#  Charged anisotropic compact objects obeying Karmarkar condition

Y. Gomez-Leyton<EMAIL_ADDRESS>Departamento de Física, Universidad Católica
del Norte, Av. Angamos 0610, Antofagasta, Chile.  Hina Javaid
<EMAIL_ADDRESS>Comsats University Islamabad, Lahore Campus, 51040,
Pakistan.  L. S. Rocha<EMAIL_ADDRESS>Universidade de São Pablo,
Rua do Matão, 1226–Butantã, São Pablo–SP, 03178–200, Brazil.  Francisco Tello-
Ortiz<EMAIL_ADDRESS>Departamento de Física, Facultad de ciencias
básicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile.

###### Abstract

This research develops a well–established analytical solution of the Einstein-
Maxwell field equations. We analyze the behavior of a spherically symmetric
and static interior driven by a charged anisotropic matter distribution. The
class I methodology is used to close the system of equations and a suitable
relation between the anisotropy factor and the electric field is imposed. The
inner geometry of this toy model is described using an ansatz for the radial
metric potential corresponding to the well–known isotropic Buchdahl space-
time. The main properties are explored in order to determine if the obtained
model is appropriate to represent a real compact body such as neutron or quark
star. We have fixed the mass and radii using the data of the compact objects
SMC X–1 and LMC X–4. It was found that the electric field and electric charge
have magnitudes of the order of $\sim 10^{21}\ [V/cm]$ and $\sim 10^{20}\
[C]$, respectively. The magnitude of the electric field and electric charge
depends on the dimensionless parameter $\chi$. To observe these effects on the
total mass, mass–radius ratio and surface gravitational red–shift, we computed
numerical data for different values of $\chi$.

##  I Introduction

It is well-known that any $n$-dimensional Riemaniann variety $V_{n}$, can be
embedded in a $m$-dimensional pseudo-Euclidean space $E_{m}$, whose dimension
is given by $m=n\left(n+1\right)/2$ eisland ; eisenhart . The minimum extra
dimension of the pseudo-Euclidean space is
$\text{dim}(V_{n})\leq\,m-n=n\,\left(n-1\right)/2$, which constitutes the
class of the embedded space. On the other hand, some known important solutions
in the arena of General Relativity (GR) are for example the Kerr space-time
which corresponds to a class $p=5$ kerr . Exterior and interior Schwarzschild
schwar  solutions are associated with classes $p=2$ and $p=1$, respectively,
while the Friedman–Lemaitre-Robertson–Walker space-time robert  is of class
$p=1$. In this regard, the so-called class I condition karmarkar ; sharma  has
become a versatile tool to find out spherically symmetric solutions of the
Einstein field equations describing the behavior of real compact structures as
white dwarf, neutron and quark stars, driven by anisotropic
(charged/uncharged) matter distributions r3 ; r5 ; r6 ; r10 ; r16 ; r17 ; r18
; r19 ; r20 ; r21 ; r22 ; r23 ; r24 ; r25 ; r26 ; r27 ; r28 ; r32 ; r33 ; r34
; r35 ; r36 ; r37 ; r38 ; tello1 ; r39  (and references contained therein). In
a more widely context, this methodology was used to explain the existence of
dark matter r40  based on the extra dimension argument. Moreover, the class
$I$ scheme spreads into arena of modified gravity theory to study the
existence of compact objects r41 ; r42 . Recently, this methodology was
generalized to obtain all possible spherically symmetric structures satisfying
Karmarkar condition r43  and extended into the gravitational decoupling by
minimal geometric deformation r44 .

Despite the plethora of articles addressing the construction of compact
configurations describing real astrophysical systems, determine admissible
models satisfying Einstein field equations remains a great challenge. For the
simplest cases the material content filling the stellar interior have an
isotropic fluid distribution ı.e, $p_{r}=p_{\perp}$ lake . Relaxing the
stringent condition ($p_{r}=p_{\perp}$) to allow local anisotropies
$\Delta\equiv p_{\perp}-p_{r}$ in the stellar medium constitutes a more
realistic situation from the astrophysical point of view. In this respect
Ruderman n1 , Canuto n2 ; n3 ; n4  and Canuto et. al n5 ; n6 ; n7 ; n8
investigations revealed that, if the density of matter overcomes the nuclear
density, this problem is anisotropic in nature and should be treated in a
relativistic way. In this direction, Bowers and Liang n9  presented one of the
first studies based on anisotropic matter distributions. Furthermore, some
works predicted that anisotropic matter distributions can occur due to the
existence of a solid stellar nucleus, by phase transitions and pion
condensation n10 , the presence of type IIIA superfluid n11 ; n12 , the
presence of rotation and electromagnetic field in the system n13 ; n14 ; n15 ,
among others.

The model proposed by Heintzmann and Hillebrandt a1  for neutron stars showed
that for cases with a large and arbitrary anisotropy there is no mass limit
for the stars. However, it has been determined that the maximum mass of a
neutron star is still beyond $3-4M_{\odot}$ n16  so that the equilibrium is
maintained. The theoretical possibility of anisotropy in strange stars was
considered, with densities greater than those in neutron stars but lower than
those in black holes.

The chances of having anisotropy in a compact star increase due to the
relativistic interaction between the constituent particles and the random
movements that they generate, resulting in a break in the uniformity of the
distribution of the entire region. The functional foundations on which these
types of structures are based are well understood and have been extensively
studied in herre ; harko ; a2 ; a4 ; a5 ; a6 ; a7 ; a8 ; a9 ; a10 ; a11 ; a12
; a13 ; a15 ; a16 ; a17 ; a19 ; a20 ; a21 ; a22 . To introduce anisotropy in
the matter distribution allows adding important benefits in the description of
the system, highlighting: $i)$ the presence of an extra gradient, repulsive in
nature when $\Delta>0$ (otherwise is attractive). This fact is relevant since
the presence of a repulsive anisotropy gradient offset the gravitational
attraction avoiding a gravitational collapse, $ii)$ the possibility to obtain
more compact objects and finally, $iii)$ the stability of the system is
enhanced.

Besides, the matter distribution might contain additional ingredients to
further improve the mentioned properties. For example, the inclusion of
electrical features help to balance the gravitational gradient, improving the
stability and balance of the system RayY . Moreover, the gravitational mass
acquires an extra portion provided by the electrical contribution. Since the
pioneering works by Bonnor e1  and Rosseland rose , the effects of electrical
properties on self-gravitating compact objects have been widely explored e2 ;
e3 ; e4 ; e5 ; e6 ; e7 ; e8 ; e9 ; e10 ; e11 ; e12  (and references contained
therein). As it is well known, Newton’s theory of gravitation and
electrostatics admit charged fluid configurations in equilibrium, where the
charge density can be as large as the mass density, in appropriate units. In a
particular case the presence of electrons in stellar matter is crucial for the
possible existence of a nuclear crust in quark–like stars, where electron
displacement is associated with an electric current. In this regard, a star
like the Sun is expected to have a net electric charge due to the escape of
electrons rather than the escape of photons.

Following these good antecedents, in the present work we obtain an analytical
charged anisotropic solution of the Einstein–Maxwell field equations for a
spherically symmetric and static space-time. To build the inner manifold we
have used the so-called class I condition. As this condition links both
potentials, the $g_{rr}$ metric component is taken to be the Buchdahl metric
potential buch , then $g_{tt}$ is obtained. With the geometrical description
at hand, it is imposed a relation between the anisotropy factor and the
electric field dey  to close the system. The relation is controlled by a
dimensionless parameter $\chi$. This constraint is well motivated since both
quantities are null at the center of the configuration an increasing functions
with increasing radial coordinate. An interesting point deduced from this link
is that the anisotropy factor increases in magnitude with increasing $\chi$,
whereas the electric field has the opposite behaviour. The effects of the
electric characteristics on the matter distribution, are evident when $\chi$
grows up in magnitude. The structure of the problem naturally discards
negative values for $\chi$, because for $\chi<0$ the radial pressure dominates
the tangential one, then $\Delta<0$, which implies an unstable system.

