# Effects of electromagnetic field on a radiating star

Suresh C<EMAIL_ADDRESS>Department of Physics &
Astronomical Sciences, Central University of Jammu, Samba, J&K- 181143, India.
Ayan<EMAIL_ADDRESS>Department of Physics & Astronomical
Science, Central University of Himachal Pradesh, Dharamshala- 176215, India.
Akshay Kumar<EMAIL_ADDRESS>Department of Physics &
Astronomical Science, Central University of Himachal Pradesh, Dharamshala-
176215, India.

In this paper we study the shear free spherical symmetric gravitational
collapse of charged radiating star. All the physical quantities including
pressure, density are regular. Energy conditions are satisfied throughout the
interior of the matter configuration. The luminosity is time independent and
mass is radiated linearly. The causal and non causal temperature remains
greater than that of the uncharged collapsing scenario.

Keywords: Electromagnetic field. Gravitational collapse. Karmarkar condition.

##  1 Introduction

The quest to understand the formation of stars, horizons, cosmic censorship
conjecture, and spacetime singularities has led to a substantial amount of
work on gravitational collapse during the last hundred years. Beginning with
the pioneering work of Oppenheimer-Snyder and Dutt on homogeneous dust
collapse, also known as the OSD model [1, 2], our knowledge of gravitational
collapse has been considerably extended through the inclusion of matter
inhomogeneities, tangential and radial pressures [3, 4]. Implications of these
features towards the collapse process have led to the development of
interesting phenomenology directed to detect whether the end result is a black
hole or a horizonless singularity [4, 5, 6, 7, 8, 9, 10, 11]. Further, a
number of exact physical solutions have been introduced to study the stability
and dynamics of stellar objects [12, 13, 14, 16, 17, 18, 19, 15, 20, 21].
However, the study of spherical stellar systems is incomplete without
considering the effects of charge on its time development, as it plays an
important role throughout its lifespan. The effects of charge in the stellar
system were started with the pioneer works of Reissner–Nordström [12, 13] and
Bonnor [22, 23]. It has been observed that the collapsing process is delayed
due to the presence of the charge in the stellar system [24, 25, 26, 27, 28,
29, 30]. Indeed, the fact that the presence of electric/ magnetic charges are
crucial in the time development of stellar as well as supermassive black holes
at the center of galaxies, has been stressed several times [31, 32, 33, 34,
35, 36, 37]. However, several facets of such solutions, like thermodynamic
stability, apparent luminosity, effect of temperature profiles and energy
conditions on these physically acceptable solutions remains unknown.
Furthermore, such solutions are context dependent, in the sense that the
mathematical techniques are dependent on the functional forms of the choices
of density, pressure and metric variables. In this paper, we show that, using
the Karmarkar conditions, it is possible to understand the collapse process of
a dissipative, shear- free collapse of a charged matter field admitting
anisotropic stress, under generic choices of the matter and geometric
variables. Indeed, the present method adapts itself naturally to anisotropy of
matter fields and the dissipative nature of the energy- momentum tensor. We
also show the precise dependence of these variables on the stability,
luminosity and adiabatic index.

Before describing our model, let us understand the precise nature of
difficulties arising in the study of dissipative stellar collapse models.
First, note that the radiating gravitational collapse of a stellar object is
an extreme dissipative process. This requires terms relating heat flux to
geometric variables be introduced which are continuous and differentiable.
Secondly, the continuous and smooth matching of the interior and exterior
spacetimes over the timelike hypersurface $\Sigma$ requires that radial
pressure to be proportional to the heat flux at the boundary of collapsing
radiant star. Since a physically consistent charged gravitationally collapsing
model must satisfy both Einstein- Maxwell field equations as well as the set
of energy conditions, the nonlinearity of the Einstein- Maxwell field
equations makes it difficult to find its exact solutions. In this paper, we
show that these complications described above may be taken care of by the
Karmarkar conditions [38, 39, 40, 41, 42, 43]. For a shear free stellar
system, the Karmarkar condition relates the two metric variables in terms of
each other, and with a single choice of metric variables, the system is
exactly solvable. We must however stress that our solution only represents
astrophysical systems which are spherical or nearly spherical/ slowly
rotating. Exact solution of non spherical collapse processes remains a major
unsolved problem.

In this paper, we extend this Karmarkar formalism for a charged radiating
collapsing stellar model. To determine the metric variables which satisfy the
solutions of nonlinear field equations and energy conditions, we consider an
adhoc but physical form of the pressure anisotropy. There are two advantages
of considering this particular form of pressure anisotropy. First is using
this form together with the Karmarkar condition make our model to be exactly
solvable. Second, we do not have to consider an adhoc form of charged density,
which was the requirement in the earlier studies of the charged stellar
systems [44, 45]. We also show that our model is physical, as all the physical
quantities are regular and also satisfy the standard energy conditions.
Furthermore, our study of the thermodynamical evolution shows that the causal
and non causal temperature for the charged case remains greater than that of
the uncharged collapsing scenario, as required.

The present article has been organized as follows. In the following section 2
we describe the shear free interior spacetime with composite matter
distribution and the Einstein- Maxwell field equations. The interior shear
free spaceime is smoothly matched with the exterior Reissner–Nordström
spacetime across the timelike surface. In the section 3, we present a new
class of exact solutions to the Einstein- Maxwell field equations and derive
the explicit expressions for physical quantities. It is then shown that the
said model is physically viable as all the physical quantities are regular.
The study of the stability of stellar system, which shows that the system is
unstable and the energy conditions are well satisfied throughout the collapse
are given in section 4. In section 5, we give the thermodynamical evolution of
the charged radiative gravitational collapse. Finally, discussion of the
result accompanied with concluding remarks are in section 6.

