ and
expand it. Then the function $W(v,r)$ (2.15) reads

$W\approx-\frac{\alpha^{\xi}r^{2}}{2}-\frac{\dot{D}(v)D(v)}{(1-\gamma)2^{\gamma}\alpha^{\gamma\xi}}r^{2\gamma}\,.$
(2.27)

The integration gives

$\int Wdr\approx-\frac{\alpha^{\xi}r^{3}}{6}-\beta(v)r^{2\gamma+1}\,.$ (2.28)

Where

$\beta(v)=\frac{\dot{D}(v)D(v)}{(1-\gamma)(2\gamma+1)2^{\gamma}\alpha^{\gamma\xi}}\,.$
(2.29)

The mass function then takes the form

$M(v,r)=C(v)-\frac{\alpha^{\xi}r^{3}}{6}-\beta(v)r^{2\gamma+1}\,.$ (2.30)

and the generalized Vaidya spacetime then reads

$ds^{2}=-\left(1-\frac{2C(v)}{r}+\frac{\alpha^{\xi}}{6}r^{2}+\beta(v)r^{2\gamma}\right)dv^{2}+2dvdr+r^{2}d\Omega^{2}\,.$
(2.31)

Where $C(v)$ and $\frac{\alpha^{\xi}}{2}$ have the same interpretation of the
mass $M(v)$ and the cosmological constant $\Lambda$. The fourth term in
$g_{00}$ can be interpreted as another cosmological field. For example, if
$\gamma=\frac{3}{2}$ then the term $\beta(v)r^{3}$ can be associated with
phantom field surrounding a Vaidya black hole [30].

We should also mention that if $\dot{\beta}<0$ then we don’t have with weak
energy condition. However, if $\dot{\beta}>0$ then there is a region near
singularity

$r\leq\left(\frac{2\dot{C}(v)}{\dot{\beta}(v)}\right)^{\frac{1}{2\gamma+1}}\,,$
(2.32)

where the weak energy condition is violated. This problem is the same as in
charged Vaidya spacetime [67] where there is also the region of weak energy
condition violation.

The apparent horizon equation gives

$1-\frac{2C(v)}{r}+\frac{\alpha^{\xi}}{6}r^{3}+\beta(v)r^{2\gamma}=0\,.$
(2.33)

Like in the previous case, for the existence of the naked singularity, one
must demand that the limit

$\lim\limits_{v\rightarrow 0,r\rightarrow 0}\frac{C(v)}{r}=X_{1}\,,$ (2.34)

should be finite.

The radial null geodesic for the metric (2.31) reads

$\frac{dv}{dr}=\frac{2}{1-\frac{2C(v)}{r}+\frac{\alpha^{\xi}}{6}r^{2}+\beta(v)r^{2\gamma}}\,.$
(2.35)

We should impose the condition of finite $\beta$ at the time $v=0$ i.e.
$\beta(0)=\beta_{0}$. Then, assuming the linear mass function

$C(v)=\mu v\quad,\quad\mu>0\,,$ (2.36)

and taking the limit of (2.35) by using the deffinition (2.24), one obtains

$\begin{split}X_{0}=\frac{2}{1-2\mu X_{0}}\rightarrow\\\
X_{0}^{\pm}=\frac{1}{4\mu}\left(1\pm\sqrt{1-16\mu}\right)\,,\end{split}$
(2.37)

the same condition for the naked singularity occurence $\mu\leq\frac{1}{16}$
like in the previous case.

