22institutetext: Department of Mathematics, Faculty of Applied Sciences,
Durban University of Technology, Durban, 4000, South Africa.
22email<EMAIL_ADDRESS>${}^{\textrm{{\char 0\relax}}}$
22email<EMAIL_ADDRESS>
22email<EMAIL_ADDRESS>
Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics
and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban
4000, South Africa.
22email<EMAIL_ADDRESS>
# Stability and Horizon Formation during Dissipative Collapse

Nolene F. Naidu† Robert S. Bogadi‡ Anand Kaisavelu∗ Megan Govender∗∗

(Received: date / Accepted: date)

###### Abstract

We investigate the role played by density inhomogeneities and dissipation on
the final outcome of collapse of a self-gravitating sphere. By imposing a
perturbative scheme on the thermodynamical variables and gravitational
potentials we track the evolution of the collapse process starting off with an
initially static perfect fluid sphere which is shear-free. The collapsing core
dissipates energy in the form of a radial heat flux with the exterior
spacetime being filled with a superposition of null energy and an anisotropic
string distribution. The ensuing dynamical process slowly evolves into a
shear-like regime with contributions from the heat flux and density
fluctuations. We show that the anisotropy due to the presence of the strings
drives the stellar fluid towards instability with this effect being enhanced
by the density inhomogeneity. An interesting and novel consequence of this
collapse scenario is the delay in the formation of the horizon.

###### Keywords:

radiative collapse anisotropic stresses density inhomogeneities

##  1 Introduction

Gravitational collapse is fundamental to the formation of the majority of
stellar objects in the universe and thus one would expect that the study of
this phenomenon should be vital to our understanding of the workings of the
universe. The pioneers in the research of gravitational collapse, Oppenheimer
and Snyder (1939), studied a spherically symmetric matter distribution in the
form of a dust sphere undergoing collapse. They obtained the first solution
for the non-adiabatic collapse of a dust ball with a Schwarzschild exterior
oppsnyd . Vaidya vaidya  obtained an exact solution to the Einstein field
equations which describes the exterior field of a radiating, spherically
symmetric fluid by noting that a radiating collapsing mass distribution has
outgoing energy and so its exterior spacetime is no longer a vacuum but
contains null radiation. The next step in improving the model was accomplished
by Santos santos  who derived the junction conditions for a collapsing
spherically symmetric, shear-free non-adiabatic fluid sphere with heat flow.
The combination of these contributions allowed for the matching of the
interior and exterior spacetimes of a collapsing star which lead the way for
studying non-adiabatic, isotropic as well as anisotropic dissipative
gravitational collapse mken -kevin .

A disturbance or perturbation of a system initially in static equilibrium
results in a change in stability which likely renders the system dynamic. The
property of a system to retain its initial stable state once perturbed, is
then referred to as its dynamical (in)stability. Hence the issue of stability
is vital in the study of self-gravitating objects as a static stellar model
which evolves towards higher instability is of little physical significance.
The dynamical instability of a spherically symmetric mass with isotropic
pressure was first investigated by Chandrasekhar chandra . He showed that for
a system to remain stable under collapse, the adiabatic index $\Gamma$ must be
greater than $\frac{4}{3}$.

Subsequently, Herrera et al. herr2  showed that for a non-adiabatic sphere
where relativistic corrections were imposed to address heat flow, the unstable
range of $\Gamma$ decreased rendering the fluid less unstable. Chan et al.
chan2  investigated the stability criteria by deviating from the perfect fluid
condition in two ways: they considered radiation in the free-streaming
approximation; and they assumed local anisotropy of the fluid. Herrera et al.
herr3  also examined the dynamical instability of expansion-free, locally
anisotropic spherical stellar bodies.

The application of Einstein’s field equations to increasingly more complex
gravitating systems with additional parameters and degrees of freedom depends
on computational techniques, which is the case when perturbative theories are
employed in which higher order terms arise. The generalization of systems,
such as the inclusion of a string density field, can increase the complexity
of expressions obtained however to first order we aim to introduce the
temporal behaviour and hence evolution of the collapse process. This method is
well established chan2 ; bon . Compact objects such as neutron stars, black
holes and the more recently proposed dark-fluid stars and strange stars
composed of quark matter invite the addition of a more complex, non-empty
stellar exterior. The Vaidya metric which is commonly used for describing the
exterior spacetime would then require modification to include both the
radiation field and a so-called string field as initially put forward by Glass
and Krisch glass ; glass2 . In this more generalized Vaidya exterior, the mass
function is augmented to acquire both temporal and spatial dependence.

In 2005, Maharaj and Govender showed that the stellar core was more unstable
than the outer regions by investigating gravitational collapse with isotropic
pressure and vanishing Weyl stresses maharaj2 . More recently, Maharaj et al.
maharaj3  showed the impact of the generalized Vaidya radiating metric on the
junction conditions for the boundary of a radiating star. Their results
describe a more general atmosphere surrounding the star, described by the
superposition of a pressure-free null dust and a string fluid. The string
density was shown to affect the fluid pressure at the surface of the star. It
was demonstrated that this string density reduces the pressure at the stellar
boundary. The usual junction conditions for the Vaidya spacetime are regained
in the absence of the string fluid.

In this study, a spherically symmetric static configuration undergoing
radiative collapse under shear-free, isotropic conditions is considered. A
boundary condition of the form
$\left(p_{r}\right)_{\Sigma}=\left(qB\right)_{\Sigma}-\rho_{s}$ is imposed
where $\rho_{s}$ is the string density and $qB$ is the heat flux. This is the
basis for developing the temporal behaviour of the self-gravitating system.
The structure of this paper is as follows: In $\S$2 the field equations
describing the geometry and matter content for a star undergoing shear-free
gravitational collapse are introduced. In $\S$3 the exterior spacetime and the
junction conditions necessary for the smooth matching of the interior
spacetime with Vaidya’s exterior solution across the boundary are presented.
In $\S$4 the perturbative scheme is described and the field equations for the
static and perturbed configurations are stated. In $\S$5 we develop the new
temporal equation employed in the perturbative scheme which includes the
effect of the string field. In $\S$6, we develop a radiating model from an
interior Schwarzschild static model. In $\S$7 dissipative collapse is
discussed, the perturbed quantities in terms of two unspecified quantities are
expressed and an equation of state which presents the perturbed quantities in
terms of radial coordinate only is introduced. The stability of the collapsing
model in the Newtonian and post-Newtonian approximations are explored in
$\S$8\. The physical analysis of the results and conclusion are discussed in
$\S$9\. The acknowledgements follows in $\S$10.