##  2 Matter distribution and the governing equations

In comoving coordinates, the general shear free spherically symmetric metric
is given by

$\displaystyle
ds^{2}=-a(r)^{2}dt^{2}+b(r)^{2}\,f(t)^{2}dr^{2}+r^{2}\,b(r)^{2}\,f(t)^{2}\left(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}\right)$
(2.1)

Also, we consider the energy-momentum tensor for the charged anisotropic
distribution with radial heat flow as

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle(p_{t}+\rho)u_{\mu}u_{\nu}+p_{t}g_{\mu\nu}+(p_{r}-p_{t})X_{\mu}X_{\nu}$
(2.2)
$\displaystyle+q_{\mu}u_{\nu}+q_{\nu}u_{\mu}+\left(F^{\lambda}_{\mu}\,F_{\nu\lambda}-\frac{1}{4}\,g_{\mu\nu}F^{\lambda\delta}\,F_{\lambda\delta}\right)$

where $\rho$, $p_{r}$ and $p_{t}$ are the density, radial pressure and
tangential pressure respectively, $q^{\mu}$, $u^{\mu}$ and $X^{\mu}$ are and
radial heat flow vector, unit time-like $4$-velocity vector and unit space-
like vector along radial $4$-vector respectively, satisfying
$u_{\mu}u^{\mu}=-X_{\mu}X^{\mu}=-1$ and $u_{\mu}X^{\mu}=u_{\mu}q^{\mu}=0$. And
$F_{\mu\nu}$ represent the Maxwell field tensor. In the comoving coordinates,
the radial heat flow vector $q^{\mu}$, $4$-velocity $u^{\mu}$ and unit space-
like vector $X^{\mu}$ of the fluid are given by

$\displaystyle\hskip 56.9055ptq^{\mu}=\frac{1}{b\,f}\,X^{\mu}\hskip
14.22636pt;\hskip 14.22636ptu^{\mu}=\frac{1}{a}\,\delta^{\mu}_{0}\hskip
14.22636pt;\hskip 14.22636ptX^{\mu}=\frac{1}{b\,f}\,\delta^{\mu}_{1}.$ (2.3)

The expansion scalar, $\Theta$ for the metric (2.1) is given by

$\displaystyle\Theta$ $\displaystyle=$
$\displaystyle\bigtriangledown_{\mu}u^{\nu}=\frac{3\,\dot{f}}{a\,f}.$ (2.4)

The Maxwell’s equations are given by

$\displaystyle F_{\mu\,\nu}$ $\displaystyle=$
$\displaystyle\phi_{\nu,\,\mu}-\phi_{\mu,\,\nu}$ (2.5) $\displaystyle
F^{\mu\,\nu}\,_{;\nu}$ $\displaystyle=$ $\displaystyle J^{\mu}$ (2.6)

where $\phi_{\mu}$ and $J^{\mu}$ represents the four potential and four
current respectively. It must be noted that the charge is assumed to be at
rest with respect to the comoving coordinates of the metric (2.1), which
implies that we have no magnetic field present, thus the four potential
$\phi_{\mu}$ and four current $J^{\nu}$ takes the form as

$\displaystyle\phi_{\mu}$ $\displaystyle=$
$\displaystyle\phi(t,r)\,\delta^{0}_{\mu}\hskip 28.45274pt;\hskip
28.45274ptJ^{\mu}=\sigma(t,r)u^{\mu}$ (2.7)

Using the choice of the potential from equation (2.7) into the equation (2.5),
the nonzero component of $F_{\mu\nu}$ are

$\displaystyle F_{01}$ $\displaystyle=$ $\displaystyle-
F_{10}=-\frac{\partial\phi}{\partial r}$ (2.8)

Now, for the interior spacetime (2.1) and (2.3), using equations (2.7) and
(2.8) into the Maxwell’s equations (2.5) and (2.6) we have

$\displaystyle\phi^{{}^{\prime\prime}}-\left(\frac{b^{{}^{\prime}}}{b}-\frac{a^{{}^{\prime}}}{a}+\frac{2}{r}\right)\phi^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle ab^{2}f^{2}\sigma$ (2.9)
$\displaystyle\dot{\phi}\,^{{}^{\prime}}+\frac{\dot{f}}{f}\phi^{{}^{\prime}}=0$
(2.10)

On integration of equation (2.9) we obtained

$\displaystyle\phi^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{s\,a}{r^{2}\,b\,f}$ (2.11)

the law of conservation of charge implies that

$\displaystyle s(r)$ $\displaystyle=$
$\displaystyle\int_{0}^{r}r^{2}\,\sigma\,b^{3}\,f^{3}\,dr$ (2.12)

where $s(r)$ is the time independent total electric charge $Q$ within a radius
$r$. The non vanishing components of Einstein-Maxwell field equations for the
metric (2.1), energy momentum tensor (2.2) and (2.3) are $($using units with
$c=1=8\pi G)$

$\displaystyle\rho+\frac{1}{2}\frac{s^{2}}{r^{4}\,b^{4}\,f^{4}}$
$\displaystyle=$
$\displaystyle\frac{3\,\dot{f}^{2}}{a^{2}\,f^{2}}-\frac{1}{b^{2}\,f^{2}}\left(\frac{2\,b^{{}^{\prime\prime}}}{b}-\frac{b^{{}^{\prime}}\,{}^{2}}{b\,^{2}}+\frac{4}{r}\frac{b^{{}^{\prime}}}{b}\right),$
(2.13) $\displaystyle{p}_{r}-\frac{1}{2}\frac{s^{2}}{r^{4}\,b^{4}\,f^{4}}$
$\displaystyle=$
$\displaystyle-\frac{1}{a^{2}}\left(\frac{2\,\ddot{f}}{f}+\frac{\dot{f}^{2}}{f^{2}}\right)+\frac{1}{b^{2}\,f^{2}}\left(\frac{2\,a^{{}^{\prime}}\,b^{{}^{\prime}}}{a\,b}+\frac{2}{r}\left(\frac{a^{{}^{\prime}}}{a}+\frac{b^{{}^{\prime}}}{b}\right)+\frac{b^{{}^{\prime}}\,{}^{2}}{b\,^{2}}\right),$
(2.14) $\displaystyle{p}_{t}+\frac{1}{2}\frac{s^{2}}{r^{4}\,b^{4}\,f^{4}}$
$\displaystyle=$
$\displaystyle-\frac{1}{a^{2}}\left(\frac{2\,\ddot{f}}{f}+\frac{\dot{f}^{2}}{f^{2}}\right)+\frac{1}{b^{2}\,f^{2}}\left(\frac{a^{{}^{\prime\prime}}}{a}+\frac{1}{r}\left(\frac{a^{{}^{\prime}}}{a}+\frac{b^{{}^{\prime}}}{b}\right)-\frac{b^{{}^{\prime}}\,{}^{2}}{b\,^{2}}+\frac{b^{{}^{\prime\prime}}}{b}\right),$
(2.15) $\displaystyle q$ $\displaystyle=$
$\displaystyle-\frac{2\,a^{{}^{\prime}}\,\dot{f}}{a^{2}\,b^{2}\,f^{3}}.$
(2.16)