###  § 2.3 The general case

We have considered some particular cases of the first integration function
$D(v)$ so far. Here, we assume that $D(v)$ is in the most general form. then
the generalized Vaidya spacetime (2.2) takes the form

$ds^{2}=-\left(1-\frac{2C(v)}{r}-\frac{1}{r}\int
W(v,r)dr\right)dv^{2}+2dvdr+r^{2}d\Omega^{2}\,.$ (2.38)

The apparent horizon equation then gives

$1-\frac{2C(v)}{r}-\frac{1}{r}\int W(v,r)dr=0\,,$ (2.39)

and the radial null geodesics reads

$\frac{dv}{dr}=\frac{2}{1-\frac{2C(v)}{r}-\frac{1}{r}\int W(v,r)dr}\,.$ (2.40)

From the condition $M(0,0)$ [40], we obtain that $D(0)=0$. Hense, we can
expand $M(v,r)$ near the singular point $(0,0)$

$M(v,r)\approx M^{\prime}_{0}r+\dot{M}_{0}v\,.$ (2.41)

Where we made the following notations

$\begin{split}\lim\limits_{v\rightarrow 0,r\rightarrow
0}\frac{dM}{dr}=M^{\prime}_{0}\,,\\\ \lim\limits_{v\rightarrow 0,r\rightarrow
0}\frac{dM}{dv}=\dot{M}_{0}\,.\end{split}$ (2.42)

From the fact that $\gamma>0$ and solution (2.15) we can conclude that

$\lim\limits_{v\rightarrow 0,r\rightarrow 0}\frac{d}{dr}\left(\int
W(v,r)dr\right)=0\,.$ (2.43)

So, like in previous cases, assuming $C(v)=\mu v$ and considering the geodesic
equation (2.40) with definition (2.24), we obtain the same condition for the
naked singularity formation $\mu\leq\frac{1}{16}$.

The integral $\int W(v,r)dr$ can be, in general, interpreted as some
cosmological field surrounding the Vaidya black hole.

## Polytropic equation of state and Maxwell field equations

In this section, we will find the solution of the Einstein field equations
coupling with electromagnetic field. We, like in the previous section, assume
the spacetime in the form (2.2)

$ds^{2}=-\left(1-\frac{2M(v,r)}{r}\right)dv^{2}+2dvdr+r^{2}d\Omega^{2}\,.$
(3.44)

However, the energy-momentum tensor (2.9) we assume to be

$T_{ik}^{total}=T_{ik}^{(ND)}+T_{ik}^{(NS)}+E_{ik}\,.$ (3.45)

Here, as earlier, the first two terms are (2.3) and (2.5). The correction
$E_{ik}$

$E_{ik}=\frac{1}{4\pi}\left(F_{ij}F^{j}_{k}-\frac{1}{4}g_{ik}F_{jl}F^{jl}\right)\,,$
(3.46)

is an electromagnetic contribution. $F_{ik}$ is antisymmetric electromagnetic
field tensor. It obeys the Maxwell equations

$\begin{split}F_{ik,l}+F_{li,k}+F_{kl,i}=0\,,\\\ F^{ik}_{;k}=-4\pi
J^{i}\,.\end{split}$ (3.47)

Without any loss of generality the electromagnetic vector potential can be
chosen as [68, 69]

$A_{i}=\frac{Q(v)}{r}\delta^{0}_{i}\,.$ (3.48)

Where $Q(v)$ is associated with a black hole dynamical electric charge. From
(3.47) one obtains non-vanishing component of the electromagnetic tensor
$F_{ik}$

$F_{10}=-F_{01}=\frac{Q(v)}{r^{2}}\,.$ (3.49)

And the electromagnetic contribution $E_{ik}$ to the total energy-momentum
tensor is

$E_{ik}=\frac{Q^{2}}{r^{2}}\eta_{ik}\,.$ (3.50)

Where $\eta_{ik}$ is the Minkowski metric tensor components.

The Einstein field equations with energy-momentum tensor (3.45) for the metric
(3.44) are given by

$\begin{split}\frac{2}{r^{2}}\dot{M}=\mu(v,r)\,,\\\
\frac{2}{r^{2}}M^{\prime}=\rho+\frac{Q^{2}}{r^{4}}\,,\\\
-\frac{1}{r}M^{\prime\prime}=P+\frac{Q^{2}}{r^{4}}\,.\end{split}$ (3.51)

Imposing the polytropic equation of state for two last equations (3.51) and
remembering the deffinitions $P$ and $\rho$ (2.7) we arrive at