##  2 Stellar Interior

In order to investigate the evolution of the radiative collapse we adopt a
spherically symmetric shear-free line element in simultaneously comoving and
isotropic coordinates given by

$ds_{-}^{2}=-A^{2}(r,t)dt^{2}+B^{2}(r,t)[dr^{2}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2})]$
(1)

where the $A(r,t)$ and $B(r,t)$ are the dynamic gravitational potentials. We
should highlight the fact that the stability of the shear-free condition may
hold for a limited period of the collapse process chan2 . Herrera and co-
workers have shown that shear-free collapse can evolve into a dynamical
process mimicking shear. The shear-like contributions can develop from
pressure anisotropy and density inhomogeneities. The stellar material for the
interior is described by an imperfect fluid with heat flux and in the form of
the energy-momentum tensor

$T_{ab}=(\rho+p_{t})u_{a}u_{b}+p_{t}g_{ab}+(p_{r}-p_{t})\chi_{a}\chi_{b}+q_{a}u_{b}+q_{b}u_{a}$
(2)

where $\rho$ is the energy density, $p_{r}$ the radial pressure, $p_{t}$ the
tangential pressure and $q_{a}$ the heat flux vector, $u_{a}$ is the timelike
four-velocity of the fluid and $\chi_{a}$ is a spacelike unit four-vector
along the radial direction. These quantities must satisfy $u_{a}u^{a}=-1$,
$u_{a}q^{a}=0$, $\chi_{a}\chi^{a}=1$ and $\chi_{a}u^{a}=0$. In co-moving
coordinates we must have

$\displaystyle u^{a}$ $\displaystyle=$ $\displaystyle A^{-1}\delta^{a}_{0}$
(3) $\displaystyle q^{a}$ $\displaystyle=$ $\displaystyle q\delta^{a}_{1}$ (4)
$\displaystyle\chi^{a}$ $\displaystyle=$ $\displaystyle B^{-1}\delta^{a}_{1}$
(5)

The nonzero components of the Einstein field equations for line element (1)
with energy-momentum tensor (2) are

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle-\frac{1}{B^{2}}\left[2\frac{B^{\prime\prime}}{B}-\left(\frac{B^{\prime}}{B}\right)^{2}+\frac{4}{r}\frac{B^{\prime}}{B}\right]+\frac{3}{A^{2}}\left(\frac{{\dot{B}}}{B}\right)^{2}$
(6) $\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{B^{2}}\left[\left(\frac{B^{\prime}}{B}\right)^{2}+\frac{2}{r}\left(\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}\right)+2\frac{A^{\prime}}{A}\frac{B^{\prime}}{B}\right]$
(7)
$\displaystyle+\frac{1}{A^{2}}\left[-2\frac{\ddot{B}}{B}-\left(\frac{\dot{B}}{B}\right)^{2}+2\frac{\dot{A}}{A}\frac{\dot{B}}{B}\right]$
$\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{A^{2}}\left[-2\frac{\ddot{B}}{B}-\left(\frac{\dot{B}}{B}\right)^{2}+2\frac{\dot{A}}{A}\frac{\dot{B}}{B}\right]$
(8)
$\displaystyle+\frac{1}{B^{2}}\left[\frac{B^{\prime\prime}}{B}-\left(\frac{B^{\prime}}{B}\right)^{2}+\frac{1}{r}\left(\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}\right)+\frac{A^{\prime\prime}}{A}\right]$
$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{2}{AB^{2}}\left[\frac{{\dot{B}}^{\prime}}{B}-\frac{\dot{B}}{B}\left(\frac{B^{\prime}}{B}+\frac{A^{\prime}}{A}\right)\right]$
(9)

where dots and primes represent partial derivatives with respect to $t$ and
$r$ respectively.

##  3 Exterior Spacetime and Matching Conditions

Since the star is radiating, the exterior spacetime can be described by the
generalized Vaidya metric vaidya  which represents a mixture of null radiation
and strings glass

$ds^{2}_{+}=-\left[1-\frac{2m(v,\mathrm{r})}{\mathrm{r}}\right]dv^{2}-2dvd\mathrm{r}+\mathrm{r}^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2})$
(10)

where $m(v,\mathrm{r})$ is the mass function which represents the total energy
within a sphere of radius $\mathrm{r}$. This is what distinguishes the
generalized Vaidya solution from the pure radiation Vaidya solution, which has
$m=m(v)$ where $v$ is the retarded time. The energy momentum tensor
corresponding to line element (10) is

$T^{+}_{ab}=\tilde{\mu}l_{a}l_{b}+(\rho+P)(l_{a}n_{b}+l_{b}n_{a})+Pg_{ab}$
(11)

where

$l_{a}=\delta^{0}_{a}$ (12)
$n_{a}=\frac{1}{2}\left[1-2\frac{m(v,\mathrm{r})}{\mathrm{r}}\right]\delta^{0}_{a}+\delta^{1}_{a}$
(13)

are null vectors such that $l_{a}l^{a}=n_{a}n^{a}=0$ and $l_{a}n^{a}=-1$. The
energy momentum tensor (11) can be interpreted as the matter source for the
exterior atmosphere of the star which is a superposition of pressureless null
dust and anisotropic null strings HusV ; WangA . The energy density of the
null dust radiation, string energy density and string pressure are
characterised by $\tilde{\mu}$, $\rho$ and $P$ respectively. We assume that
the string diffusion is equivalent to point particle diffusion where the
number density diffuses from higher to lower numbers subjected to the
continuity equation

$\dot{\rho}=\frac{\cal{D}}{\mathrm{r}^{2}}\frac{\partial}{\partial\mathrm{r}}\left(\mathrm{r}^{2}\frac{\partial\rho}{\partial\mathrm{r}}\right)$
(14)

where $\cal D$ is the positive coefficient of self-diffusion govin . Following
de Oliveira et al. oliv2 , we obtain the boundary conditions which include a
string density $\rho_{s}$,