where dot and prime are the derivatives with respect to $t$ and $r$
respectively. From equations (2.13)-(2.16) we can see that the number of field
equations are less than that of the number of unknowns. Also, the form of the
three metric potentials fixes all the unknown physical quantities of the
system. So, in order to completely study the collapsing phenomena we need to
find the forms of these metric potentials.

First, let us give the Israel-Darmois junction conditions. The interior
manifold is $\it\bf{M^{-}}$ and exterior manifold is $\it\bf{M^{+}}$ to be
matched across the bounding timelike three space $\Sigma$, at $r=r_{b}$. The
exterior spacetime is described by the Vaidya– Reissner– Nordstrm spacetime
having outgoing radial flow of the radiation around a charged spherically
symmetric source of gravitational field is given by [12, 13]

$\displaystyle
ds^{2}_{+}=-\left(1-\frac{2M(v)}{{\bf{r}}}+\frac{Q^{2}}{\bf{r}^{2}}\right)dv-2dvd{\bf{r}}+{\bf{r}}^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)$ (2.17)

where $M(v)$ and $Q$ are the total mass and total charge respectively. The
junction conditions require the matching of metric as well as the extrinsic
curvatures

$\displaystyle ds^{2}_{\Sigma}$ $\displaystyle=$
$\displaystyle(ds_{-})^{2}_{\Sigma}=(ds_{+})^{2}_{\Sigma}$ (2.18)
$\displaystyle\left[K_{ij}\right]_{\Sigma}$ $\displaystyle=$ $\displaystyle
K_{ij}^{+}=K_{ij}^{-}$ (2.19)

where $K_{ij}^{\pm}$ is the extrinsic curvature to $\Sigma$. The junction
condition (2.18) at the hypersurface $\Sigma$ gives

$\displaystyle dt$ $\displaystyle=$ $\displaystyle
a(r)_{{}_{\Sigma}}^{-1}\,d\tau,$ (2.20)
$\displaystyle{\bf{r}}_{{}_{\Sigma}}(v)$ $\displaystyle=$
$\displaystyle(r\,b\,f)_{{}_{\Sigma}},$ (2.21)
$\displaystyle\left(\frac{dv}{d\tau}\right)_{\Sigma}^{-2}$ $\displaystyle=$
$\displaystyle\left(1-\frac{2M}{\bf{r}}+\frac{Q^{2}}{\bf{r}^{2}}+2\frac{d\bf{r}}{dv}\right)_{\Sigma}.$
(2.22)

where $\tau$ is the time coordinate defined only on the hypersurface $\Sigma$.
The unit normal vectors on the hypersurface $\Sigma$ for the interior and
exterior spacetime are given by

$\displaystyle n^{-}_{l}$ $\displaystyle=$
$\displaystyle\left[0,(b\,f)_{{}_{\Sigma}},0,0\right]$ (2.23) $\displaystyle
n^{+}_{l}$ $\displaystyle=$
$\displaystyle\left(1-\frac{2M(v)}{{\bf{r}}}+\frac{Q^{2}}{\bf{r}^{2}}+2\frac{d{\bf{r}}}{dv}\right)^{-\frac{1}{2}}_{\Sigma}\left(-\frac{d{\bf{r}}}{dv}\delta^{0}_{l}+\delta^{1}_{l}\right)_{\Sigma}$
(2.24)

The non vanishing components of the extrinsic curvature for metrics (2.1) and
(2.17) are given by

$\displaystyle K^{-}_{\tau\tau}$ $\displaystyle=$
$\displaystyle-\left[\frac{a^{{}^{\prime}}}{a\,b\,f}\right]_{\Sigma},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
K^{-}_{\theta\theta}=\left[r\,b\,f\left(1+\frac{r\,b^{{}^{\prime}}}{b}\right)\right]_{\Sigma},$
(2.25) $\displaystyle K^{+}_{\tau\tau}$ $\displaystyle=$
$\displaystyle\left[\frac{d^{2}v}{d\tau^{2}}\left(\frac{dv}{d\tau}\right)^{-1}-\left(\frac{dv}{d\tau}\right)\left(\frac{M}{{\bf{r}}^{2}}-\frac{Q^{2}}{{\bf{r}}^{3}}\right)\right]_{\Sigma},$
(2.26) $\displaystyle K^{+}_{\theta\theta}$ $\displaystyle=$
$\displaystyle\left[\left(\frac{dv}{d\tau}\right)\left(1-\frac{2M}{\bf{r}}+\frac{Q^{2}}{{\bf{r}}^{2}}\right){\bf{r}}-{\bf{r}}\frac{d{\bf{r}}}{d\tau}\right]_{\Sigma},$
(2.27) $\displaystyle K^{-}_{\phi\phi}$ $\displaystyle=$
$\displaystyle\sin^{2}{\theta}K^{-}_{\theta\theta}\,\,,\,\,K^{+}_{\phi\phi}=\sin^{2}{\theta}K^{+}_{\theta\theta}.$