$\frac{1}{r}M^{\prime\prime}+\frac{Q^{2}}{r^{2}}+\alpha\left(\frac{2}{r^{2}}M^{\prime}-\frac{Q^{2}}{r^{2}}\right)^{\gamma}=0\,.$
(3.52)

Let’s introduce the function $W(v,r)$ by

$W(v,r)=M^{\prime}-\frac{Q^{2}}{2r^{2}}\,.$ (3.53)

Then (3.52) becomes

$\frac{1}{r}W^{\prime}+\alpha\left(\frac{2W}{r^{2}}\right)^{\gamma}=0\,.$
(3.54)

The solution to this equation is

$W(v,r)=\left(D(v)-2^{\gamma-1}\alpha
r^{2-2\gamma}\right)^{\frac{1}{1-\gamma}}\,.$ (3.55)

Remembering the deffinition of $W$ (3.53), one obtains the mass function
$M(v,r)$ in the form

$M(v,r)=\int Wdr-\frac{Q^{2}}{2r}+C(v)\,.$ (3.56)

Note, that the integral $\int Wdr$ is the same as in the uncharged Vaidya
case. In the previous section, we found out that it might be interpreted as a
cosmological field and it doesn’t affect the condition for the naked
singularity occurence. Here the situation is the same. Thus we don’t need to
consider all three cases of general $D(v)$, negigible small or zero $D(v)$.
Instead, we will consider the simplest case $D(v)\equiv 0$ and find out the
conditions for the naked singularity formation. These conditions will be the
same in all other cases.

With the mass function of the form (3.56), the generalized Vaidya spacetime
(3.44) takes the form

$ds^{2}=-\left(1-\frac{2C(v)}{r}+\frac{Q^{2}}{r^{2}}-\int
Wdr\right)dv^{2}+2dvdr+r^{2}d\Omega^{2}\,.$ (3.57)

Here we assume $D(v)\equiv 0$. Then this spacetime reads

$ds^{2}=-\left(1-\frac{2C(v)}{r}+\frac{Q^{2}}{r^{2}}+\frac{\alpha^{\xi}}{6}r^{2}\right)dv^{2}+2dvdr+r^{2}d\Omega^{2}\,.$
(3.58)

This spacetime is the charged Vaidya-(anti) de Sitter spacetime [31]. Let’s
denote $M(v)\equiv M(v)$ and $\Lambda=\frac{\alpha^{\xi}}{2}$. In these
notations the apparent horizon equation reads

$1-\frac{2M(v)}{r}+\frac{Q^{2}(v)}{r^{2}}+\frac{\Lambda}{3}r^{2}=0\,.$ (3.59)

This spacetime, as we mentioned above, has the region

$r\leq\frac{Q\dot{Q}}{\dot{M}}\,,$ (3.60)

where the weak energy condition is violated. However, particles can’t get into
this region due to the repulsive Lorentz force [70].

This spacetime, under some values of $Q\,,M\,,\Lambda$ might have the eternal
naked singularity formation111Under notion ’eternal’ we mean that the
singularity might form during the gravitational collapse and will never be
covered with the apparent horizon and only one cosmological horizon. The
radial null geodesic has the form

$\frac{dv}{dr}=\frac{2}{1-\frac{2M(v)}{r}+\frac{Q^{2}}{r^{2}}+\frac{\Lambda}{3}r^{2}}\,.$
(3.61)

If we assume the mass and charge functions of the form

$M(v)\equiv\mu v\quad,\quad Q^{2}=\nu^{2}v^{2}\,.$ (3.62)

and taking into account the deffinition (2.24), we arrive at

$\begin{split}X_{0}=\frac{2}{1-2\mu X_{0}+\nu^{2}X_{0}^{2}}\rightarrow\\\
\nu^{2}X_{0}^{3}-2\mu X_{0}^{2}+X_{0}-2=0\,.\end{split}$ (3.63)

The last equation is the cubic one. It means that in general, it admits at
least one real root. Note that the last equation (3.63) is negative at
$X_{0}=0$ and positive at $X_{0}=2+\frac{2\mu}{\nu^{2}}$ i.e. the last
equation has the positive finite root in the interval
$X_{0}^{r}\in\left(0\,,\frac{2\mu}{\nu^{2}}\right)$. Hence, the limit is
positive and finite and the naked singularity might form during the
gravitational collapse. This result coincides with the one obtained in [71].
Note that regardless of the value $\gamma$, the naked singularity formation
will depend only on conditions obtained in (3.63) because the integral (3.56)
doesn’t influence the condition of the naked singularity occurrence.