$\hskip
28.45274pt\left(p_{r}\right)_{\Sigma}=\left(qB\right)_{\Sigma}-\left(\rho_{s}\right)_{\Sigma}$
(15) $\hskip
28.45274pt\left(qB\right)_{\Sigma}=-\left(\frac{2}{{\mathrm{r}}^{2}}{{\dot{v}}^{2}}\frac{dm}{dv}\right)_{\Sigma}$
(16) $\hskip 28.45274pt\left(rB\right)_{\Sigma}=\mathrm{r}_{\Sigma}$ (17)
$\hskip
28.45274pt\left(Adt\right)_{\Sigma}=\left(1-\frac{2m}{\mathrm{r}}+2\frac{d\mathrm{r}}{dv}\right)^{1/2}_{\Sigma}dv$
(18)

Equation (15) represents the conservation of momentum flux across the stellar
boundary which we will employ in $\S$5 to determine the temporal evolution of
our model. The total energy entrapped within a radius $r$ inside $\Sigma$ is
given by

$m(r,t)=\frac{r^{3}B{\dot{B}}^{2}}{2A^{2}}-r^{2}{B^{\prime}}-\frac{r^{3}{{B^{\prime}}^{2}}}{2B}$
(19)

At the boundary, this is given by

$\hskip 28.45274ptm(v,\mathrm{r})=m(r,t)|_{\Sigma}$ (20)

and included as a boundary condition.

##  4 Perturbative Scheme

Following the method in Herrera et al. herr2 , as well as the works of Chan
et. al. chan2  and Govender et. al. gov2 , we present our model in this
section. To begin, we will assume that the fluid is in static equilibrium. The
system is then perturbed and undergoes slow shear-free dissipative collapse.
Thermodynamical quantities in the static system are represented by a zero
subscript, while those in the perturbed fluid are represented by an overhead
bar. The metric functions $A(r,t)$ and $B(r,t)$ are taken to have the same
temporal dependence, which extends to the perturbed material quantities. The
time-dependent metric functions and material quantities are given by

$\displaystyle A(r,t)$ $\displaystyle=$ $\displaystyle A_{0}(r)+\epsilon
a(r)T(t)$ (21) $\displaystyle B(r,t)$ $\displaystyle=$ $\displaystyle
B_{0}(r)+\epsilon b(r)T(t)$ (22) $\displaystyle\rho(r,t)$ $\displaystyle=$
$\displaystyle\rho_{0}(r)+\epsilon\bar{\rho}(r,t)$ (23) $\displaystyle
p_{r}(r,t)$ $\displaystyle=$ $\displaystyle
p_{r0}(r)+\epsilon{\bar{p}}_{r}(r,t)$ (24) $\displaystyle p_{t}(r,t)$
$\displaystyle=$ $\displaystyle p_{t0}(r)+\epsilon{\bar{p}}_{t}(r,t)$ (25)
$\displaystyle m(r,t)$ $\displaystyle=$ $\displaystyle
m_{0}(r)+\epsilon\bar{m}(r,t)$ (26) $\displaystyle q(r,t)$ $\displaystyle=$
$\displaystyle\epsilon\bar{q}(r,t)$ (27)

where we assume that $0<\epsilon<<1$. We observe that the temporal dependence
of the perturbative quantities, $T(t)$ is the same for both the gravitational
potentials and the thermodynamical variables. The imposition of spherical
symmetry alone implies that we have a very large gauge (coordinate) freedom to
write the line element. In adopting the form of the line element given by (1)
we exhaust all coordinate freedom with the exception of re-scaling the radial
coordinate and/or the temporal coordinates. It is clear that such re-scaling
would not change the form of (21)-(27). The choice of the perturbed variables
as given in the perturbative scheme is not unique. However, once the line
element has been chosen, the choice of the perturbed variables cannot be
varied to produce the same physics kevin1 .

The Einstein field equations for the static configuration are given by

$\displaystyle\rho_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{B_{0}^{2}}\left[2\frac{B_{0}^{\prime\prime}}{B_{0}}-\left(\frac{B_{0}^{\prime}}{B_{0}}\right)^{2}+\frac{4}{r}\frac{B_{0}^{\prime}}{B_{0}}\right]$
(28) $\displaystyle p_{r0}$ $\displaystyle=$
$\displaystyle\frac{1}{B_{0}^{2}}\left[\left(\frac{B_{0}^{\prime}}{B_{0}}\right)^{2}+\frac{2}{r}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{B_{0}^{\prime}}{B_{0}}\right)+2\frac{A_{0}^{\prime}}{A_{0}}\frac{B_{0}^{\prime}}{B_{0}}\right]$
(29) $\displaystyle p_{t0}$ $\displaystyle=$
$\displaystyle\frac{1}{B_{0}^{2}}\left[\frac{B_{0}^{\prime\prime}}{B_{0}}-\left(\frac{B_{0}^{\prime}}{B_{0}}\right)^{2}+\frac{1}{r}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{B_{0}^{\prime}}{B_{0}}\right)+\frac{A_{0}^{\prime\prime}}{A_{0}}\right]$
(30)