To support our analysis we have used the mass and radii of the real compact
objects SMC X–1 and LMC X–4 r47 . Highlighting that quarks are small
fundamental fragments of matter, which are interestingly combined to form the
heaviest particles Ivanenko , constituted by the three lightest flavors (top,
bottom, and strange quarks Farhi ). These elementary particles refuse to be
seen individually, and the physics associated with grouped quarks is quite
complex. In the framework of stellar interiors, these types of structures have
extremely high pressure and temperature, forcing the nuclear constituents to
form quark matter or strange matter. Currently, there is no solid scientific
basis to confirm or reject this hypothesis and thus explain various
astrophysical phenomena. Nevertheless, from the phenomenological and
theoretical point of view the study of quark or strange stars is fundamental
to understand the fast radio burst process. Despite the possibility is still
theoretical, there are good antecedents confirming that the collapse of these
strange star crusts could be the origin of fast radio bursts mechanism l1 ; l2
.

To verify the viability of the proposed model, a complete graph is made,
studying the thermodynamic variables, the electrical properties, the
hydrostatic balance and the stability using a relativistic adiabatic index,
subliminal sound velocities and the Harrison–Zeldovich–Novikov procedure. In
addition, we discussed the impact of the electric field on macro-physical
observables, such as total gravitational mass, compactness factor, and
gravitational red surface displacement.

The document is organized as follows: The Sec. II is dedicated to presenting
the Class I methodology, The Einstein–Maxwell field equations and the proposed
model. In Sec. III the toy model is matched in a smooth way with the exterior
space-time described by the Reissner-Nordström solution, in order to determine
the space parameter that characterizes the model. Sec. IV is devoted to
physical and mathematical analysis of the main features, such as the
thermodynamic variables, energy conditions and electric properties. In Sec. V
the hydrostatic balance and stability analysis is performed to check the
reliability of the solution. Sec. VI discuss the impact of electric component
on the main macro-physical observables mentioned before. Finally, Sec. VII
provides some remarks and conclusions for the reported study. It worth
mentioning that throughout the article we shall employ relativistic
geometrized units ı.e, $G=c=1$, the mostly negative signature $\\{+,-,-,-\\}$
and the following definition of the Riemann tensor: $R^{\gamma}_{\
\mu\nu\rho}=\partial_{\nu}\Gamma^{\gamma}_{\
\mu\rho}-\partial_{\mu}\Gamma^{\gamma}_{\ \nu\rho}+\Gamma^{\gamma}_{\
\mu\lambda}\Gamma^{\lambda}_{\ \nu\rho}-\Gamma^{\gamma}_{\
\nu\lambda}\Gamma^{\lambda}_{\ \mu\rho}$.

##  II Class I Approach Revisited and The Toy Model

In this section, we provide a detailed revision about class I condition and
how it works in the framework of GR. Let us start by describing the usual
mathematical complications in dealing with Einstein–Maxwell field equations to
find out solutions describing compact objects such as neutron stars. Then,
class I approach is presented as an auxiliary condition coming from purely
geometric considerations, to reduce the mathematical issues and finally solve
the system of equations. The salient toy model representing compact structures
is also presented.

###  II.1 Class I Methodology

Since Einstein field equations constitute a complex set of couple partial
non–linear differential equations, given by

$R_{\mu\,\nu}-\frac{R}{2}\,g_{\mu\,\nu}=8\,\pi\,T_{\mu\,\nu},$ (1)

finding solutions that satisfy this intricate system of coupled equations is
very difficult, whether the problem is analyzed analytically or numerically.
In the simplest case, when the geometry of the space-time is described by a
spherically symmetric and static line element, expressed in canonical
coordinates $x^{\mu}\equiv\left(t,r,\theta,\phi\right)$ as

$ds^{2}=e^{\eta}\,dt^{2}-e^{\lambda}\,dr^{2}-r^{2}\,d\Omega^{2},$ (2)

where $\left[\eta=\eta\left(r\right),\lambda=\lambda\left(r\right)\right]$
ı.e, purely radial functions, the problem is greatly reduced from the
mathematical point of view. Nevertheless, the complexity in solve this set of
equations also depends on the form of the matter distribution. In this
concern, we shall assume an imperfect fluid coupled to an electromagnetic
field

$T^{\mu}_{\nu}={M}^{\mu}_{\nu}+E^{\mu}_{\nu}=\underbrace{\left({\rho}+{\bar{p}}_{\perp}\right)\,\chi^{\mu}\,\chi_{\nu}-\delta^{\mu}_{\nu}\,{\bar{p}}_{\perp}+\left({\bar{p}}_{r}-{\bar{p}}_{\perp}\right)\,u^{\mu}\,u_{\nu}}_{M^{\mu}_{\nu}}+\underbrace{\left(-F^{\mu\,\alpha}F_{\nu\,\alpha}+\delta^{\mu}_{\nu}F_{\beta\,\alpha}\,F^{\beta\,\alpha}/4\right)/4\,\pi}_{E^{\mu}_{\nu}},$
(3)

with $\rho$ being the energy density, and ${\bar{p}}_{r}$ and
${\bar{p}}_{\perp}$ being the pressure waves along the main directions of the
fluid sphere ı.e, the radial and transverse ones, respectively. The time–like
vector $\chi^{\omega}=e^{-\eta/2}\delta^{\omega}_{\ t}$, satisfying
$\chi^{\omega}\chi_{\omega}=1$, represents the velocity of the fluid. Moreover
$u^{\omega}=e^{-\lambda/2}\delta^{\omega}_{\ r}$ is a unit space-like vector
in the radial direction (orthogonal to $\chi^{\omega}$), satisfying
$u^{\omega}u_{\omega}=-1$. The tensor $F_{\mu\nu}$, represents the
skew–symmetric electromagnetic tensor defined as usual by

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ (4)

and in the present context satisfying the covariant vacuum Maxwell equations

$\nabla_{[\alpha}F_{\beta\gamma]}=0,\quad\mbox{and}\quad\nabla_{\beta}\,F^{\beta\,\alpha}=4\,\pi\,J^{\alpha},$
(5)

being $J^{\alpha}$ the four electric current. In the static and spherically
symmetric case, the vectorial potential $A^{\mu}$ is expressed as follows

$A^{\mu}=\left(\Phi(r),0,0,0\right)=\Phi(r)\delta^{\mu}_{\ t},$ (6)

where $\Phi(r)$ is the electric scalar potential, and the four electric
current is expressed by

$J^{\mu}=\left(e^{-\nu/2}\,\sigma,0,0,0\right)=e^{-\nu/2}\,\sigma\,\delta^{\mu}_{\
t},$ (7)

where $\sigma$ is representing the surface charge density. Now integrating the
expression at the left hand side in (5) one gets

$F^{tr}(r)=-F^{rt}(r)=\frac{q(r)}{r^{2}}\,e^{-(\lambda(r)+\nu(r))/2}.$ (8)

As usual the electric charge $q(r)$ has been defined by using the relativistic
Gauss’s law as follows bek ; dio

$q\left(r\right)=4\pi\int^{r}_{0}\sigma(x)x^{2}e^{\lambda(x)/2}dx=r^{2}\,\sqrt{-F^{tr}F_{tr}}.$
(9)

Furthermore, as we are dealing with a spherically symmetric and static
configuration, this implies that the only non–vanishing components of the
electromagnetic tensor $F^{\mu\nu}$ are $F^{tr}=-F^{rt}$. These components are
just the electric field $E=E(r)$ along the radial direction. Thus, from Eq.
(8) one obtains

$E(r)=\frac{q(r)}{r^{2}}\,e^{-(\lambda(r)+\nu(r))/2}.$ (10)

Thus the energy–momentum tensor (3) can be cast as follows

$T^{\mu}_{\nu}=\text{diag}\left(\rho+\frac{E^{2}}{8\pi},-p_{r}+\frac{E^{2}}{8\pi},-p_{\perp}-\frac{E^{2}}{8\pi},-p_{\perp}-\frac{E^{2}}{8\pi}\right).$
(11)

It should be noted that the form of the energy–momentum tensor (3) arises from
the variation of the full action (gravitational field minimally coupled to the
electromagnetic field, source and particles) with respect to the metric
tensor. For further details about this point see appendix A.