Now, from the second junction condition (2.19)
$K^{+}_{\theta\theta}=K^{-}_{\theta\theta}$ at hypersurface $\Sigma$, and
using equations (2.20), (2.21) and (2.22) gives

$\displaystyle\left[r\,b\,f\left(1+\frac{r\,b^{{}^{\prime}}}{b}\right)\right]_{\Sigma}$
$\displaystyle=$
$\displaystyle\left[\left(\frac{dv}{d\tau}\right)\left(1-\frac{2M}{{\bf{r}}}+\frac{Q^{2}}{{\bf{r}}^{2}}\right){\bf{r}}-{\bf{r}}\frac{d{\bf{r}}}{d\tau}\right]_{\Sigma}$
(2.28) $\displaystyle m_{\Sigma}$ $\displaystyle=$
$\displaystyle\left[\frac{r^{3}\,\dot{f}^{2}\,b^{3}f}{2\,a^{2}}-\frac{r^{3}\,f\,b^{{}^{\prime}}\,{}^{2}}{2\,b}-r^{2}\,f\,b^{{}^{\prime}}+\frac{Q^{2}}{2rbf}\right]_{\Sigma}$
(2.29)

where $2M$ is the total energy entrapped inside the hypersurface $\Sigma$ [24,
46, 47]. Now, again from the matching condition (2.19), the matching of the
$K^{+}_{\tau\tau}=K^{-}_{\tau\tau}$ component together with the equations
(2.20) we have

$\displaystyle-\left[\frac{a^{{}^{\prime}}}{a\,b\,f}\right]_{\Sigma}$
$\displaystyle=$
$\displaystyle\left[\frac{d^{2}v}{d\tau^{2}}\left(\frac{dv}{d\tau}\right)^{-1}-\left(\frac{dv}{d\tau}\right)\left(\frac{M}{{\bf{r}}^{2}}-\frac{Q^{2}}{{\bf{r}}^{3}}\right)\right]_{\Sigma}$
(2.30)

Substituting equations (2.20), (2.21) and (2.29) into the equation (2.28) we
have

$\displaystyle\left(\frac{dv}{d\tau}\right)_{\Sigma}$ $\displaystyle=$
$\displaystyle\left(1+\frac{r\,b^{{}^{\prime}}}{b}+\frac{r\,b\,\dot{f}}{a}\right)_{\Sigma}^{-1}$
(2.31)

This is the gravitational red shift as viewed by an observer at rest at
infinity. Now, differentiating (2.31) with respect to the $\tau$ and using
equations (2.29) and (2.31), we can write the equation (2.30) and comparing
with equations (2.14) and (2.16) we have

$\displaystyle(p\,_{r})_{{}_{\Sigma}}$ $\displaystyle=$
$\displaystyle(q\,b\,f)_{{}_{\Sigma}}.$ (2.32)

We also require the total Luminosity for an observer at rest at infinity is
given by [9]

$\displaystyle L_{\infty}$ $\displaystyle=$
$\displaystyle-\left(\frac{dm}{dv}\right)_{\Sigma}=-\left[\frac{dm}{dt}\frac{dt}{d\tau}\left(\frac{dv}{d\tau}\right)^{-1}\right]_{\Sigma}.$
(2.33)

##  3 Exact solutions of the field equations

All the physical quantities in the Einstein- Maxwell field equations
(2.13)-(2.16) depends upon the forms of metric variables. In our previous
work, we have developed an adhoc formalism to obtained the exact solutions of
the field equations, and the model obtained is shown to be physically viable
both for the GR [44] and $f(R)$ regime [45]. We consider an adhoc form of the
pressure anisotropy so that we can use its equation to obtained the solution
of one of metric variable. In this work we will generalize it to the charged
collapsing scenarios. We used the Karmarkar condition to find the other metric
variable. From equations (2.14), (2.15), the pressure anisotropy
$\Delta=p_{t}-p_{r}$ has the form

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{1}{f^{2}\,b^{2}}\left[\frac{a^{{}^{\prime\prime}}}{a}+\frac{b^{{}^{\prime\prime}}}{b}-\frac{1}{r}\left(\frac{a^{{}^{\prime}}}{a}+\frac{b^{{}^{\prime}}}{b}\right)-\frac{2\,a^{{}^{\prime}}\,b^{{}^{\prime}}}{a\,b}-\frac{2\,b^{{}^{\prime}\,{}^{2}}}{b^{2}}\right]-\frac{s^{2}}{r^{4}\,b^{4}\,f^{4}}$
(3.1)

To find the exact solutions of the metric variable, we consider the following
adhoc form of the pressure anisotropy $\Delta$ as

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{\alpha}{f^{2}\,b^{2}}\left[\frac{a^{{}^{\prime\prime}}}{a}-\frac{a^{{}^{\prime}}}{r\,a}-\frac{2\,a^{{}^{\prime}}\,b^{{}^{\prime}}}{a\,b}\right]-\frac{s^{2}}{r^{4}\,b^{4}\,f^{4}}$
(3.2)

The above form of the anisotropy parameter (3.2) reduces the total pressure
anisotropy equation (3.1) into a differential equation of one variable and the
solution of metric variable $b(r)$ is given by

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{f^{2}\,b^{2}}\left[\frac{b^{{}^{\prime\prime}}}{b}-\frac{b^{{}^{\prime}}}{r\,b}-\frac{2\,b^{{}^{\prime}\,{}^{2}}}{b^{2}}\right]$
$\displaystyle b(r)$ $\displaystyle=$
$\displaystyle-\frac{2}{C_{3}\,r^{2}+2\,C_{4}}$ (3.3)

where $C_{3}$ and $C_{4}$ are constant of integration. The solution obtained
for the $b(r)$ is the same as that of obtained for our earlier work, however
the difference is in the choice of the pressure anisotropy equation (3.2).
This choice of the charged pressure anisotropy give us the freedom to have the
time dependent solutions.