The perturbed field equations up to first order in $\epsilon$ can be written
as

$\displaystyle\bar{\rho}$ $\displaystyle=$
$\displaystyle-3\rho_{0}\frac{b}{B_{0}}T+\frac{1}{B_{0}^{3}}\left[-\left(\frac{B_{0}^{\prime}}{B_{0}}\right)^{2}b+2\left(\frac{B_{0}^{\prime}}{B_{0}}-\frac{2}{r}\right)b^{\prime}-2b^{\prime\prime}\right]T$
(31) $\displaystyle\bar{p}_{r}$ $\displaystyle=$
$\displaystyle-2p_{r0}\frac{b}{B_{0}}T+\frac{2}{B_{0}^{2}}\bigg{[}\left(\frac{B_{0}^{\prime}}{B_{0}}+\frac{1}{r}+\frac{A_{0}^{\prime}}{A_{0}}\right)\left(\frac{b}{B_{0}}\right)^{\prime}$
(32)
$\displaystyle+\left(\frac{B_{0}^{\prime}}{B_{0}}+\frac{1}{r}\right)\left(\frac{a}{A_{0}}\right)^{\prime}\bigg{]}T-2\frac{b}{{A_{0}^{2}}B_{0}}{\ddot{T}}$
$\displaystyle\bar{p}_{t}$ $\displaystyle=$
$\displaystyle-2p_{t0}\frac{b}{B_{0}}T+\frac{1}{B_{0}^{2}}\bigg{[}\left(\frac{b}{B_{0}}\right)^{\prime\prime}+\frac{1}{r}\left(\frac{b}{B_{0}}\right)^{\prime}+2\frac{A_{0}^{\prime}}{A_{0}}\left(\frac{a}{A_{0}}\right)^{\prime}$
(33)
$\displaystyle+\left(\frac{a}{A_{0}}\right)^{\prime\prime}+\frac{1}{r}\left(\frac{a}{A_{0}}\right)^{\prime}\bigg{]}T-2\frac{b}{{A_{0}^{2}}B_{0}}{\ddot{T}}$
$\displaystyle\bar{q}B$ $\displaystyle=$
$\displaystyle\frac{2}{B_{0}}\left(\frac{b}{A_{0}B_{0}}\right)^{\prime}{\dot{T}}$
(34)

The total energy enclosed within $\Sigma$ is obtained by using (19) and (26).
We separate the static and time-dependent/perturbed components and are shown
as follows

$\displaystyle m_{0}(r_{\Sigma})$ $\displaystyle=$
$\displaystyle-\left(r^{2}{B_{0}^{\prime}}+\frac{r^{3}{{B_{0}^{\prime}}^{2}}}{2B_{0}}\right)_{\Sigma}$
(35)

and

$\displaystyle\bar{m}(r_{\Sigma},t)$ $\displaystyle=$
$\displaystyle-\left(\left[{r^{2}}b^{\prime}+\frac{r^{3}{{B_{0}^{\prime}}^{2}}}{2B_{0}}\left(2\frac{b^{\prime}}{B_{0}^{\prime}}-\frac{b}{B_{0}}\right)\right]T(t)\right)_{\Sigma}$
(36)

In the case where the radial and tangential stresses are equal, $p_{r}=p_{t}$,
the condition of pressure isotropy for the static model is $p_{r0}=p_{t0}$
which gives

$\displaystyle\left(\frac{A^{\prime}_{0}}{A_{0}}+\frac{B^{\prime}_{0}}{B_{0}}\right)^{\prime}-\left(\frac{A^{\prime}_{0}}{A_{0}}+\frac{B^{\prime}_{0}}{B_{0}}\right)^{2}-\frac{1}{r}\left(\frac{A^{\prime}_{0}}{A_{0}}+\frac{B^{\prime}_{0}}{B_{0}}\right)+2\left(\frac{A^{\prime}_{0}}{A_{0}}\right)^{2}=0$
(37)

The pressure isotropy condition for the perturbed model is
$\bar{p}_{r}=\bar{p}_{t}$ which gives

$\displaystyle\left[\left(\frac{a}{A_{0}}\right)^{\prime}+\left(\frac{b}{B_{0}}\right)^{\prime}\right]^{\prime}-2\left[\left(\frac{a}{A_{0}}\right)^{\prime}+\left(\frac{b}{B_{0}}\right)^{\prime}\right]\left(\frac{A^{\prime}_{0}}{A_{0}}+\frac{B^{\prime}_{0}}{B_{0}}\right)$
$\displaystyle-\frac{1}{r}\left[\left(\frac{a}{A_{0}}\right)^{\prime}+\left(\frac{b}{B_{0}}\right)^{\prime}\right]+4\frac{A^{\prime}_{0}}{A_{0}}\left(\frac{a}{A_{0}}\right)^{\prime}=0$
(38)

This completes the outline of the perturbative scheme as applied to our choice
of metrics (1) and (10). In the next section we will examine the temporal
aspect more closely.

##  5 Explicit Form of the Temporal Function

We employ the junction conditions derived by Maharaj et al. maharaj3  to
determine the temporal evolution of our model.

$\left({p}_{r}\right)_{\Sigma}=\left(qB\right)_{\Sigma}-\rho_{s}({\mathrm{r}},v)|_{\Sigma}$
(39)

It is important to point out that (39) holds only at the boundary of the star.
We require that the static pressure vanishes at the surface via the condition
$\left(p_{r0}\right)_{\Sigma}=0$, so that the following equation is obtained
in $T(t)$, namely

$\hskip
28.45274pt\alpha_{\Sigma}T-{\ddot{T}}=2\beta_{\Sigma}{\dot{T}}+\lambda_{\Sigma}$
(40)

where $\alpha$, $\beta$ and $\lambda$ are given by

$\displaystyle\alpha(r)$ $\displaystyle=$
$\displaystyle\frac{A_{0}^{2}}{B_{0}b}\bigg{[}\left(\frac{B_{0}^{\prime}}{B_{0}}+\frac{1}{r}+\frac{A_{0}^{\prime}}{A_{0}}\right)\left(\frac{b}{B_{0}}\right)^{\prime}+\left(\frac{B_{0}^{\prime}}{B_{0}}+\frac{1}{r}\right)\left(\frac{a}{A_{0}}\right)^{\prime}$
(41)
$\displaystyle+B_{0}^{2}\bigg{(}\left(\frac{3a}{2A_{0}}+\frac{b}{B_{0}}\right)p_{r0}+\left(\frac{3a}{2A_{0}}+\frac{2b}{B_{0}}\right)\rho_{s0}+$
$\displaystyle\left(\frac{a}{A_{0}}+\frac{b}{B_{0}}\right)\frac{3k}{2rB_{0}}\bigg{)}\bigg{]}$

where $\rho_{s0}$ is the constant string density.