## The generalized polytropic equation of state

In this section we consider generalized Vaidya spacetime with type-II matter
field with the equation of the state

$P=k\rho+\alpha\rho^{\gamma}\,.$ (4.64)

Here, if one considers barotropic equation of state $P=k\rho$ then :

  * •

$k=1$ corresponds to the stiff fluid and solution is the charged Vaidya
spacetime;

  * •

$k=\frac{1}{3}$ corresponds to the radiation;

  * •

$k=0$ is the equation of state describing dust;

  * •

$k=-\frac{2}{3}$ corresponds to the quentesence;

  * •

$k=-1$ is the vaccuum energy and

  * •

$k=-\frac{4}{3}$ corresponds to phantom field.

The pure polytropic part $P=\alpha\rho^{\gamma}$ may be due to Bose-Einstein
condensates with repulsive ($\alpha>0$) or attractive ($\alpha<0$) self-
interaction, or have another origin [73].

The Einstein equations for the generalized Vaidya spacetime (2.2) with energy
density $\rho$ and pressure $P$ defined in (2.7) with the generalized
polytropic equation of state (4.64) lead to the following second-order
differential equation:

$-\frac{M^{\prime\prime}}{r}=k\left(\frac{2M^{\prime}}{r^{2}}\right)+\alpha\left(\frac{2M^{\prime}}{r^{2}}\right)^{\gamma}\,.$
(4.65)

Let’s introduce the function $W$ by

$W(v,r)\equiv\frac{2M^{\prime}}{r^{2}}\,.$ (4.66)

Then the Einstein equation (4.65) reads in terms of new function

$-W^{\prime}r=(2k+2)W+2\alpha W^{\gamma}\,.$ (4.67)

To solve this equation one needs to consider two cases.

###  § 4.1 The case $k\neq-1$

In this case, the solution of the differential equation (4.67) is

$W=\frac{(2k+2)^{-\xi}D(v)r^{-(2k+2)}}{\left(2k+2-2\alpha
D(v)r^{-(2k+2)(\gamma-1)}\right)^{-\xi}}\,.$ (4.68)

Remembering the definition of the function $W$ (4.66) and integrating one more
time, we obtain the mass function $M(v,r)$ of the form:

$M(v,r)=C(v)+\frac{1}{2}\int\frac{(2k+2)^{-\xi}D(v)r^{-2k}}{\left(2k+2-2\alpha
D(v)r^{-(2k+2)(\gamma-1)}\right)^{-\xi}}dr\,.$ (4.69)

Note that in the case of $\alpha=0$ i.e. one considers only barotropic
equation of state $P=k\rho$, the mass function $M(v,r)$ becomes

$M(v,r)=C(v)+D(v)r^{1-2k}\,,$ (4.70)

which corresponds to the Husain solution [37, 43].

Here we need to evaluate the integral in the right hand-side of the equation
(4.69). Note if $D(v)\equiv 0$ then the mass function

$M(v,r)=C(v)\quad,\quad\text{for}D(v)\equiv 0\,,$ (4.71)

is the usual Vaidya mass function. If we consider the small integration
function $D(v)\ll 1$ then the mass function $M(v,r)$ reduces to the Husain
solution (4.70). The gravitational collapse, in this case, can lead to the
naked singularity formation [43].

###  § 4.2 The case $k=-1$

This case, despite the simpler differential equation, leads to the solution
which can be hardly analysed. The solution of the differential equation (4.67)
is given by

$W(v,r)=\left(D(v)+\ln r^{-2\alpha(1-\gamma)}\right)^{\xi}\,.$ (4.72)

And we can formaly write the solution in the form

$M(v,r)=C(v)+\int W(v,r)dr\,.$ (4.73)

However, this solution can’t be analysed within this paper.