Next, putting together Eqs. (1)–(3), one obtains the following set of
equations bek ; dio

$\displaystyle 8\,\pi\,{\rho}+E^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}-e^{-\lambda}\left(\frac{1}{r^{2}}-\frac{\lambda^{\prime}}{r}\right),$
(12) $\displaystyle 8\,\pi\,{p}_{r}-E^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{r^{2}}+e^{-\lambda}\left(\frac{1}{r^{2}}+\frac{\eta^{\prime}}{r}\right),$
(13) $\displaystyle 8\,\pi\,{p}_{\perp}+E^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{-\lambda}\left(2\eta^{\prime\prime}+\eta^{\prime
2}-\lambda^{\prime}\eta^{\prime}+2\frac{\eta^{\prime}-\lambda^{\prime}}{r}\right).$
(14)

It is evident that the above system (12)–(14) contains six unknowns namely,
the geometry $\\{\eta,\lambda\\}$, the thermodynamic variables
$\\{\rho,p_{r},p_{t}\\}$ and the electric field $E$, but only three equations.
The conservation of the energy–momentum tensor,

$\nabla_{\mu}\,T^{\mu}_{\
\nu}=0\,\Rightarrow-p^{\prime}_{r}-\frac{\eta^{\prime}}{2}\left(\rho+p_{r}\right)+\frac{2}{r}\left(p_{\perp}-p_{r}\right)+\sigma\,E\,e^{\lambda/2}=0,$
(15)

does not provide any additional information, since (15) is a linear
combination of Eqs. (12)–(14). Is in this sense that the so-called class I
condition acts as an auxiliary constraint to solve the given set of equations,
although in this case to close the system it is necessary to impose for
example an equation of state or another relation to determine the full
energy–momentum tensor. This is due to the electric contribution.
Nevertheless, if $E=0$ one recovers the usual Einstein equations for an
anisotropic fluid distribution. Therefore, the class I condition is enough to
close the problem at least from the mathematical point of view.

In general the 4–dimensional manifold given by Eq. (2) corresponds to a
space–time of class II. This means that is necessary a 6–dimensional
pseudo–Euclidean space to encrust it. However, under a suitable
parametrization the space–time (2) can be immersed into a 5–dimensional
pseudo-Euclidean space, turning into a class I eisland ; eisenhart ; karmarkar
. In general any variety $V_{n}$ can be embedded in a flat space of
$n\left(n+1\right)/2$ dimensions eisland ; eisenhart . Nevertheless, if the
lowest order of this flat space is $n+p$, we say that $V_{n}$ is of class $p$.
Necessary and sufficient conditions must be satisfied for any spherically
symmetric space–time (static and non–static) so it can be assigned as class I.
Those are:

  * •

A system of symmetric quantities $b_{\mu\,\nu}$ must be established, such that

$R_{\mu\,\nu\,\alpha\,\beta}=\epsilon\,\left(b_{\mu\,\alpha}\,b_{\nu\,\beta}-b_{\mu\,\beta}\,b_{\nu\,\alpha}\right)\quad\left(\text{Gauss's
equation}\right),$ (16)

where $\epsilon=\pm 1$ whenever the normal to the manifold is space–like (+1)
or time–like (-1).

  * •

The system $b_{\mu\nu}$ must satisfy the differential equations

$\nabla_{\alpha}b_{\mu\,\nu}-\nabla_{\nu}\,b_{\mu\,\alpha}=0\quad\left(\text{Codazzi's
equation}\right).$ (17)

In general Codazzi’s equation (17) is not zero. This form of Eq. (17) is
implied by Eq. (16) eisenhart .

From (2) the non vanishing elements of the Riemann tensor are:

$\displaystyle R_{trtr}$ $\displaystyle=$
$\displaystyle-\frac{e^{\eta}}{4}\left[2\,\eta^{\prime\prime}+\eta^{\prime\
2}-\lambda^{\prime}\,\eta^{\prime}\right],$ $\displaystyle
R_{\theta\,\phi\,\theta\,\phi}$ $\displaystyle=$
$\displaystyle-e^{-\lambda}\,r^{2}\,sin^{2}\theta\left(e^{-\lambda}-1\right),$
$\displaystyle R_{r\,\phi\,r\,\phi}$ $\displaystyle=$ $\displaystyle
sin^{2}\theta\,R_{r\,\theta\,r\,\theta}=\frac{r}{2}\,\lambda^{\prime},$ (18)
$\displaystyle R_{t\,\theta\,t\,\theta}$ $\displaystyle=$ $\displaystyle
sin^{2}\theta\,R_{t\,\phi\,t\,\phi}=-\frac{r}{2}\,\eta^{\prime}\,e^{\eta-\lambda}.$

So, by using the set of equations (18) into Eq. (16) one gets:

$\displaystyle b_{tr}\,b_{\phi\,\phi}=R_{r\phi
t\phi}=0;~{}~{}b_{tr}\,b_{\theta\,\theta}=R_{r\theta t\theta}=0;$ (19)
$\displaystyle
b_{tt}\,b_{\phi\,\phi}=R_{t\phi\,t\,\phi};~{}~{}b_{tt}\,b_{\theta\,\theta}=R_{t\theta
t\theta};~{}~{}b_{rr}\,b_{\phi\,\phi}=R_{r\phi r\phi};$ $\displaystyle
b_{\theta\,\theta}\,b_{\phi\,\phi}=R_{\theta\,\phi\,\theta\,\phi};~{}~{}b_{rr}\,b_{\theta\,\theta}=R_{r\theta\,r\theta};~{}~{}b_{tt}\,b_{rr}=R_{trtr}.$

The above relations leads to:

$\left(b_{tt}\right)^{2}=\frac{\left(R_{t\,\theta\,t\,\theta}\right)^{2}}{R_{\theta\,\phi\,\theta\,\phi}}\,sin^{2}\theta,\quad\left(b_{rr}\right)^{2}=\frac{\left(R_{r\,\theta\,r\,\theta}\right)^{2}}{R_{\theta\,\phi\,\theta\,\phi}}\,sin^{2}\theta,\quad\left(b_{\theta\,\theta}\right)^{2}=\frac{R_{\theta\,\phi\,\theta\,\phi}}{sin^{2}\theta},\quad\left(b_{\phi\,\phi}\right)^{2}=sin^{2}\theta\,R_{\theta\,\phi\,\theta\,\phi}.$
(20)

Upon replacing (20) into expression (19) one gets:

$R_{t\,\theta\,t\,\theta}\,R_{r\,\phi\,r\,\phi}=R_{trtr}\,R_{\theta\,\phi\,\theta\,\phi},$
(21)

subject to $R_{\theta\,\phi\,\theta\,\phi}\neq\,0$ sharma . It should be noted
that eq. (19) satisfies Codazzi’s equation (17). On the other hand, in the
case of a general non–static spherically symmetric space–time, the second and
last equality in (19) become:

$b_{tr}\,b_{\theta\,\theta}=R_{r\,\theta\,t\,\theta}\quad\mbox{and}\quad
b_{tt}\,b_{rr}-\left(b_{tr}\right)^{2}=R_{trtr},$ (22)

where $\left(b_{tr}\right)^{2}=sin^{2}\theta\left(R_{r\theta
t\theta}\right)^{2}/R_{\theta\,\phi\,\theta\,\phi}$. So, the class I condition
becomes eisland ; karmarkar :

$R_{t\,\theta\,t\,\theta}\,R_{r\,\phi\,r\,\phi}=R_{trtr}\,R_{\theta\,\phi\,\theta\,\phi}+R_{r\theta
t\theta}\,R_{r\,\phi\,t\,\phi}.$ (23)

In this particular case, where the space–time is given by eq. (2) the
condition (21) (or equivalently (23)) leads to:

$2\frac{\eta^{\prime\prime}}{\eta^{\prime}}+\eta^{\prime}=\frac{\lambda^{\prime}\,e^{\lambda}}{e^{\lambda}-1},$
(24)

with $e^{\lambda}\neq 1$. This equation can be solved to express
$\eta=\eta\left(\lambda\right)$ or $\lambda=\lambda\left(\eta\right)$. The
results are:

$e^{\lambda}=1+C\,\eta^{\prime\,2}\,e^{\eta},$ (25)

or

$e^{\eta}=\left[A+B\int\,\sqrt{\left(e^{\lambda}-1\right)}\,dr\right]^{2},$
(26)

being $\\{A,B,C\\}$ integration constants. At this stage some comments are
pertinent. As we pointed out before, the class I condition helps to reduce the
mathematical complexity in order to tackle the set of equations (14)–(16).
This fact is reflected by Eq. (25) or (26), so with a suitable choice of one
of the metric potentials the geometry of the space–time is completely
determined. Therefore, with this information in hand the energy–momentum
tensor is obtained from Eqs. (14)–(16). It should be noted that the class I
condition also serves to find out spherical solutions whose matter
distribution is described by an isotropic fluid $p_{r}=p_{\perp}=p$. From Eqs.
(15)–(16) and (26) the anisotropy factor $\Delta$ reads r3 ; r5 ; r6 :

$8\,\pi\,\Delta=\frac{\eta^{\prime}\,e^{-\lambda}}{4}\left(\frac{\eta^{\prime}e^{\eta}}{2\,B^{2}\,r}-1\right)\left(\frac{2}{r}-\frac{\lambda^{\prime}\,e^{-\lambda}}{1-e^{-\lambda}}\right)-2\,E^{2}.$
(27)

Imposing the isotropic condition $\Delta=0$ and $E=0$ one obtains the
following:

$\frac{\eta^{\prime}\,e^{-\lambda}}{4}\left(\frac{\eta^{\prime}\,e^{\eta}}{2\,B^{2}\,r}-1\right)\left(\frac{2}{r}-\frac{\lambda^{\prime}\,e^{-\lambda}}{1-e^{-\lambda}}\right)=0.$
(28)

It is observed from Eq. (28), that if the first parenthesis is zero, the
corresponding solution will be the Kohlar–Chao koler  solution, while if the
second parenthesis is null, the interior Schwarzschild schwar  solution is
obtained. Notwithstanding, if $E\neq 0$, it is possible to build charged
isotropic fluid spheres.

###  II.2 The Model

As explained before, additional constraints are necessary to close the
Einstein–Maxwell fields equations. To determine the geometrical sector we have
imposed the $g_{rr}$ metric potential to be

$e^{\lambda(r)}=\frac{2\left(1+Cr^{2}\right)}{2-Cr^{2}}.$ (29)

This metric potential corresponds to one of the solutions provided by Buchdahl
buch , and was already employed in the context of embedding class I
methodology to build compact stars r10 . The motivation of this choice relies
on the well physical and mathematical properties that it has ı.e, finite at
$r=0$, positive defined and increasing function with increasing radial
coordinate. These features, in principle111In this case the metric potential
(29) does not determine completely the behavior of the density $\rho$, because
the electric field contribution plays and important role on the behaviour of
$\rho$. assure a well posed energy density $\rho$ within the stellar medium.
So, by inserting Eq. (29) into (26) one arrives to

$e^{\eta(r)}=\left(A-\bar{B}\sqrt{2-Cr^{2}}\right)^{2},$ (30)

where $A$ and $\bar{B}$ are dimensionless parameters and $C$ has units of
$\text{length}^{-2}$. In order to present the results in a more compact
fashion, in Eq. (30) we have defined $\bar{B}$ to be equal to
$\sqrt{3}B/\sqrt{C}$. Thus, the space–time describing the stellar interior is
given by

$ds^{2}=\left(A-\bar{B}\sqrt{2-Cr^{2}}\right)^{2}dt^{2}-\frac{2\left(1+Cr^{2}\right)}{2-Cr^{2}}dr^{2}-r^{2}d\Omega^{2}.$
(31)

To close the system (14)–(16) we shall employ the following ansatz

$\Delta=2\chi E^{2},$ (32)

where $\chi$ is a dimensionless constant. This relation was recently employed
to build higher dimensional fluid spheres in the background of Finch–Skea
space–time dey . The main physical reason behind (32) is that both $\Delta$
and $E$ have the same behavior inside the compact structure ı.e, positive
defined and increasing functions with increasing radial coordinate (as we will
see later, these requirements are necessary to describe a well posed charged
compact configuration), implying that at $r=0$: $\Delta(0)=E(0)=0$. Now
subtracting Eqs. (13) from (14) we obtain

$8\pi\left(p_{\perp}-p_{r}\right)(r)+2E^{2}(r)=f(r).$ (33)

Remembering that the anisotropy factor is defined as $\Delta\equiv
p_{\perp}-p_{r}$, the Eq. (33) becomes

$8\pi\Delta(r)+2E^{2}(r)=f(r).$ (34)

Therefore, putting together (32) and (34) one gets

$E^{2}(r)=\frac{f(r)}{2\left(1+8\pi\chi\right)}.$ (35)

The anisotropy factor is then expressed by

$\Delta(r)=\frac{f(r)\chi}{\left(1+8\pi\chi\right)},$ (36)

where

$f(r)=\frac{e^{-\lambda}}{4r^{2}}\left[4\left(e^{\lambda}-1\right)+r\left(\eta^{\prime}\left(r\eta^{\prime}-2\right)-\lambda^{\prime}\left(2+r\eta^{\prime}\right)+2r\eta^{\prime\prime}\right)\right]=\frac{C^{2}r^{2}\left[4\bar{B}\left(Cr^{2}-2\right)+3A\sqrt{2-Cr^{2}}\right]}{2\left[1+Cr^{2}\right]^{2}\left[\bar{B}\left(Cr^{2}-2\right)+A\sqrt{2-Cr^{2}}\right]}$
(37)

for the present model described by the metric potentials (29) and (30). At
this stage some comments are pertinent

  1. 1.

Equation (36) restricts $\chi$ to be positive defined, since $\chi<0$ implies
$\Delta(r)<0$, then the system will be unstable.

  2. 2.

In the case $\chi=0$ one obtains an isotropic charged solution
($\Delta(r)=0$).

  3. 3.

If $E^{2}(r)=0$ one gets an anisotropic fluid ball with anisotropy factor
given $\Delta(r)=f(r)/8\pi$.

In what follows we shall assume $\chi\neq 0$, $E^{2}(r)\neq 0$ and
$\Delta(r)\neq 0$ with the aim to build a toy model representing a compact
charged anisotropic fluid sphere. Replacing equations (29)–(30) and (35) into
the system (12)–(14) one yields at the following thermodynamic variables

$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\frac{3C\left[3+Cr^{2}\right]}{16\pi\left[1+Cr^{2}\right]^{2}}-\frac{E^{2}(r)}{8\pi},$
(38) $\displaystyle p_{r}(r)$ $\displaystyle=$
$\displaystyle\frac{C\left[5\bar{B}\left(2-Cr^{2}\right)-3A\sqrt{2-Cr^{2}}\right]}{16\pi\left[1+Cr^{2}\right]\left[\bar{B}\left(Cr^{2}-2\right)+A\sqrt{2-Cr^{2}}\right]}+\frac{E^{2}(r)}{8\pi},$
(39) $\displaystyle p_{\perp}(r)$ $\displaystyle=$
$\displaystyle\frac{C\left[\bar{B}\left(2-Cr^{2}\right)\left(5+Cr^{2}\right)-3A\sqrt{2-Cr^{2}}\right]}{16\pi\left[1+Cr^{2}\right]^{2}\left[\bar{B}\left(Cr^{2}-2\right)+A\sqrt{2-Cr^{2}}\right]}-\frac{E^{2}(r)}{8\pi}.$
(40)

In the next section the junction condition formalism is performed in order to
determine the space parameter $\\{A,\bar{B},C\\}$ characterizing the model.