$\hskip
28.45274pt\beta(r)=\frac{A_{0}^{2}}{2b}\left(\frac{b}{A_{0}B_{0}}\right)^{\prime}$
(42) $\lambda(r)=-\frac{A_{0}^{2}B_{0}}{2\epsilon b}\left(\rho_{s}\right)$
(43)

to be evaluated at the boundary $r_{\Sigma}$. It should be noted that $p_{r0}$
vanishes at the boundary. The diffusion equation (14) has been extensively
studied and exact several solutions have been obtained glass ; glass2 ,
maharaj3 , govin , Ghosh1 , byron1  and NaiduNF1 . One such solution of the
diffusion equation (14) for which the string density is a function of the
external radial coordinate is given by

$\rho_{s}({\mathrm{r}})=\rho_{0}+\frac{k}{{\mathrm{r}}}$ (44)

where $\rho_{0}$ and $k$ are constants. The string density profile in (44) was
utilised by Naidu et al. NaiduNF1  to study the effect of an anisotropic
atmosphere on the temperature profiles during radiative collapse. The above
choice of string profile generalizes earlier work by Govender and Thirukannesh
gov2  and Govender et al. BrasselB  in which the string density was constant
($k=0$). The choice of a constant string density not only makes the problem
mathematically tractable but also simplifies the underlying physics. The
constant string distribution gives rise to pressure anisotropy in the exterior
while any inhomogeneities are suppressed. Our choice (44) allows for pressure
anisotropy and inhomogeneities due to density fluctuations. At the boundary of
the star the string density (44) can be written as

$\left(\rho_{s}\right)_{\Sigma}=\rho_{0}+\frac{k}{rB}|_{\Sigma}$ (45)

where we have invoked the junction condition (17).

It is necessary to highlight the connection between $r$ and ${\mathrm{r}}$ at
this point. The boundary of the collapsing star divides spacetime into two
distinct regions ${\cal{M}}^{-}$ and ${\cal{M}}^{+}$, respectively. The
coordinates in the interior spacetime ${\cal{M}}^{-}$ are $(t,r,\theta,\phi)$
while the coordinates in ${\cal{M}}^{+}$ are $(v,{\mathrm{r}},\theta,\phi)$.
The boundary $\Sigma$, is a time-like hypersurface described by the line
element

$ds^{2}_{\Sigma}=-d\tau^{2}+{\cal{R}}^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2})$
(46)

endowed with coordinates ${\xi}^{i}=(\tau,\theta,\phi)$ and
${\cal{R}}={\cal{R}}(\tau)$. Note that the time coordinate $\tau$ is defined
only on the surface $\Sigma$. The junction condition (17) is a consequence of
requiring the smooth matching of the line elements (1) and (10) across
$\Sigma$, ie.,

$(ds^{2}_{-})_{\Sigma}=(ds^{2}_{+})_{\Sigma}=ds^{2}_{\Sigma}$ (47)

For ${\cal{M}}^{-}$ we obtain

$\displaystyle A(r_{\Sigma},t)\dot{t}$ $\displaystyle=$ $\displaystyle 1$ (48)
$\displaystyle r_{\Sigma}B(r_{\Sigma},t)$ $\displaystyle=$
$\displaystyle\cal{R}(\tau)$ (49)

where dots represent differentiation with respect to $\tau$ while for
${\cal{M}}^{+}$ we have

$\displaystyle{{\mathrm{r}}}_{\Sigma}(v)$ $\displaystyle=$
$\displaystyle{\cal{R}}(\tau)$ (50)
$\displaystyle\left(1-\displaystyle\frac{2m}{{\mathrm{r}}}+2\displaystyle\frac{d{{\mathrm{r}}}}{dv}\right)_{\Sigma}$
$\displaystyle=$ $\displaystyle\left(\frac{1}{{\dot{v}}^{2}}\right)_{\Sigma}$
(51)

We observe that (49) and (50) relate $r$ and ${\mathrm{r}}$.

We complete the expression for the temporal function $T(t)$ by solving (40).
This gives

$\hskip
28.45274ptT(t)=\frac{\lambda_{\Sigma}}{\alpha_{\Sigma}}-\exp\left[\left(-\beta_{\Sigma}+\sqrt{\alpha_{\Sigma}+\beta^{2}_{\Sigma}}\right)t\right]$
(52)

which, together with (41) and (44) and $\alpha_{\Sigma}>0$ as well as
$\beta_{\Sigma}<0$ describes a system in static equilibrium that starts to
collapse at $t=-\infty$ and continues to collapse as $t$ increases.

##  6 Dynamical Model

In order to investigate the properties of the extended form of the temporal
function, we make use of the simple Schwarzschild interior metric in isotropic
coordinates bon ; ray  given by

$\displaystyle A_{0}(r)$ $\displaystyle=$ $\displaystyle
c_{1}-\frac{1}{2}\frac{\left(1-r^{2}\right)}{(1+r^{2})}$ (53) $\displaystyle
B_{0}(r)$ $\displaystyle=$ $\displaystyle\frac{2R}{1+r^{2}}$ (54)

where $c_{1}$ and $R$ are constants. Then (28) and (29) can be written as

$\displaystyle\mu_{0}$ $\displaystyle=$ $\displaystyle\frac{3}{R^{2}}$ (55)
$\displaystyle p_{r0}$ $\displaystyle=$
$\displaystyle-\frac{2c_{1}(1+r^{2})-3(1-r^{2})}{R^{2}\left[2c_{1}(1+r^{2})-(1-r^{2})\right]}$
(56)

The constant $R$ is easily determined from (55), given the initial static
energy density, and parameter $c_{1}$ is obtained from (56) by evaluation at
the boundary, giving

$c_{1}=\frac{3(1-r_{\Sigma}^{2})}{2(1+r_{\Sigma}^{2})}$ (57)

Restrictions on $r_{\Sigma}$ as given by Santos bon  are noted, namely

$\frac{2m_{0}}{r_{\Sigma}}=\frac{4r_{\Sigma}^{2}}{(1+r_{\Sigma}^{2})^{2}}<\frac{3}{4}$
(58)

and

$0\leq r<r_{\Sigma}<\frac{1}{\sqrt{3}}.$ (59)

We also note that in the case $p_{r}=p_{t}$, the anisotropy parameter $\Delta$
vanishes.