## Conclusion

In this paper we have obtained the three solutions of the Einstein field
equations. The first one describes the generalized Vaidya spacetime with the
type-II matter field satisfying the polytropic equation of the state. The mass
function of a new solution has the general form (2.16). To consider the
gravitational collapse, one should consider the integral $\int Wdr$. In the
case of uncharged spacetime, we have considered three cases. The first one
leads to the Vaidya-de Sitter spacetime, the second one leads to Vaidya
spacetime surrounded by the combination of the cosmological fields. In both
cases, there is not influence of the integral $\int Wdr$ on the naked
singularity formation. In general case when the integration function $D(v)$ is
supposed to be general, we have concluded that this integral $\int Wdr$ may
act as a cosmological field.

By considering the generalized Vaidya spacetime coupling with the
electromagnetic field, the solution has the form (3.56). In the limiting case
$D(v)=0$ the resulting spacetime is charged Vaidya-de Sitter one. Again,
considering the gravitational collapse to find the conditions of the naked
singularity occurrence, we have found out that these conditions don’t depend
upon the integral $\int Wdr$ i.e. the polytropic index $\gamma$ doesn’t have
any impact on these conditions.

The third solution corresponds to the generalized equation of state (4.64).
The solution for zero or negligible small integration function $D(v)$ is
either the usual Vaidya spacetime or Husain solution respectively.

The generalized Vaidya spacetime with the polytropic equation of state
violates the cosmic censorship hypothesis. However, the singularities appear
in a wide variety of forms and it seems at present not feasible to rule out
all singularities with only a single theorem. If, however, cosmic censorship
holds, it should be in a highly refined and fine-tuned form only. The naked
singularity formation, on the other hand, might play an important role in
astrophysics because such ultra-dense regions can tell us about the effects of
quantum gravity which are possible and might be visible to the faraway
observer.

The naked singularity formation can be considered as a source of the gamma ray
bursts [72, 8].