##  III Junction Conditions at the Star Surface

As we are dealing with a finite configuration. In order to guaranteed that the
toy model represented by (31) which is describing the inner space–time
$\mathcal{M}^{-}$ is well established, one needs to join it in a smoothly way
at the surface interface $\Sigma:r=R$ (or equivalently the compact object
surface), with external space–time $\mathcal{M}^{+}$. In this case the
exterior manifold is no longer empty since we are facing a charged compact
configuration, thus exterior variety $\mathcal{M^{+}}$ is given by the
Reissner–Nordström solution

$ds^{2}=\left(1-\frac{2\bar{M}}{r}+\frac{Q^{2}}{r^{2}}\right)dt^{2}-\left(1-\frac{2\bar{M}}{r}+\frac{Q^{2}}{r^{2}}\right)^{-1}dr^{2}-r^{2}d\Omega^{2}.$
(41)

This process known as the matching condition procedure, serves to obtain the
complete set of constant parameters $\\{A,\bar{B},C\\}$. To do this, it is
employed the Israel–Darmois (ID) is ; dar  mechanism. This method is supported
by the continuity of the temporal $g_{tt}$ and radial $g_{rr}$ metric
potentials throughout the surface interface $\Sigma$. Technically, this
condition is known as the first fundamental form. Explicitly it reads

$g^{-}_{tt}\bigg{|}_{r=R}=g^{+}_{tt}\bigg{|}_{r=R}\quad\mbox{and}\quad
g^{-}_{rr}\bigg{|}_{r=R}=g^{+}_{rr}\bigg{|}_{r=R}.$ (42)

In our scenario we have

$\displaystyle\left(A-\bar{B}\sqrt{2-CR^{2}}\right)^{2}$ $\displaystyle=$
$\displaystyle 1-\frac{2\bar{M}}{R}+\frac{Q^{2}}{R^{2}},$ (43)
$\displaystyle\frac{2-CR^{2}}{2\left(1+CR^{2}\right)}$ $\displaystyle=$
$\displaystyle 1-\frac{2\bar{M}}{R}+\frac{Q^{2}}{R^{2}}.$ (44)

At junction surface $\Sigma$ both $M$ and $\bar{M}$ coincide. Thus, the total
mass $M$ inside the charged ball is determined by the external space–time. As
we are gluing the interior geometry $\mathcal{M}^{-}$ with outer one
$\mathcal{M}^{+}$, at the surface $r=R$ is induced by $g^{-}_{\mu\nu}$ and
$g^{+}_{\mu\nu}$ an intrinsic geometry described by the extrinsic curvature
tensor $K_{ij}$. Precisely, the continuity of $K_{rr}$ across $\Sigma$, the
so–called second fundamental form, guaranteed a completely null radial
pressure

$\leavevmode\resizebox{441.57083pt}{}{
$p_{r}(R)=0\Rightarrow\frac{2C\left[3A\sqrt{2-CR^{2}}+5\bar{B}\left(CR^{2}-2\right)\right]\left[1+8\pi\chi\right]+C^{2}R^{2}\left[2\bar{B}\left(CR^{2}-2\right)\left(3+40\pi\chi\right)+3A\sqrt{2-CR^{2}}\left(1+16\pi\chi\right)\right]}{32\pi\left[1+CR^{2}\right]^{2}\left[\bar{B}\left(CR^{2}-2\right)+A\sqrt{2-CR^{2}}\right]\left[1+8\pi\chi\right]}=0$}.$
(45)

A null radial pressure at the boundary $\Sigma$ is a necessary mechanism to
confine the matter content inside a bound space–time region $0\leq r\leq R$,
which in turn determines the size of the collapsed configuration ı.e, its
radii $R$. Furthermore, the continuity $K_{\phi\phi}$ and $K_{\varphi\varphi}$
fixes the total mass of the compact structure

$\left[K^{-}_{\phi\phi}-K^{+}_{\phi\phi}\right]\bigg{|}_{\Sigma}=\left[K^{-}_{\varphi\varphi}-K^{+}_{\varphi\varphi}\right]\bigg{|}_{\Sigma}=0\Rightarrow
m(R)=M.$ (46)

Besides, it is also necessary to impose the electric charge $q(r)$ continuity
across the boundary $\Sigma$

$q(R)=Q=R^{2}E(R)=R^{2}\sqrt{\frac{C^{2}R^{2}\left[4\bar{B}\left(CR^{2}-2\right)+3A\sqrt{2-CR^{2}}\right]}{4\left[1+8\pi\chi\right]\left[1+CR^{2}\right]^{2}\left[\bar{B}\left(CR^{2}-2\right)+A\sqrt{2-CR^{2}}\right]}}.$
(47)

Equations (43)–(45) and (47) are the necessary conditions to close the problem
and obtain the arbitrary constants $\\{A,\bar{B},C\\}$. In table 1 are
exhibited the numerical values by the mentioned space parameter for total
gravitational mass $M$ and radius $R$ corresponding with the compact stars SMC
X–1 and LMC X–4 r47 .

Table 1: Numerical values of constant parameters $A$, $\bar{B}$ and $C$ for different values of $M$ and $R$ and $\chi=0.008$.  Strange Star | $M/M_{\odot}$ | $R\ [km]$ | $C\ [\text{km}^{-2}]$ |  $A$ (Dimensionless) |  $\bar{B}$ (Dimensionless)
---|---|---|---|---|---
SMC X–1 r47  | 1.04 | 8.301 | 0.004436 | $-1.951878$ | $-0.880693$
LMC X–4 r47  | 1.29 | 8.831 | $0.004761$ | $-1.857516$ | $-0.851659$

##  IV On the physical and mathematical feasibility

Here the principal physical and mathematical properties representing the model
are thoroughly analyzed. This point is carried out by using the usual analysis
provided by Herrera et. al and Mak et. al herre ; harko .

###  IV.1 Geometry and Thermodynamic Description

Let us start by analyzing the behavior of the inner geometry given by metric
potentials (29)–(30). From the space–time generated by these potentials we can
remark the following features:

  * •

It is observed that, $e^{\eta}$ and $e^{\lambda}$ are free from pathologies
ı.e, both are continuous functions everywhere, for all $r\in[0,R]$.

  * •

At the center of the star, equation (30) yields to $e^{\eta(0)}=B^{2}$ and
equation (29) to $e^{\lambda(0)}=1$. This information corroborates that the
geometry of the inner space–time is being described by two monotonically
increasing metric functions whose minimum values are reached at $r=0$.

As the upper left panel in Fig. 1 illustrates, both metric potential behaved
as expected. It is important to note that we plotted the inverse of
$e^{\lambda}$ in order to show that both functions match at the boundary. This
confirms that the junction procedure is well–posed. Besides, in order to avoid
undesirable physical behaviors the radial coordinate in Eq. (30) should
satisfy $r\neq\pm\sqrt{\frac{1}{C}\left(1-\frac{A^{2}}{\bar{B}^{2}}\right)}$
for all $r\in[0,R]$. To build this panel we have used the numerical data
listed in table 1. In considering the thermodynamic observables
$\\{\rho,p_{r},p_{\perp}\\}$, they all should satisfy some rules. First, all
of them must have their maximum values attained at $r=0$. This means that
these functions are monotonous decreasing with increasing $r$ inside the
compact object. Second, a positive defined behavior is required, in order to
avoid a non–physical situation. Furthermore, for the radial and tangential
pressures is required: $p_{t}>p_{r}\Rightarrow\Delta>0$, assuring a healthy
stellar interior. As said before, a positive anisotropy factor in principle
enhances the stability and hydrostatic equilibrium of the configuration (see
below for further details). Additionally, the radial pressure must vanish at
the surface $p_{r}(R)=0$, determining the size of the object. The previous
conditions are subject to some restrictions imposed on the parameters
$\\{A,\bar{B},C\\}$. At the center, the density $\rho$ and the radial pressure
$p_{r}$ provide the following constraints