##  7 Radiating Collapse

We note that (31)-(34) contain two unspecified quantities, namely $a(r)$ and
$b(r)$, which modulate the temporal part of the gravitational potentials. Thus
it is important that these are determined carefully in order to obtain a
physically meaningful dynamical model. Following Chan et al. chan2  we adopt
the following form for $b(r)$,

$\hskip 28.45274ptb(r)=\left(1+\xi f(r)\right)A_{0}B_{0}$ (60)

This choice for $b(r)$ has been widely used to investigate stability of
radiating stars undergoing dissipative collapse in the form of a radial heat
flux Kich , herr2  and chan2 . Furthermore, we follow narenee  and choose the
following form $f(r)=r^{2}$. Using (60) in (38) above, we obtain an explicit
form for $a(r)$ as

$\displaystyle a(r)$ $\displaystyle=$
$\displaystyle\left(\frac{2c_{1}+1}{2}-\frac{1}{1+r^{2}}\right)\times\bigg{[}c_{2}-\frac{1-\xi}{1+r^{2}}-\frac{\xi}{2}(2c_{1}+1)(1+r^{2})\bigg{]}$
(61)
$\displaystyle+\frac{(2c_{1}+1)(1+2c_{1}(1-\xi)-11\xi)-8c_{3}R^{2}}{2(2c_{1}+1)(1+r^{2})}+$
$\displaystyle 2\xi\left(2c_{1}+1\right)\log(1+r^{2})$

where $c_{2}$ and $c_{3}$ are constants of integration. These may be set by
considering the work of Govender et al. gov1  with the simple case of the
relationship between $a(r)$ and $b(r)$ being employed, namely

$\frac{a(r)}{A_{0}(r)}=\frac{b(r)}{B_{0}(r)}$ (62)

At this stage, we point out that the radial and temporal evolution of our
model is fully determined. We use numerical data given in Table 1. for
performing graphical analyses of stability and horizon formation comparisons
which follow.

Table 1: Values of constants used in model Name | Value | Reference
---|---|---
$\mu_{0}$ | $2.36321\times 10^{8}g/cm^{3}$ | bon
$r_{\Sigma}$ | $2.159\times 10^{8}cm$ | bon
R | $26.08\times 10^{8}cm$ | (55)
$c_{1}$ | 1.49984 | (57)
$\xi$ | -0.2 | chan2 ; narenee

###  7.1 Luminosity

The luminosity for an observer at rest at infinity is given by

$L_{\infty}(v)=-\left(\frac{dm(v)}{dv}\right)_{\Sigma}$ (63)

where $v$ is the retarded time. The luminosity can then be written as

$-\frac{dm(v)}{dv}=-\frac{dm}{dt}\frac{dt}{d\tau}\frac{d\tau}{dv}$ (64)

where

$\frac{dt}{d\tau}=\frac{1}{A}\hskip 17.07182pt\mathrm{and}\hskip
17.07182pt\frac{d\tau}{dv}=\frac{1}{\dot{v}}=\frac{r\dot{B}}{A}+\frac{B+rB^{\prime}}{B}$
(65)

and $\tau$ is the proper time defined on $\Sigma$ pin  with $A$ and $B$ given
by (21) and (22). For our model, this is calculated to be

$L_{\infty}(t)=-\frac{\epsilon g_{r_{\Sigma}}\dot{T}(t)}{A_{0}+\epsilon
aT(t)}\left(1+\frac{rB_{0}^{\prime}}{B_{0}+\epsilon
bT(t)}\right)|_{r=r_{\Sigma}}$ (66)

where

$g_{r_{\Sigma}}=-r^{2}b^{\prime}-\frac{r^{3}{{B_{0}^{\prime}}^{2}}}{2B_{0}}\left(2\frac{b^{\prime}}{B_{0}^{\prime}}-\frac{b}{B_{0}}\right)|_{r=r_{\Sigma}}$
(67)

###  7.2 Horizon Formation

From (63) and (64) we note that the luminosity vanishes when

$\frac{1}{\dot{v}}=0$ (68)

which determines the time of formation of the horizon as the collapse process
proceeds from $-\infty<t\leq t_{H}$ which corresponds to $-\infty<v<\infty$.
For our model we have the following instances where the luminosity vanishes.
By examining (66) we must have $\dot{T}=0$ or $g_{r_{\Sigma}}=0$.

####  7.2.1 Case 1: $\dot{T}=0$

In the case $\dot{T}=0$, we have from the derivative of (52)

$-\eta e^{\eta t}=0$ (69)

where

$\eta=-\beta_{\Sigma}+\sqrt{\alpha_{\Sigma}+\beta_{\Sigma}^{2}}$ (70)

Thus $\dot{T}=0$ implies $\eta=0$ which forces $\alpha_{\Sigma}=0$. From the
expression for $T(t)$ (52), this is only possible if $\lambda_{\Sigma}=0$ (ie.
vanishing of the string density). We observe that removing the string density
gives a pure radiation solution and the horizon is able to form. However, the
inclusion of strings inhibits the formation of the horizon.

####  7.2.2 Case 2: $g_{r_{\Sigma}}=0$

The second case $g_{r_{\Sigma}}=0$ gives

$\frac{2r^{3}Rh(r)}{(1+r^{2})^{4}}=0$ (71)

using $B_{0}(r)$ from (54) and $b(r)$ from (60) where we have used

$\displaystyle h(r)$ $\displaystyle=$
$\displaystyle(r^{2}-1)(3+\xi(r^{4}+3r^{2}-1))+2c_{1}(1+r^{2}-\xi+2r^{4}\xi+r^{6}\xi)$
(72)

Examining (71) we see that $g_{r_{\Sigma}}=0$ when either $R=0$ or $h(r)=0$.
For our model, we have chosen an initial density $\mu_{0}$ as given in Table
1. following numerical work by Santos bon . By (55) this means $R\neq 0$ hence
we consider the second possibility of $h(r)=0$. Given $R$, this places a
restriction on $c_{1}$ thus creating a relationship between constants $c_{1}$
and $R$.

##  8 Stability Analysis

In order to provide insight into the stability of the star, we begin with the
second law of thermodynamics, and follow the approach of herr2 . This leads to
the adiabatic parameter $\Gamma$ which is the ratio of specific heats at
constant pressure and constant volume, and taken to be constant throughout the
distribution (or at least in the region being studied). In literature
referring to this ratio in herr2 , it was considered an indicator of
stability, and called the stability factor narenee . From the expressions for
$\Gamma$ given below chan2 , it is clear that pressure anisotropy and the
presence of radiation within the stellar core affect the stability factor. For
example, if the sign of the anisotropy parameter $\Delta=(p_{t0}-p_{r0})$
changes, the stellar core becomes unstable. We observe this is in the
Newtonian limit,

$\displaystyle\Gamma$ $\displaystyle<$
$\displaystyle\frac{4}{3}+\left[\frac{1}{3|p_{r0}^{\prime}|}\left(4\frac{(p_{t0}-p_{r0})}{r}+2\alpha_{\Sigma}\xi|f^{\prime}|\right)\right]_{max}$
(73)

In agreement with classical fluid dynamics, the fluid sphere becomes more
unstable (increasing the unstable range of $\Gamma$) as a result of the
Newtonian contribution due to dissipation. Relativistic contributions from the
energy density lead to a stability factor different from its Newtonian
counterpart.