## References

  * 1\.  B. P. Abbott et al. (LIGO Scientific, Virgo), Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc].
  * 2\.  Abbott, B.P. et al. [LIGO Scientific Collaboration and Virgo Collaboration]. GW170814: A Three-Detector Observation of Gravitational Waves from a Binary Black Hole Coalescence. _Phys. Rev. Lett._ 2017, _119_ , 141101.
  * 3\.  The Event Horizon Telescope Collaboration, firstM87 Event Horizon Telescope results. I. The shadow of the supermassive black hole, Astrophys. J. Lett. 875 (2019)L1
  * 4\.  The Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope results. II. Array and instrumentation, Astrophys. J. Lett. 875 (2019) L2
  * 5\.  Akiyama, K. et al. [Event Horizon Telescope Collaboration]. First Sagittarius A$*$ Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way. _Astrophys. J. Lett._ 2022, _930_ , L12.
  * 6\.  Pankaj S. Joshi Gravitational collapse and spacetime singularities. Cambridge University Press. 2007.p.273.
  * 7\.  Pankaj S. Joshi, D. Malafarina, ”Recent development in gravitational collapse and spacetime singularitits”. Int. J. Mod. Phys. D, 20 2641 (2011).
  * 8\.  T. Harko. Gravitational collapse of a Hagedorn fluid in Vaidya geometry, Phys.Rev. D68 (2003) 064005.
  * 9\.  Sergio M. C. V. Goncalves, Sanjay Jhingan. Singularities in gravitational collapse with radial pressure, Gen.Rel.Grav.33:2125-2149,2001 [arXiv:gr-qc/0107054]
  * 10\.  Karim Mosani, Dipanjan Dey, Pankaj S. Joshi. Global visibility of a strong curvature singularity in non-marginally bound dust collapse, Physical Review D 102(4):044037 [arXiv:2003.07092 [gr-qc]]
  * 11\.  Dipanjan Dey, Karim Mosani, Pankaj Joshi, Vitalii Vertogradov. Causal structure of singularity in non-spherical gravitational collapse. Eur. Phys. J. C 82, 431 (2022) [arXiv:2103.07190]
  * 12\.  Nolene F. Naidu, Robert S. Bogadi, Anand Kaisavelu, Megan Govender. Stability and Horizon Formation during Dissipative Collapse, General Relativity and Gravitation 52(8) [arXiv:2102.06063 [gr-qc]]
  * 13\.  Rajibul Shaikh, Prashant Kocherlakota, Ramesh Narayan, Pankaj S. Joshi. Shadows of spherically symmetric black holes and naked singularities, MNRAS 482, 52 (2019) [arXiv:1802.08060 [astro-ph.HE]]
  * 14\.  J.R. Oppenheimer, H.Snyder, Phys. Rev. 56, 455-459 (1939).
  * 15\.  T. P. Singh, P. S. Joshi. The Final Fate of Spherical Inhomogeneous Dust Collapse, Class.Quant.Grav. 13 (1996) 559-572 [arXiv:gr-qc/9409062]
  * 16\.  Sanjay Jhingan, P. S. Joshi, T. P. Singh. The final fate of spherical inhomogeneous dust collapse II: Initial data and causal structure of singularity, Class.Quant.Grav.13:3057-3068,1996 [arXiv:gr-qc/9604046]
  * 17\.  P. C. Vaidya. Nonstatic solutions of Einstein’s field equations for spheres of fluids radiating energy. Phys. Rev., 83:10, 1951.
  * 18\.  A. Papapetrou, In: A Random Walk in Relativity and Cosmology. Wiley Eastern, New Delhi (1985).
  * 19\.  Santos, N.O. Non-adiabatic radiating collapse. _Mon. Not. R. Astron. Soc._ 1985, _216_ , 403.
  * 20\.  Herrera, L.; Prisco, A.D.; Ospino, J.Some analytical models of radiating collapsing spheres. _Phys. Rev. D_ 2006, _74_ , 044001.
  * 21\.  Herrera, L.; Denmat, G.L.; Santos, N.O. Dynamical instability and the expansion-free condition. _Gen. Relativ. Gravit._ 2012, _44_ , 1143.
  * 22\.  Dwivedi, I.H.; Joshi, P.S. On the nature of naked singularities in Vaidya spacetimes Class. _Quantum Grav._ 1989, _6_ , 1599.
  * 23\.  E. Babichev, V. Dokuchaev, Yu. Eroshenko. Backreaction of accreting matter onto a black hole in the Eddington-Finkelstein coordinates, Classical and Quantum Gravity 29(11) 2012 [arXiv:1202.2836 [gr-qc]]