It has been shown that an $(n+1)$-dimensional space $V^{n+1}$ can be embedded
into a pseudo Euclidean space $E^{n+2}$ of dimension $(n+2)$ if there exists a
symmetric tensor $b_{\mu\nu}$ which satisfies the following Gauss- Codazzi
equations [38].

$\displaystyle R_{\mu\,\nu\,\lambda\,\delta}$ $\displaystyle=$ $\displaystyle
2\,e\,b_{\mu[\lambda}\,b_{\delta]\nu}$ (3.4) $\displaystyle 0$
$\displaystyle=$ $\displaystyle
b_{\mu\,[\nu;\,\lambda]}-\Gamma^{\sigma}_{[\nu\,\lambda]}\,b_{\mu\,\sigma}+\Gamma^{\sigma}_{\mu\,[\nu}\,b_{\lambda]\,\sigma}$
(3.5)

where $e=\pm 1$ ($+$ or $-$, when the normal to the manifold is spacelike or
timelike respectively) and $b_{\mu\,\nu}$ are the coefficient of the second
differential form. The necessary and sufficient condition for any Riemannian
space to be embedding class $\bf{I}$, is also known as the Karmarkar
conditions [39, 40]

$\displaystyle R_{rtrt}\,R_{\theta\phi\theta\phi}=R_{r\theta
r\theta}\,R_{t\phi t\phi}-R_{\theta rt\theta}\,R_{\phi rt\phi}$ (3.6)

As shown in [44, 45], for the shear free spherically symmetric metric (2.1),
the Karmarkar class I condition becomes

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
r\,b^{3}{\dot{f}}^{2}\,\left(rab\,{a{{}^{\prime\prime}}}-ab\,{a{{}^{\prime}}}+{rb\,a{{}^{\prime}}^{2}}-2\,ra\,a^{\prime}b^{\prime}\right)+r\,a^{2}b^{2}f\ddot{f}\left(b\,b{{}^{\prime}}+2r{a{{}^{\prime}}^{2}}-rb\,{b{{}^{\prime\prime}}}\right)$
(3.7) $\displaystyle+$ $\displaystyle
ra^{3}a{{}^{\prime}}b{{}^{\prime\prime}}\left(rb^{\prime}+ba^{\prime}\right)-ra^{3}a{{}^{\prime\prime}}\left(2ba^{\prime}+rb^{\prime}{{}^{2}}\right)+a^{3}a^{\prime}b^{\prime}\left(b+2rb^{\prime}\right)$

The Karmarkar condition equation (3.7) is a nonlinear differential in its
temporal and radial behavior. A physical model of collapse must simultaneously
satisfy both the conditions (2.32) and (3.7). One of the solution, which
satisfies both these conditions, is to have a linear temporal dependence in
the model as considered in [41]

$\displaystyle f(t)$ $\displaystyle=$ $\displaystyle-C_{Z}\,t$ (3.8)

where $C_{Z}>0$ for the collapsing phenomena. Using the forms of $b(r)$ and
$f(t)$ in equation (3.7) give rise to the form of $a(r)$ as

$\displaystyle a(r)$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2}\sqrt{C_{3}C_{4}}}\sqrt{C_{4}^{2}\left(C_{1}b(r)+4C_{2}C_{3}\right)^{2}-4C_{Z}^{2}}$
(3.9)

where $C_{1}$ and $C_{2}$ are integration constants. These set of solutions
are similar to as found in [44], and there it has been shown that combination
of the Karmarkar condition with the pressure isotropy leads to only two set of
exact solutions which are: the Schwarzschild [48] or Kohler and Chao [49]
solution. We knows that the Kohler- Chao solution is only physical for unbound
configuration such as cosmological model as the radial pressure vanishes at
$r\rightarrow\infty$. Also, in the static case, from karmarkar condition and
pressure isotropy we must have $C_{1}=0$. Thus, the Schwarzschild like form of
the metric variables are

$\displaystyle
a(r)^{2}=\frac{4C_{2}^{2}C_{3}^{2}C_{4}^{2}-1}{2C_{3}C_{4}}\hskip
8.5359pt;\hskip
28.45274ptb(r)^{2}=\frac{4}{\left(2\,C_{4}+C_{3}\,r^{2}\right)^{2}}\hskip
8.5359pt;\hskip 28.45274ptf(t)^{2}=1.$ (3.10)

Recently Ospino et.al. [43], derived the Karmarkar scalar condition, and used
for the nonstatic system, where they found two class of solutions. One of
their solution represent the earlier found solution of horizon free radiating
collapse as found by the Naidu et. al. [42]. However, the other solution is
the completely new class of solution. The form of second class of solutions
are

$\displaystyle A(t,r)$ $\displaystyle=$
$\displaystyle-\dot{\bar{b}}(t)\frac{\sqrt{C\left(-2+\bar{a}C\right)+\bar{a}r^{4}\left(-1+\bar{a}\,^{2}C^{2}\right)+2r^{2}\left(-1-\bar{a}C+\bar{a}\,^{2}C^{2}\right)}}{{\sqrt{2}(1+\bar{a}r^{2})}}$
(3.11) $\displaystyle B(t,r)$ $\displaystyle=$
$\displaystyle\frac{\bar{b}(t)}{(1+\bar{a}r^{2})}\hskip 14.22636pt;\hskip
28.45274ptR(t,r)=rB(t,r)$ (3.12)

It must be noted that, for
$C_{4}=1,\,C_{Z}=1,\,C_{1}=2C_{Z},\,\bar{b}(t)=\dot{f(t)}=-C_{Z}\,t,\,\bar{a}=C_{3}/2,\,C(t)=-C_{2}$
these forms of the gravitational potentials obtained in [43] (3.11)-(3.12)
reduces to the obtained by the combination of this adhoc form of the
anisotropy together with Karmarkar condition. In the collapsing phenomena, the
metric variables $a(r)$ and $b(r)$ should remains positive and nonzero. The
condition $a(r)>0$ and $b(r)>0$ during the collapse implies that from equation
(3.9) we must have
$C_{Z}^{2}<C_{4}^{2}\left[\frac{C_{1}}{C_{3}r^{2}+2C_{4}}-2C_{2}C_{3}\right]^{2}$.