$\displaystyle\Gamma$ $\displaystyle<$
$\displaystyle\frac{4}{3}+\bigg{[}\frac{1}{3|p_{r0}^{\prime}|}\bigg{(}4\frac{(p_{t0}-p_{r0})}{r}+2\alpha_{\Sigma}\xi|f^{\prime}|+\mu_{0}\bigg{(}p_{r0}r-\xi|f^{\prime}|\bigg{)}\bigg{)}\bigg{]}_{max}$
(74)

Equation (74) shows that the unstable range of $\Gamma$ is increased by the
Newtonian term due to dissipation, as in the Newtonian limit. Furthermore, the
unstable range of $\Gamma$ is increased by the relativistic correction due to
the static background fluid configuration; however the relativistic correction
due to dissipation decreases the unstable range of $\Gamma$. Bonnor et. al.
bon  state that dissipation, by diminishing that total mass entrapped inside
the fluid sphere, renders the systems less unstable.

In order to investigate the stability of our model in both the Newtonian and
post-Newtonian limits, we graphed $\Gamma$ for the case of pure radiation
(absence of string density), radiation plus constant density ($k=0$) and the
radiation and inhomogeneous string density ($\rho_{s}\neq 0,k\neq 0$). Since
our static model is described by the interior Schwarzschild solution, the
anisotropy parameter $\Delta=p_{t0}-p_{r0}$ vanishes in (73) and (74). The
modified effects due to pure string density and inhomogeneity are encoded in
$\alpha_{\Sigma}$. We have also graphed the luminosity as a function of time
for both the pure and generalized Vaidya exterior. It is important to note
that the graphs are plotted using geometrized units, where $G$ and $c$ are
taken to be unity BrasselB .

Figure 1: Adiabatic parameter in the Newtonian limit for various string
densities, $\rho_{s}=\rho_{s}(\rho_{s0},k)$ from the centre $r=0$ to the
boundary $r=r_{\Sigma}$, $r$ in units of seconds. Figure 2: Adiabatic
parameter in the post-Newtonian limit for various string densities,
$\rho_{s}=\rho_{s}(\rho_{s0},k)$ from the centre $r=0$ to the boundary
$r=r_{\Sigma}$, $r$ in units of seconds.

Figure 3: Luminosity at infinity (geometrized) versus time (seconds) for
various string densities, $\rho_{s}=\rho_{s}(\rho_{s0},k)$

.

##  9 Physical Analysis and Conclusion

Figure 1 shows the stability factor when the star is close to hydrostatic
equilibrium in the Newtonian limit. We observe that the different matter
configurations exhibit instability with $\Gamma<\frac{4}{3}$ which signifies
the onset of collapse. The inclusion of the string field drives the stellar
fluid towards instability with this effect being enhanced by inhomogeneity
($k>0$). Figure 2 displays $\Gamma$ for the post-Newtonian regime. It is clear
that the collapse process drives the fluid towards stability. The presence of
the strings and their associated anisotropy and inhomogeneity make the fluid
more stable at late times. This could be due to trapping of heat within the
stellar core due to an inhomogeneous atmosphere, thus resulting in higher core
temperatures. An increase in the core temperature results in an increase in
outward pressure thus hindering gravitational collapse. In Figure 3 we note
that inclusion of the string density field promotes an earlier time of horizon
formation (luminosity vanishes), with this effect being particularly sensitive
to string inhomogeneity. The effect of the string density on the time of
formation of the horizon was also observed by Govender megan . They reasoned
that the presence of the anisotropic strings in the exterior lowered the rate
of heat dissipation to the exterior thus leading to a lower heat production
rate within the core. This results in a lower outward radial pressure thus
allowing gravity to dominate within the stellar interior. This results in a
higher collapse rate and eventually to the horizon forming earlier.
Luminosities for collapsing neutron star models were studied by de Oliveira et
al. de1  in which they considered profiles in the $\gamma$-ray, X-ray and
visible bandwidths for core masses of $2M_{\odot}$ and $3M_{\odot}$ and a
radius of 10 km. The luminosity profiles for all three bandwidths are similar
to the profiles depicted in Figure 3. They noted that the radiation pulses do
not occur simultaneously for an observer placed at infinity from the
collapsing body. Furthermore, they show that nearly all the energy is emitted
in the form of $\gamma$-rays. It is well known that the luminosity profile
depends on the increasing gravitational redshift as the star collapses and the
increase in the effective temperature. Our study provides a possible mechanism
to explain the temperature changes within the core which manifests as the
luminosity profiles displayed in Figure 3.

It is important to note that while the dynamical (in)stability of radiating
spheres has been extensively studied in the Newtonian and post-Newtonian
approximations, our investigation is the first attempt at considering the
dynamics of the collapse process with a generalised Vaidya exterior. The
generalised Vaidya atmosphere alters the temporal evolution of the model which
impacts on the stability and time of formation of the horizon. While our study
has considered an imperfect fluid interior with heat dissipation and pressure
anisotropy it would be interesting to include shear viscosity within the
framework of extended irreversible thermodynamics.

##  10 Acknowledgments

NFN wishes to acknowledge funding from the National Research Foundation (Grant
number: 116629) as well as the DSI-NRF Centre of Excellence in Mathematical
and Statistical Sciences (CoE-MaSS). RB and MG acknowledge financial support
from the office of the DVC: Research and Innovation at the Durban University
of Technology. MG is indebted to Dr A Barnes for providing the excellent
facilities at Glenwood Boys High School where part of this work was completed.