  * 24\.  Jay Solanki, Volker Perlick Phys. Rev. D, 105:064056, (2022), arXiv:2201.03274 (gr-qc)
  * 25\.  Yasutaka Koga, Nobuyuki Asaka, Masashi Kimura, Kazumasa Okabayashi, Dynamical photon sphere and time evolving shadow around black holes with temporal accretion, Phys. Rev. D 105, 104040 (2022)
  * 26\.  Vitalii Vertogradov. The influence of the charge on a dynamical photon sphere, [arXiv:2311.08930 [gr-qc]]
  * 27\.  A. B. Nielsen. Revisiting Vaidya horizons. Galaxies, 2:62, 2014.
  * 28\.  Nielsen, A.B.; Yoon, J.H. Dynamical surface gravity. Class. Quant. Gravity 2008, 25, 085010, doi:10.1088/0264-9381/25/8/085010.
  * 29\.  V. Vertogradov and D. Kudryavcev, Generalized vaidya spacetime: horizons, conformal symmetries, surfacegravity and diagonalization, Modern Physics Letters A, p. 2350119, 2023.
  * 30\.  Y. Heydarzade, F. Darabi Surrounded Vaidya black holes: apparent horizon properties, Eur. Phys. J. C (2018) 78:342
  * 31\.  Heydarzade, Y.; Darabi, F. Surrounded Vaidya solution by cosmological fields. _Eur. Phys. J. C_ 2018, _78_ , 582.
  * 32\.  Heydarzade, Y.; Darabi, F. Surrounded Bonnor Vaidya solution by cosmological fields. _Eur. Phys. J. C_ 2018, _78_ , 1004.
  * 33\.  Reddy, K.P.; Govender, M.; Maharaj,S.D. Impact of anisotropic stresses during dissipative gravitational collapse _Gen. Relativ. Gravit._ 2015, _47_ , 35.
  * 34\.  Thirukkanesh, S.; Moopanar, S.; Govender, M. The final outcome of dissipative collapse in the presence of $\Lambda$. _Pramana J. Phys._ 2012, _79_ , 223–232.
  * 35\.  Thirukkanesh, S.; Govender, M. The role of the electromagnetic field in dissipative collapse. _Int. J. Mod. Phys. D_ 2013, _22_ , 1350087.
  * 36\.  Wang, A.; Wu, Y. Generalized Vaidya solutions. _Gen Relativ. Gravit._ 1999, _31_ , 107.
  * 37\.  Husain, V. Exact solutions for null fluid collapse. _Phys. Rev. D_ 1996, _53_ , R1759.
  * 38\.  Glass, E.N.; Krisch, J.P. Radiation and string atmosphere for relativistic stars. _Phys. Rev. D_ 1998, _57_ , 5945.
  * 39\.  Glass, E.N.; Krisch, J.P. Classical and Quantum Gravity Two-fluid atmosphere for relativistic stars. _Class. Quant. Grav._ 1999, _16_ , 1175.
  * 40\.  Mkenyeleye, M.D.; Goswami, R.; Maharaj, S.D. Vaidya and generalized Vaidya solutions by gravitational de- coupling. _Phys. Rev. D_ 2015, _92_ , 024041.
  * 41\.  Mkenyeleye, M.D.; Goswami, R.; Maharaj, S.D. Thermodynamics of gravity favours Weak Censorship Conjecture. _Phys. Rev. D_ 2014, _90_ , 064034.
  * 42\.  Vertogradov, V. The eternal naked singularity formation in the case of gravitational collapse of generalized Vaidya spacetime. _Int. J. Mod. Phys. A_ 2018, _33_ , 1850102. arXiv:2210.16131 [gr-qc]
  * 43\.  Vertogradov, V. Naked singularity formation in generalized Vaidya space-time. _Grav. Cosmol._ 2016, _22_ , 220–223.
  * 44\.  Vertogradov, V. Gravitational collapse of Vaidya spacetime. _Int. J. Mod. Phys. Conf. Ser._ 2016, _41_ , 1660124.
  * 45\.  Saibal Ray, Arijit Panda, Bivash Majumder, Md. Rabiul Islam, Collapsing scenario for the k-essence emergent generalised Vaidya spacetime in the context of massive gravity’s rainbow Chinese Physics C 46(12) DOI: 10.1088/1674-1137/ac8868
  * 46\.  Dey, D.; Joshi, P.S. Gravitational collapse of baryonic and dark matter. _Arab. J. Math._ 2019, _8_ , 269.
  * 47\.  Samson Ojako, Rituparno Goswami, Sunil D. Maharaj, Rivendra Narain. Conformal symmetries in generalised Vaidya spacetimes, 2020 Class. Quantum Grav. 37 055005 [arXiv:1904.08120 [gr-qc]]