$\displaystyle p_{r}(0)$ $\displaystyle>$ $\displaystyle
0\Rightarrow\bar{B}>\frac{3\sqrt{2}}{10}A,$ (48)
$\displaystyle\frac{p_{r}(0)}{\rho(0)}$ $\displaystyle\leq$ $\displaystyle
1\Rightarrow\bar{B}\leq\frac{3\sqrt{2}}{7}A,$ (49) $\displaystyle\rho(0)$
$\displaystyle>$ $\displaystyle 0\Rightarrow C>0.$ (50)

So from Eqs. (48)–(49) one has

$\frac{3\sqrt{2}}{10}A<\bar{B}\leq\frac{3\sqrt{2}}{7}A.$ (51)

In table 2 we have placed the central values for density and radial pressure
and the surface density of mentioned objects. The order of magnitude of these
quantities is consistent with stars containing a core with strange matter
(quark stars) r47 ; nature . The right upper panel in Fig. 1 shows the trend
of the density $\rho$ inside the star. As can be seen, this observable is
positive defined everywhere and has its maximum attained at the origin. The
lower panel in the same figure display the behaviour of the pressures waves
$p_{r}$ and $p_{\perp}$ and anisotropy factor $\Delta$. It is observed that
the transverse pressure $p_{\perp}$ dominates the radial one $p_{r}$ inside
the star, ensuring $\Delta>0$ everywhere. These features shows that the
stellar interior is well–behaved. Notwithstanding, the matter distribution
must satisfy additional restrictions to confirm its viability in describing a
stellar interior. Those are: i) positive and well behaved energy–momentum
tensor and ii) preservation of causality condition. The former indicates that
any well behaved energy–momentum tensor should meet the following inequalities
known as energy conditions (e4, ; visserbook, )

  1. 1.

Null energy condition (NEC): $\rho+p_{r}\geq 0$,
$\rho+p_{t}+\frac{E^{2}}{4\pi}\geq 0$.

  2. 2.

Weak energy condition (WEC): $\rho+\frac{E^{2}}{8\pi}\geq 0$, $\rho+p_{r}\geq
0$, $\rho+p_{t}+\frac{E^{2}}{4\pi}\geq 0$ .

  3. 3.

Strong energy condition (SEC): $\rho+p_{r}\geq 0$,
$\rho+p_{t}+\frac{E^{2}}{4\pi}\geq 0$,
$\rho+2p_{t}+p_{r}+\frac{E^{2}}{4\pi}\geq 0$.

  4. 4.

Dominant energy condition (DEC):
$\rho+\frac{E^{2}}{8\pi}-|p_{r}-\frac{E^{2}}{8\pi}|\geq 0$,
$\rho+\frac{E^{2}}{8\pi}-|p_{t}+\frac{E^{2}}{8\pi}|\geq 0$.

  5. 5.

Trace energy condition (TEC): $\rho-p_{r}-2p_{t}\geq 0$.

From Fig. 2 (left panel) is clear that SEC and TEC are satisfied everywhere,
while the right panel exhibiting the DEC shows that this condition is also
satisfied. Hence, the stellar interior is described by a well defined and
positive energy–momentum tensor. It is worth mentioning that the NEC and WEC
are contained by SEC and DEC.

Table 2: Numerical values for central and surface density, central pressure, critical adiabatic index and central adiabatic index for different values listed in table 1. Strange | $\rho(0)$ | $\rho(R)$ | $p_{r}(0)$ | $\Gamma_{\text{crit}}$ | $\Gamma$
---|---|---|---|---|---
Star | $\times 10^{15}\ [\text{g}/\text{cm}^{3}]$ | $\times 10^{14}\ [\text{g}/\text{cm}^{3}]$ | $\times 10^{34}\ [\text{dyne}/\text{cm}^{2}]$ |  |
SMC X–1 r47  | 1.07168 | 6.78682 | 5.64072 | 1.5003 | 2.8551
LMC X–4 r47  | $1.15024$ | $6.70688$ | $7.91838$ | $1.5280$ | $2.5726$

Figure 1: Metric potential behaviours are shown in left panel of the upper row
while the graphical behavior of density is shown in right panel of the upper
row against radial coordinate $r/R$ by taking different values from table 1.
The left side of lower row shows the path of the radial and tangential
pressures against the dimensionless radial variable $r/R$ while the right side
shows the plot of anisotropy function against $r/R$.

Figure 2:  Left panel of the figure shows the behavior of the SEC and TEC
against the dimensionless radial coordinate $r/R$ while the right panel shows
the plot of DEC versus radial coordinate $r/R$. The graphs has plotted by
using different values mentioned in table 1.

###  IV.2 Electric Properties

As we are dealing with a charged configuration, electric properties should
also meet some requirements. Both electric charge $q(r)$ and electric field
$E(r)$ must be strictly positive and increasing functions with radius, meaning
that at the origin both must be null ı.e, $q(0)=E(0)=0$. From expressions (35)
and (37) the electric field is given by

$E(r)=\sqrt{\frac{C^{2}r^{2}\left[4\bar{B}\left(Cr^{2}-2\right)+3A\sqrt{2-Cr^{2}}\right]}{4\left[1+8\pi\chi\right]\left[1+Cr^{2}\right]^{2}\left[\bar{B}\left(Cr^{2}-2\right)+A\sqrt{2-Cr^{2}}\right]}},$
(52)

and from Eq. (9) the electric charge is given by

$q(r)=r^{2}\,\sqrt{-F^{tr}F_{tr}}=r^{2}\,\sqrt{E^{2}(r)}=r^{2}E(r)=r^{2}\sqrt{\frac{C^{2}r^{2}\left[4\bar{B}\left(Cr^{2}-2\right)+3A\sqrt{2-Cr^{2}}\right]}{4\left[1+8\pi\chi\right]\left[1+Cr^{2}\right]^{2}\left[\bar{B}\left(Cr^{2}-2\right)+A\sqrt{2-Cr^{2}}\right]}}.$
(53)

From the above expressions is clear that at $r=0$ both quantities are zero, as
can be confirmed in left and middle panels of Fig. 3. Additionally, we have
checked the trend of charge density $\sigma$ inside the star. The general
expression to obtain $\sigma$ is

$\sigma(r)=e^{-\lambda/2}\frac{\left(r^{2}E\right)^{\prime}}{4\pi r^{2}}.$
(54)

The right panel of Fig. 3 illustrates the behaviour of this quantity. Unlike
the electric charge, it has its maximum attained at the center and decreases
monotonously towards the boundary of the compact object. In table 3 are shown
the values reached by $q$ and $E$ at the surface of the structure. As was
point out in e6 , to appreciate any effect on the phenomenology of compact
stars, the electric field have to be huge. Specifically, the order of
magnitude of the electric field in the compact object should be $\sim 10^{21}\
[V/m]$, implying $Q\sim 10^{20}\ [C]$. As it is observed, the order of
magnitude of these quantities is enough to appreciate the impact of electric
components in the current scenario. Moreover, these values are within the
upper limits reported in previous studies f1 ; f2 ; f3 ; f4 . To account the
effects of parameter $\chi$ in electric properties, we have computed the
numerical data presented in table 4 for different values of $\chi$. As can be
seen, as $\chi$ decreases both $E(R)$ and $q(R)$ reach greater values.

Table 3: The surface gravitational red–shift, electric field and electric charge for values depicted in table 1.  Strange | Lower bound | $z_{s}$ | Upper bound | $E(R)$ | $q(R)$
---|---|---|---|---|---
Star | $z_{s}$ |  | $z_{s}$ | $\times 10^{21}[V/cm]$ | $\times 10^{20}[C]$
SMC X–1 r47  | 0.013763 | 0.259026 | 2.38478 | 2.18009 | 1.66914
LMC X–4 r47  | $0.018692$ | $0.324997$ | $2.55665$ | 2.37971 | 1.82197
Table 4: The surface gravitational red–shift, electric field and electric charge for values depicted in table 1 and different values of $\chi$.  Strange | $\chi$ | Lower bound | Upper bound | $E(R)$ | $q(R)$
---|---|---|---|---|---
Star |  | $z_{s}$ | $z_{s}$ | $\times 10^{21}[V/cm]$ | $\times 10^{20}[C]$
SMC X–1 r47  | 0.2 | 0.002809 | 2.06939 | 0.992808 | 0.760123
SMC X–1 r47  | 1.2 | 0.000546 | 2.01318 | 0.438413 | 0.335661
LMC X–4 r47  | 0.2 | 0.003818 | 2.09531 | 1.08717 | 0.832372
LMC X–4 r47  | 1.2 | 0.000742 | 2.01795 | 0.480407 | 0.367814

Figure 3:  The electric field trend against dimensionless radial coordinate
$r/R$ is shown in the left panel. The trend of electric charge versus the
dimensionless radial coordinate $r/R$ is shown in the middle panel while right
side of the figure shows the graphical behavior of charge density versus
$r/R$. These plots were built using different values mentioned at table 1.