(a)

(b)

Figure 1: (a) $\&$ (b) Shows the plots of the density $\rho$ (2.13) and radial
pressure $p_{r}$ (2.14), with respect to $t$ and $r$, with center at $r=0$ and
surface at $r=1$. It is observed that both density and radial pressure remains
regular and positive throughout the collapse.

(a)

(b)

Figure 2: (a) Plot of the tangential pressure $p_{t}$ (2.15), with respect to
$t$ and $r$, with center at $r=0$ and surface at $r=1$. (b) Plot of the radial
heat flux $q$ (2.16), with respect to $t$ and $r$. It is observed that as the
stellar system deviates from the equilibrium, it starts radiating more and
more heat flux. Plots (a) $\&$ (b) shows that both tangential pressure and
radial heat flux remains regular and positive throughout the collapse.

(a)

(b)

Figure 3:  (a) $\&$ (b) Shows the plots of the pressure anisotropy $\Delta$,
equation (3.2) and the Electric field with respect to $t$ and $r$. Both the
pressure anisotropy and Electric field remains regular and positive throughout
the collapse.

(a)

(b)

Figure 4:  (a) Plot of the expansion scalar $\Theta$, equation (3.13), with
respect to $t$ and $r$. As expected for the collapsing phenomena, the
expansion scalar $\Theta$ remains negative throughout the interior of stellar
system. (b) Plot of the mass $m$, equation (3.14), of the collapsing stellar
system with respect to $t$ for both chargeless and charged collapse. It is
observed that the mass radiates linearly for both the cases, however, it has
large value for the charged case than that of its counterpart.

The expansion scalar (2.4) and the Misner sharp mass function (2.29) have the
form

$\displaystyle\Theta$ $\displaystyle=$
$\displaystyle\frac{6\sqrt{C_{3}C_{4}}}{t\sqrt{C_{4}^{2}\left(2C_{2}C_{3}-\frac{C_{1}}{C_{3}r^{2}+2C_{4}}\right)-2C_{Z}^{2}}}$
(3.13) $\displaystyle m$ $\displaystyle=$
$\displaystyle\frac{8tr^{3}C_{3}C_{4}^{3}C_{Z}}{\left(C_{3}r^{2}+2C_{4}\right)^{3}}\,\left[\frac{2C_{2}C_{3}\left(C_{3}r^{2}+2C_{4}\right)-C_{1}}{2\left(C_{3}r^{2}+2C_{4}\right)\left(C_{2}C_{3}C_{4}^{2}-C_{Z}^{2}\right)-C_{1}C_{4}^{2}}\right]$
(3.14)

The boundary condition (2.32) in the view of (2.14)-(2.16) becomes

$\displaystyle 2f\,\ddot{f}+\dot{f}^{2}-2x\dot{f}$ $\displaystyle=$
$\displaystyle y$ (3.15)

where

$\displaystyle x$ $\displaystyle=$
$\displaystyle\left(\frac{a^{{}^{\prime}}}{b}\right)_{\Sigma}$ (3.16)
$\displaystyle y$ $\displaystyle=$
$\displaystyle\left(\frac{a^{2}}{b^{2}}\left[\frac{b^{{}^{\prime}\,{}^{2}}}{b^{2}}+\frac{2}{r}\left(\frac{b^{{}^{\prime}}}{b}+\frac{a^{{}^{\prime}}}{a}\right)+\frac{2a^{{}^{\prime}}b^{{}^{\prime}}}{ab}\right]\right)_{\Sigma}$
(3.17)

##  4 Stability and Energy conditions

To check the stability of the shear free condition for our model, we consider
the stability criteria given in [50, 51, 52]. Here, from the study of the
behavior of adiabatic index and speed of sound one can infer the stability of
the stellar system. More precisely, a stellar model is considered to be stable
(unstable) if adiabatic index $\Gamma$ is greater than 4/3 ($\Gamma<4/3$). In
this range the pressure remains greater (less) than that of the weight of
stellar system and hence, it remains in equilibrium, any departure from
equilibrium leads to collapse. Other way to study the stability of a stellar
system is by using the difference between the propagation of the radial
$V_{r}^{2}$ and transverse $V_{t}^{2}$ speeds of sound in a stellar system.
Both the speeds of sound should always be less than that of velocity of light
and lies in the range $0<V_{r}^{2}$ and $\,V_{t}^{2}<1$. Also, as shown in
[51, 52], we can check whether a particular region of a matter configuration
is potentially stable or unstable. The regions where the radial speed of sound
remains greater than that of the transverse speed are called as the
potentially unstable regions and viceversa [52].

It can be seen from adiabatic index $\gamma$, Fig 5(b) that it remains a
constant and less than $4/3$ implying the stellar system under consideration
is unstable and represents a collapsing phenomena. The unstable and collapsing
nature of the stellar system is also confirmed from the behavior of radial
$V_{r}^{2}$ and transverse $V_{t}^{2}$ speeds of sound. Throughout the
interior of the stellar system, the radial speed of sound remains greater than
that of transverse speed and it shows that the region of matter configuration
is potentially unstable.

(a)

(b)

(c)

Figure 5:  (a) $\&$ (b) shows the plots of the luminosity and adiabatic index
with respect to $t$ for both the chargeless and charged collapse. It has been
found that both luminosity and adiabatic index are time independent. The
adiabatic index $\Gamma$ has value less than 4/3 implying that the stellar
system under consideration is representing unstable and collapsing phenomena.
(c) shows the plots of the speed of sound and can be seen that both radial
$V_{r}^{2}$ and tangential $V_{t}^{2}$ speed of sound lies in the range of the
$0<V_{r}^{2}\,;\,V_{t}^{2}<1$. The stellar system is unstable as the radial
speed remains greater than that of the tangential speed. Thus, the unstable
and collapsing nature of the stellar system is confirmed by the behavior of
adiabatic index Fig. (b) and speed of sound Fig (c).