  * 48\.  Seoktae Koh, Miok Park, Abbas M. Sherif. Thermodynamics with conformal Killing vector in the charged Vaidya metric, [arXiv:2309.17398 [gr-qc]]
  * 49\.  Nikolaev, A.V.; Maharaj, S.D. Embedding with Vaidya geometry. _Eur. Phys. J. C_ 2020, _80_ , 648.
  * 50\.  Faraoni, V.; Giusti, A.; Fahim, B.H. Vaidya geometries and scalar fields with null gradients. _Eur. Phys. J. C_ 2021, _81_ , 232.
  * 51\.  Brassel, B.P.; Maharaj, S.D.; Goswami, R. Charged radiation collapse in Einstein Gauss Bonnet gravity. _Eur. Phys. J. C_ 2022, _82_ , 359.
  * 52\.  R. Penrose and R. M. Floyd, Nature Physical Science 229, 177 (1971).
  * 53\.  V. Vertogradov, Universe 6(9), 155 (2020), arXiv:2209.10976 (gr-qc)
  * 54\.  Denardo Ruffini, Phys. Lett., vol 45B, 259 (1973)
  * 55\.  Vitalii Vertogradov 2023 Commun. Theor. Phys. 75 045404
  * 56\.  Vitalii Vertogradov, Forces in Schwarzschild, Vaidya and generalized Vaidya spacetimes, 2021 J. Phys.: Conf. Ser. 2081 012036 DOI 10.1088/1742-6596/2081/1/012036
  * 57\.  V.D. Vertogradov, Non-linearity of Vaidya spacetime and forces in the central naked singularity. Physics of Complex Systems, 2022, vol. 3, no. 2 [arXiv:2203.05270]
  * 58\.  Vitalii Vertogradov, Maxim Misyura ”Vaidya and Generalized Vaidya Solutions by Gravitational Decoupling”Universe 2022, 8(11), 567; doi:10.3390/universe8110567 arXiv:2209.07441 [gr-qc]
  * 59\.  Isayev, A.A. (2017). Relativistic anisotropic stars with the polytropic equation of state in general relativity. J. Phys. Conf.Ser. 934, 012039.
  * 60\.  Manuel Malaver de la Fuente. CLASSES OF CHARGED ANISOTROPIC STARS WITH POLYTROPIC EQUATION OF STATE, 2021, International Journal of Research and Reviews in Applied Sciences, Vol.46, No.1, pp. 38-51
  * 61\.  Komathiraj, K. and Maharaj,S.D. (2008). Classes of exact Einstein-Maxwell solutions, Gen. Rel.Grav. 39(12), 2079-2093.
  * 62\.  Malaver, M. (2013). Analytical model for charged polytropic stars with Van der Waals Modified Equation of State,American Journal of Astronomy and Astrophysics, 1(4), 37-42.
  * 63\.  Mafa Takisa, P. and Maharaj, S.D. (2013). Some charged polytropic models. Gen.Rel.Grav, 45, 1951-1969.
  * 64\.  Poisson, E. _A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics_ ; Cambridge University Press: Cambridge, UK, 2007.
  * 65\.  P C Vaidya and K B Shah (1957) Proc. Nat. Inst. of Sci. (India) 23, 534
  * 66\.  Wagh, S.M., Maharaj, S.D. Naked Singularity of the Vaidya-de Sitter Spacetime and Cosmic Censorship Conjecture. General Relativity and Gravitation 31, 975-982 (1999).
  * 67\.  A. Ori, ”‘Charged null fluid and the weak energy cond ition,”’ Class. Quantum Grav. 8 1559 (1991).
  * 68\.  W. B. Bonnor and P. C. Vaidya, ”‘Spherically symmetric radiation of charge in Einstein-Maxwell theory”’ Gen. Rel. Grav. 1, 127 (1970).
  * 69\.  K. Lake and T. Zannias, Structure of singularities in the spherical gravitational collapse of a charged null fluid, Phys. Rev. D 43, 1798 (1991).
  * 70\.  Chatterjee, S., Ganguli, S. and Virmani, A. Charged Vaidya solution satisfies weak energy condition. Gen Relativ Gravit 48, 91 (2016), arXiv:1512.02422 (gr-qc)
  * 71\.  A. Beesham, S.G. Ghosh, Naked Singularities in the Charged Vaidya DeSitter Spacetime, International Journal of Modern Physics D 12 801 (2003);arXiv:gr-qc/0003044.1
  * 72\.  Sandip K. Chakrabarti, Pankaj S. Joshi. Naked Singularities as Possible Candidates for Gamma-ray Bursters, [arXiv:hep-th/9208060]
  * 73\.  Pierre-Henri Chavanis. Models of universe with a polytropic equation of state: I. The early universe, Eur. Phys. J. Plus, 129, 38 (2014)