##  V Hydrostatic Equilibrium and Stability

For the isotropic scenario, the hydrostatic equilibrium of the structure is
subject to the gravitational and hydrostatic gradients. However, in presence
of extra ingredients, such as anisotropies and electric charge, the
hydrostatic balance changes. Now, the configuration is under the action of
gravitational $F_{g}$, hydrostatic $F_{h}$, anisotropy $F_{a}$ and electric
$F_{e}$ gradients. If the structure is in equilibrium, all gradients must
satisfy

$F_{h}+F_{a}+F_{g}+F_{e}=0,$ (55)

or, in terms of the main physical variables

$\begin{split}-p^{\prime}_{r}-\frac{1}{2}\,\eta^{\prime}\left({\rho}+{p}\right)+\frac{2}{r}\left(p_{\perp}-p_{r}\right)+\sigma\,E\,e^{\lambda/2}=0.\end{split}$
(56)

Eq. (56) is just the conservation equation (16). It should be noted that when
$\Delta\equiv p_{t}-p_{r}=0$ and $E=0$, the original
Tolman–Oppenheimer–Volkoff equation tolman ; oppen  used to study the
hydrostatic balance of compact perfect fluids, is regained. The new pieces add
into the system positive gradients ($F_{a}$ and $F_{e}$) ı.e, repulsive in
nature. It is worth mentioning that the electric gradient is repulsive due to
the Coulomb electric repulsion. These repulsive gradients help to offset the
gravitational one. The presence of these new components prevents the
contraction of the object by gravitational action. In Fig. 4 is depicted the
action of the mentioned gradient at all points within the structure. As can be
seen, the configuration is in hydrostatic equilibrium. Is remarkable to note
that the electric component overcome the anisotropy one. This situation is
connected with the inverse behavior that $\Delta$ and $E$ have with respect to
the magnitude of parameter $\chi$. In this opportunity we have taken a small
$\chi$, concretely $0.008$. This is so because the anisotropy gradient
represented by the green curve has less contribution than the electric
gradient (cyan curve) in the balance of the compact object. However the most
important thing here is that both gradients in accordance with the hydrostatic
gradient produce the balance against gravitational attraction.

Figure 4:  The trend of the hydrostatic $F_{h}$, gravitational $F_{g}$,
anisotropic $F_{a}$ and electric force gradients for each strange star
candidate are shown in both panels of the figure. These plots were built for
values mentioned in table 1.

Next, we analyze if the hydrostatic balance is stable or unstable. To do so,
the following criteria are considered: i) the relativistic adiabatic index
$\Gamma$, ii) the Harrison–-Zeldovich–-Novikov harrison ; nikolov  condition
and iii) the criterion based on sound speed of the matter distribution a19 .

As it is well known, in the classical domain (Newtonian fluid) taking into
account a perfect fluid content, the stability condition is $\Gamma>4/3$ a1 ;
a8 . Notwithstanding, in contrast with the relativistic regime subject to an
imperfect matter distribution, the scene changes radically. In this concern
local imperfections (or anisotropies) within the material content insert harsh
modifications for the stability condition a9 ; a10 . Under this situation, the
stability condition becomes

$\Gamma>\frac{4}{3}+\left[\frac{1}{3}\,\kappa\,\frac{\rho_{0}\,p_{r0}}{|p^{\prime}_{r0}|}r+\frac{4}{3}\frac{\left(p_{\perp
0}-p_{r0}\right)}{|p^{\prime}_{r0}|r}\right]_{max},$ (57)

where $\left[\rho_{0},p_{r0},p_{t0}\right]$ are the initial values of the
thermodynamic observables, when the matter distribution is in static
equilibrium. The terms enclosed in the brackets are representing the
relativistic adjustment and the point anisotropies contributions. It should be
noted that local anisotropies can be seen as a stabilizer mechanism. However,
as Chandrasekhar warned chandra1 ; chandra2 , these relativistic corrections
to $\Gamma$ could insert some instabilities within the matter distribution. To
overcome this problem, new constraints on $\Gamma$ were imposed mousta .
Specifically, a critical adiabatic index $\Gamma_{\text{crit}}$ depending on
the critical value of the amplitude of the Lagrangian displacement from
equilibrium and the compactness factor $u\equiv M/R$ were fixed. Particularly,
this constraint reads as

$\Gamma_{\text{crit}}=\frac{4}{3}+\frac{19}{21}\,u,$ (58)

being the stability condition $\Gamma\geq\Gamma_{\text{crit}}$, where $\Gamma$
is computed from a7

$\Gamma=\frac{\rho+p_{r}}{p_{r}}\frac{d\,p_{r}}{d\,\rho}.$ (59)

From the left panel of Fig. 5 it is appreciated that in all cases the
relativistic adiabatic index $\Gamma$ is greater than $4/3$. Commonly, it is
assumed that when $\Gamma>4/3$ at $r=0$ the system is stable under this
criterion. However, as pointed out before, a more precise analysis indicates
$\Gamma\geq\Gamma_{\text{crit}}$ at $r=0$. The fifth and sixth columns of
table 2 presents the corresponding values for $\Gamma$ and
$\Gamma_{\text{crit}}$ for each considered compact structure. The Harrison et
al. harrison  and Zeldovich–-Novikov nikolov  methods propose that any fluid
configuration is stable if the mass is a increasing function with respect to
the central density $\rho(0)=\rho_{c}$ ı.e, $\frac{\partial
M(\rho_{c})}{\partial\rho_{c}}>0$, otherwise the model is unstable. For the
present model, the mass as a function of the central density has the following
form

$M(\rho_{c})=\frac{1}{8}R\left(6-\frac{6}{1+\frac{16}{9}\pi\rho_{c}R^{2}}+\frac{\left[\frac{16}{9}\pi\rho_{c}\right]^{2}R^{4}\left[4\bar{B}\left(\frac{16}{9}\pi\rho_{c}R^{2}-2\right)+3A\sqrt{2-\frac{16}{9}\pi\rho_{c}R^{2}}\right]}{\left[1+\frac{16}{9}\pi\rho_{c}R^{2}\right]^{2}\left[\bar{B}\left(\frac{16}{9}\pi\rho_{c}R^{2}-2\right)+A\sqrt{2-\frac{16}{9}\pi\rho_{c}R^{2}}\right]\left[1+8\pi\chi\right]}\right).$
(60)

Right panel of Fig. 5 is illustrates the trend of total mass against the
central density. As can be seen the criteria is satisfied, thus the model is
stable under this criteria.

Figure 5:  The relativistic adiabatic index path against the dimensionless
radial coordinate $r/R$ is shown in left panel, whereas the graphical behavior
of mass function versus $r/R$ is shown in right panel of the figure for
different values mentioned in previous tables.