## References

  * (1) Oppenheimer, J.R., Snyder, H.: On continued gravitational contraction. Phys. Rev. D. 56, 455 (1939)
  * (2) Vaidya, P.C.: The gravitational field of a radiating star. Proc. Indian Acad. Sc. A. 33, 264 (1951)
  * (3) Santos, N.O.: Non-adiabatic radiating collapse. Mon. Not. R. Astron. Soc. 216, 403 (1985)
  * (4) Mkenyeleye, M.D., Goswami, R., Maharaj, S.D.: Gravitational collapse of generalised Vaidya spacetime. Phys. Rev. D. 92 024041 (2015)
  * (5) Sharma, R., Tikekar, R.: Non-adiabatic radiative collapse of a relativistic star under different initial conditions. Pramana-J. Phys. 79, 501 (2012)
  * (6) Sharma, R., Tikekar R.: Space-time inhomogeneity, anisotropy and gravitational collapse. Gen. Relativ. Gravit. 44, 2503-2520 (2012)
  * (7) Sharma, R., Das, S., Rahaman, F., Shit, G.C.: Gravitational collapse of a circularly symmetric star in an anti-de Sitter spacetime. Astrophys. Space Sci. 259, 40 (2015)
  * (8) Govender, M., Bogadi, R., Sharma, R., Das, S.: Gravitational collapse in spatially isotropic coordinates. Astrophys. Space Sci. 361, 33 (2016)
  * (9) Govender, M., Reddy, K.P., Maharaj, S.D.: The role of shear in dissipative gravitational collapse. Int. J. Mod. Phys. D. 23, 1450013 (2014)
  * (10) Chandrasekhar, S.: The dynamical instability of gaseous masses approaching the Schwarzschild limit in general relativity. Astrophys. J. 140, 417 (1964)
  * (11) Herrera, L., Le Denmat, G., Santos, N.O.: Dynamical instability for non-adiabatic spherical collapse. Mon. Not. R. Astron. Soc. 237, 257 (1989)
  * (12) Chan, R., Herrera, L., Santos, N.O.: Dynamical Instability for Radiating Anisotropic Collapse. Mon. Not. R. Astron. Soc. 265, 533 (1993)
  * (13) Herrera, L., Le Denmat, G., Santos, N.O.: Dynamical instability and the expansion-free condition. Gen. Relativ. Gravit. 44, 1143 (2012)
  * (14) Bonnor, W.B., de Oliveira, A.K.G., Santos, N.O.: Radiating spherical collapse. Phys. Rep. 181, 269 (1989)
  * (15) Glass, E.N., Krisch, J.P.: Radiation and string atmosphere for relativistic stars. Phys. Rev. D. 57, 5945 (1998)
  * (16) Glass, E.N., Krisch, J.P.: Two-fluid atmosphere for relativistic stars. Class. Quantum Grav. 16, 1175 (1999)
  * (17) Maharaj, S.D., Govender, M.: Radiating Collapse with Vanishing Weyl stresses. Int. J. Mod. Phys. D. 14, 667-676 (2005)
  * (18) Maharaj, S.D., Govender, G., Govender, M.: Radiating stars with generalised Vaidya atmospheres. Gen. Relativ. Gravit. 44, 1089-1099 (2012)
  * (19) Husain, V.: Exact solutions for null fluid collapse. Phys. Rev. D. 53, 1759 (1996)
  * (20) Wang, A., Wu, Y.: Generalized Vaidya Solutions. Gen. Relativ. Gravit. 31, 107 (1999)
  * (21) Govinder, K.S., Govender, M.: Gravitational collapse of null radiation and a string fluid. Phys. Rev. D. 68, 024034 (2003)
  * (22) de Oliveira, A.K.G., Santos, N.O., Kolassis, C.A.: Collapse of a radiating star. Mon. Not. R. Astron. Soc. 216, 1001 (1985)
  * (23) Govender, M., Thirukkanesh, S.: Anisotropic static spheres with linear equation of state in isotropic coordinates. Astrophys. Space Sci. 358, 16 (2015)
  * (24) Reddy, K.P., Govender, M., Maharaj, S.D.: Impact of anisotropic stresses during dissipative gravitational collapse. Gen. Relativ. Gravit. 47, 35 (2015)
  * (25) Ghosh, S.G., Deshkar, D.W.: Exact Nonspherical Radiating Collapse. Int. J. Mod. Phys. A. 22, 2945 (2007)
  * (26) Brassel, P.B., Goswami, R., Maharaj, S.D.: Collapsing radiating stars with various equations of state. Phys. Rev. D. 95, 124051 (2017)
  * (27) Naidu, N.F., Govender, M., Thirukkanesh, S., Maharaj, S.D.: Radiating fluid sphere immersed in an anisotropic atmosphere. Gen. Relat. Gravit. 49, 95 (2017)
  * (28) Govender, G., Brassel, P.B., Maharaj, S.D.: The effect of a two-fluid atmosphere on relativistic stars. Eur. Phys. J. C. 75, 324 (2015)
  * (29) Raychaudhuri, A.K., Maiti, S.R.: Conformal flatness and the Schwarzschild interior solution. J. Math. Phys. 20, 245 (1979)
  * (30) Chan, R., Kichenassamy, S., Le Denmat, G., Santos, N.O.: Heat flow and dynamical instability in spherical collapse. Mon. Not. R. Astron. Soc. 239, 91 (1989)
  * (31) Govender, M., Mewalal, N., Hansraj, S.: The role of an equation of state in the dynamical (in)stability of a radiating star. Eur. Phys. J. C. 79, 24 (2019)
  * (32) Govender, M., Govinder, K.S., Maharaj, S.D., Sharma, R., Mukherjee, S., Dey, T.K.: Radiating Spherical Collapse with Heat Flow. Int. J. Mod. Phys. D. 12, 667 (2003)
  * (33) Pinheiro, G., Chan, R.: Radiating gravitational collapse with shear viscosity revisited. Gen. Relativ. Gravit. 40, 2149 (2008)
  * (34) Govender, M.: Nonadiabatic Spherical Collapse with a Two-Fluid Atmosphere. Int. J. Mod. Phys. D. 22, 1350049 (2013)
  * (35) de Oliveira, A.K.G., de F. Pacheco, J.A., Santos, N.O.: More about collapse of a radiating star. MNRAS. 220, 405 (1986)