It can be seen from the Figs. 1(a), 1(b) and 2(a) that density, radial
pressure and tangential pressure remains regular and positive throughout the
charged collapse. Also, radial heat flux is regular throughout the collapse as
can be seen from Fig. 2(b). It can be seen that pressure anisotropy 3(a) and
electric field intensity 3(b) remains regular and positive. The expansion
scalar $\Theta$ shows the contracting behavior as show in Fig. 4(a) it remains
regular and negative throughout the collapse. Fig. 4(b) shows mass $m$,
equation (3.14), of the collapsing stellar system with respect to $t$ for both
chargeless and charged collapse. It is observed that the mass radiates
linearly for both the cases, however, it has large value for the charged case
than that of its counterpart. Similar to our earlier work of noncharged case
[44], the Luminosity (LABEL:LInfinity) of the charged radiating collapse is
time independent. Figs. 5(b) and 5(c) shows that the stellar system to be
unstable and representing collapsing phenomena.

Let us now ask if the solutions also satisfy the energy conditions as well.
Energy conditions plays important role in the study of the astronomical
phenomena like collapsing stellar models. In this section we will analyze the
physical evidences of our model by verifying the energy conditions. The energy
conditions namely weak energy condition (WEC), null energy condition (NEC),
dominant energy condition (DEC) and strong energy conditions (SEC) will be
satisfied at all points in the stellar model if the following inequalities are
satisfied simultaneously [9]
E1 : $\left(\rho+p_{r}\right)^{2}-4q^{2}\,\geq\,0$ (SEC/DEC/WEC)
E2 : $\rho-p_{r}\,\geq\,0\,\,\,\,$ (DEC)
E3 : $\rho-p_{r}-2p_{t}+\sqrt{\left(\rho+p_{r}\right)^{2}-4q^{2}}\,\geq\,0$
(DEC)
E4 : $\rho-p_{r}+\sqrt{\left(\rho+p_{r}\right)^{2}-4q^{2}}\,\geq\,0\,\,\,\,$
(DEC/WEC)
E5 : $\rho-p_{r}+2p_{t}+\sqrt{\left(\rho+p_{r}\right)^{2}-4q^{2}}\,\geq\,0$
(SEC/DEC/WEC)
E6 : $2p_{t}+\sqrt{\left(\rho+p_{r}\right)^{2}-4q^{2}}\,\geq\,0$ (SEC)
Beside these energy conditions, a physically reasonable stellar model should
also satisfy
E7 : $\rho>0$, $p_{r}>0$, $p_{t}>0$, and $\rho^{{}^{\prime}}<0$,
$p_{r}^{{}^{\prime}}<0$, $p_{t}^{{}^{\prime}}<0$.
Here, we can see that the validity of the E1 and E2 inequalities implies that
the E4 inequality is satisfied. In the same fashion, the validity of the E1,
E2 and E7 inequalities ensures that the E5 and E6 inequalities are satisfied.
So, in general we only need to check the validity of the E1, E2, E3 and E7.
For our radiating stellar model, it can be seen from the figures 6(a), 6(b)
and 6(c) that all these energy conditions are well satisfied throughout the
interior of the collapsing star.

(a)

(b)

(c)

Figure 6:  (a), (b) $\&$ (c) shows the plots of energy condition
$E1,\,E2,\,E3$ for the charged radiating star with respect to $t$ and $r$
respectively. It shows that this stellar system satisfies these above
conditions.

##  5 Thermal properties

The study of the thermodynamical evolution and the temperature profiles of the
radiating stars plays prominent role during dissipative gravitational collapse
as they decide the departure from the thermodynamical equilibrium. Previous
studies of the shear free and shearing dissipative gravitational collapses
shows that the relaxation effects plays significant role in the temperature
profiles towards the end state of the dissipative gravitational collapse see
[53, 54, 55, 56, 57, 58, 59] and references therein. To study the
thermodynamical evolution and the temperature profiles of inside the
collapsing star, we will use the causal transport equation for the metric
(2.1) given by [53, 54, 57, 58]

$\displaystyle\tau h_{\mu}^{\nu}\dot{q}_{\nu}+q_{\mu}$ $\displaystyle=$
$\displaystyle-k\left(h_{\mu}^{\nu}\nabla_{\nu}T+T\dot{u}_{\mu}\right)$ (5.1)
$\displaystyle\tau\left(qbf\right)_{,t}+q\,a\,b\,f$ $\displaystyle=$
$\displaystyle-\frac{k\left(aT\right)_{,r}}{bf}$ (5.2)

where, $\alpha>0$, $\beta>0$, $\gamma>0$ and $\sigma>0$ are constants and
$h^{\mu\nu}=g^{\mu\nu}+u^{\mu}u^{\nu}$. Also,

$\displaystyle\tau_{c}=\left(\frac{\alpha}{\gamma}\right)\,T^{-\sigma}\hskip
5.69046pt,\hskip 17.07182ptk=\gamma\,T^{3}\,\tau_{c}\hskip 5.69046pt,\hskip
17.07182pt\tau=\left(\frac{\beta\,\gamma}{\alpha}\right)\,\tau_{c}$ (5.3)

are physically reasonable choices of the mean collision time, between massive
and massless particles $\tau_{c}$ , thermal conductivity $k$ and the
relaxation time $\tau$ respectively [54, 60]. $\tau$ represents the causality
index, measures the strength of relaxational effects and $\tau=0$ or $\beta=0$
represents the noncausal case. Using these forms of the physical quantities
given in equation (5.3), the form of the causal heat transport equation (5.2)
becomes

$\displaystyle\beta T^{-\sigma}\left(qbf\right)_{,t}+q\,a\,b\,f$
$\displaystyle=$
$\displaystyle-\frac{\alpha\,\left(aT\right)_{,r}}{bf}\,\,T^{3-\sigma}$ (5.4)

The noncausal solution of heat equations are obtained by setting $\beta=0$
i.e. $\tau=0$ in the above transport equation (5.4) [60]

$\displaystyle\left(a\,T\right)^{4}$ $\displaystyle=$
$\displaystyle-\frac{4}{\alpha}\int a^{4}\,q\,b^{2}\,f^{2}\,dr+F(t),\hskip
45.52458pt\sigma=0$ (5.5) $\displaystyle\ln\left(a\,T\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{\alpha}\int q\,b^{2}\,f^{2}\,dr+F(t),\hskip
56.9055pt\sigma=4$ (5.6)

The causal solution of the above transport equation (5.4) are given by [60]

$\displaystyle\left(a\,T\right)^{4}$ $\displaystyle=$
$\displaystyle-\frac{4}{\alpha}\left[\beta\int
a^{3}\,b\,f(q\,b\,f)_{,t}\,dr+\int
a^{4}\,q\,b^{2}\,f^{2}\,dr\right]+F(t),\hskip 17.07182pt\sigma=0$ (5.7)
$\displaystyle\left(a\,T\right)^{4}$ $\displaystyle=$
$\displaystyle-\frac{4\beta}{\alpha}exp\left(-\int\frac{4\,q\,b^{2}\,f^{2}}{\alpha}\,dr\right)\int
a^{3}\,b\,f(q\,b\,f)_{,t}\,dr\,exp\left(\int\frac{4\,q\,b^{2}\,f^{2}}{\alpha}\,dr\right)$
(5.8)
$\displaystyle+F(t)exp\left(-\int\frac{4\,q\,b^{2}\,f^{2}}{\alpha}\,dr\right),\hskip
119.50148pt\sigma=4$

where $F(t)$ is the function of integration. The function $F(t)$ is determined
by invoking boundary conditions

$\displaystyle\left(T^{4}\right)_{\Sigma}$ $\displaystyle=$
$\displaystyle\left(\frac{L_{\infty}}{4\pi\delta
r^{2}b^{2}f^{2}}\right)_{\Sigma}$ (5.9)

where $L_{\infty}$ is the total luminosity for an observer at infinity and
$\delta>0$ is constant.

(a)

(b)

Figure 7: (a) $\&$ (b) shows the plots of causal and non- causal temperatures
$T$ of stellar system with respect to $t$ and $r$. It is observed that both
causal and non- causal temperatures for the charged collapse remains greater
than that of their counterpart, chargeless collapse. It is also observed that
the causal temperature remains greater than that of non- causal temperature
throughout the interiors of both the charged and chargeless collapse.

As with previous investigations [58, 60], Fig.7(a) shows that both the causal
and noncausal temperature are same at the boundary of the star. However, at
later stages of the collapse, relaxation effects plays significant role and
they differ from the noncausal case. This behavior can be seen from the Fig.
7(a) that with $\beta>0$ the relaxations effects grows and the causal
temperature remains greater than that of noncausal temperature throughout the
interior of the star. These results are in agreement with the earlier results
obtained for the shear free collapse [57, 59].

##  6 Conclusion

In this paper we have studied the shear free spherical symmetric gravitational
collapse of charged radiating astrophysical systems including compact
configurations on the stellar or galactic scales. The interior and exterior
spacetimes of the solution has been smoothly matched over the timelike
hypersurface. To exactly solve for the system, we have utilised the pressure
anisotropy (3.2) together with the Karmarkar condition [40], where the metric
functions are taken to be linear in time dependence.

Let us describe our observations and conclusions obtained here: First, the
model under consideration has all the characteristics of a physically viable
compact object, since all the physical quantities like density (2.13) pressure
profiles (2.14), (2.15), radial heat flux (2.16), pressure anisotropy (3.2)
and Electric filed intensity are regular and positive throughout the collapse.
The mass (3.14) is a linear function of time. It is observed that the mass for
the charged collapse is greater than that of the chargeless case. One may also
show that the luminosity of this system is time independent and radiates
uniformly throughout the collapse. Secondly, we confirm that the system is
truly unstable. To verify, we determined the stability of the charged
radiating stellar system by studying the behavior of the adiabatic index. This
value equals the difference of the radial and transverse speeds of sound in
the matter configuration. It is known that a matter configuration is stable
(unstable) if the adiabatic index is greater than $4/3$ ($\Gamma<4/3$) [50,
51] . For the present system, the adiabatic index for the stellar system under
consideration is less $4/3$, implying that the system is unstable and
represents a collapsing phenomena. Thirdly, as can be seen from the Figs 6(a),
6(b) and 6(c) that the model under consideration satisfies the energy
conditions for the charge radiating stellar system. This also strengthens our
claim that the model is physically viable. Furthermore, the thermal properties
of the system show that the causal and non- causal temperature for the charged
collapse are greater than that of their counterpart of chargeless collapse;
the causal temperature remains higher than that of the non- causal temperature
throughout the collapse.

The presence of charge has great importance in the study of radiating stellar
system. It affects significantly the behavior of all the physical quantities
like density, pressure and mass profiles during the collapsing phenomena. The
stability analysis also gets affected by the presence of charge in the
collapse, as adiabatic index and radial $\&$ transverse speeds of sound has
significant difference to their counterpart chargeless case. The study of
thermal properties shows that both causal and non- causal temperatures for
charge collapse has higher value than that of chargeless collapse. Thus
collapse with charge does affects the collapsing phenomena as it delays the
collapse and it takes more time to reach the central singularity than that of
the chargeless
collapse.https://www.overleaf.com/project/64f19e779bcc28ba58f98a68

Acknowledgments
The author AC is supported through the DAE-BRNS project 58/14/25/2019-BRNS.

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