Finally, we check the stability of the system by means of the subliminal sound
speeds of the pressure waves. Before to proceed with the criteria it is
important to see if the matter distribution is fulfilling the causality
condition

$0\leq v^{2}_{r}=\frac{dp_{r}}{d\rho}\leq 1\quad\mbox{and}\quad 0\leq
v^{2}_{\perp}=\frac{dp_{\perp}}{d\rho}\leq 1.$ (61)

Eq. (61) states that no signal can travel faster than the speed of light
$c=1$. From left panel in Fig. 6 we check that causality condition is
preserved along the principal directions of the fluid sphere. Next, based on
this important fact, the stable/unstable regions within the stellar, interior,
when local anisotropies are there can be found as a19 :

$\frac{\delta{\Delta}}{\delta{\rho}}\sim\frac{\delta\left({p}_{\perp}-{p}_{r}\right)}{\delta{\rho}}\sim\frac{\delta{p}_{\perp}}{\delta{\rho}}-\frac{\delta{p}_{r}}{\delta{\rho}}\sim
v^{2}_{\perp}-v^{2}_{r}.$ (62)

Taking into account equation (62) one gets $0\leq|v^{2}_{\perp}-v^{2}_{r}|\leq
1$ or equivalently

$\begin{split}-1\leq v^{2}_{\perp}-v^{2}_{r}\leq
1=\left\\{\begin{array}[]{ll}-1\leq v^{2}_{\perp}-v^{2}_{r}\leq
0{}{}&\mathrm{Potentially\ stable\ }\\\ 0<v^{2}_{\perp}-v^{2}_{r}\leq
1{}{}&\mathrm{Potentially\ unstable}\end{array}\right\\}.\end{split}$ (63)

Hence, the compact object is stable under radial perturbation if and only if
the radial sound speed $v^{2}_{r}$ dominates at all points the transverse
sound speed $v^{2}_{\perp}$. In the present case, it is evident that the
difference of the square sound speeds is bounded between -1 and 0 and the
absolute value of this quantity is between 0 and 1, as showed by middle and
right panel of Fig.6. Thus, all the regions inside the collapse structure are
stable under this criterion.

Figure 6: The path of the radial and transverse speed of sounds against the
dimensionless quantity $r/R$ is shown in left panel of the figure. The
difference of square of sound speed velocities and its absolute value versus
$r/R$ are shown in middle and right panels. These plots were built for
different values mentioned in table 1.

##  VI Astrophysical Observables

Since it is not possible to directly measure the properties of a stellar
object, from astrophysical techniques one can obtain a relevant quantity: the
surface gravitational red–shift $z_{s}$, to infer important properties such as
the total mass $M$ and the radius $R$ of the star, its chemical composition,
etc. In considering the total mass of the compact structure, when the matter
distribution contains a charge component it increases by a certain amount,
which is provided by the electric field. Integrating the field equation (12)
one gets e2 ; e3 ; e8

$e^{-\lambda\left(r\right)}=1-\frac{1}{r}\int^{r}_{0}\left[8\,\pi\,\rho\left(x\right)\,x^{2}+E^{2}\left(x\right)\right]\,dx,$
(64)

this expression must coincide with exterior space–time at the boundary
$\Sigma$, yielding to

$1-2\,\frac{M}{R}+\frac{Q^{2}}{R^{2}}=1-\frac{1}{r}\int^{r}_{0}\left[8\,\pi\,\rho\left(x\right)\,x^{2}+E^{2}\left(x\right)\right]\,dx,$
(65)

being $Q=q\left(R\right)$ and $M$ the total mass of the object. So, from (65)
it is possible to obtain

$M=\frac{1}{2}\int^{R}_{0}\left[8\,\pi\,\rho\left(r\right)\,r^{2}+E^{2}\left(r\right)\right]\,dr+\frac{1}{2}\,\frac{Q^{2}}{R}.$
(66)

This total mass $M$ correspond to the gravitational mass evaluated at the
surface $m_{g}\left(r\right)=M$. So, from (65) it is clear that the
gravitational function mass is given by,

$m_{g}\left(r\right)=m_{i}\left(r\right)+\frac{1}{2}\int_{0}^{r}{\frac{q^{2}(x)}{x^{2}}\,dx}+\frac{1}{2}\,\frac{q^{2}}{r}.$
(67)

Here, $m_{i}\left(r\right)$ is the usual definition of the mass within a
radius $r$,

$m_{i}\left(r\right)=4\,\pi\int_{0}^{r}{{\rho(x)}\,x^{2}\,dx}.$ (68)

In the above expression we have denoted $m_{i}$ as the mass containing both
rest and internal energy. This prescription allows us to distinguish between
$m_{i}\left(r\right)$ from the gravitational mass $m_{g}\left(r\right)$ e8 .
On the other hand, associated with gravitational mass is the so–called
compactness factor or mass–radius ratios $u\equiv M/R$. For this particular
model the gravitational mass functions and compactness factor are given by

$m_{g}\left(r\right)=\frac{r}{8}\left[6-\frac{6}{1+C\,r^{2}}+\frac{C^{2}\,r^{4}\left(4\,\bar{B}\left(C\,r^{2}-2\right)+3\,A\,\sqrt{2-C\,r^{2}}\right)}{\left(1+C\,r^{2}\right)^{2}\left(\bar{B}\,\left(C\,r^{2}-2\right)+A\,\sqrt{2-C\,r^{2}}\right)\left(1+8\,\pi\,\alpha\right)}\right],$
(69)
$u\left(r\right)=\frac{1}{8}\left[6-\frac{6}{1+C\,r^{2}}+\frac{C^{2}\,r^{4}\left(4\,\bar{B}\left(C\,r^{2}-2\right)+3\,A\,\sqrt{2-C\,r^{2}}\right)}{\left(1+C\,r^{2}\right)^{2}\left(\bar{B}\,\left(C\,r^{2}-2\right)+A\,\sqrt{2-C\,r^{2}}\right)\left(1+8\,\pi\,\alpha\right)}\right].$
(70)

As can be seen the electric component modified the mass and mass–radius ratio.
This implies that the surface gravitational red–shift $z_{s}$ is also altered.
In fact, $z_{s}$ depends on $u$ in the following way

$z_{s}=\frac{1}{\sqrt{1-2\,u}}-1.$ (71)

In Fig. 7 the trend of (69)–(71) is displayed for numerical values given in
table 1 for different real compact objects. Furthermore, in tables 3 and 5 are
exhibited the vales for $z_{s}$ and $u$ corresponding to these stars.
Nevertheless, the electric field modifies the mass–radius relation, in such a
way that the well–known Buchdahl limit $u=M/R=4/9$ buch  for isotropic
uncharged fluid spheres can be overcome. This means that the upper bound
exceeds the value $u=0.\bar{4}$ e9 , and also acquires a lower bound e7

$\frac{Q^{2}\left(18R^{2}+Q^{2}\right)}{2R^{2}\left(12R^{2}+Q^{2}\right)}\leq\frac{M}{R}\leq\frac{2R^{2}+3Q^{2}+2R\sqrt{R^{2}+3Q^{2}}}{9R^{2}}.$
(72)

As it is appreciated from table 5 the upper bound of $u$ surpasses the
Buchdahl bound. For both lower and upper limits we have computed the
corresponding $z_{s}$ shown by table 3. In this respect, Ivanov iva  pointed
out that for an anisotropic star the constraint on the surface gravitational
red–shift is $z_{s}=5.211$. As can be seen the obtained results are bounded by
this value. Moreover, we have checked the impact of parameter $\chi$ on the
lower and upper bounds of the surface gravitational red–shift (see table 4)
and compactness factor (see table 6). It is observed that a increasing $\chi$
in magnitude dismiss both $z_{s}$ and $u$ (the lower and upper limits).

Table 5: Lower and upper bounds for the mass-radius ratio and the mass-radius relation for $\chi=0.008$ and values mentioned in table 1 Strange | Lower bound |  Mass–Radius ratio | Upper bound
---|---|---|---
Star | $\frac{Q^{2}\left(18R^{2}+Q^{2}\right)}{2R^{2}\left(12R^{2}+Q^{2}\right)}$ | $\frac{M}{R}$ | $\frac{2R^{2}+3Q^{2}+2R\sqrt{R^{2}+3Q^{2}}}{9R^{2}}$
SMC X–1 r47  | 0.0134841 | 0.1846 | 0.45636
LMC X–4 r47  | 0.0181804 | 0.2152 | 0.46047

Figure 7:  Left panel: The mass function trend against the dimensionless radial coordinate $r/R$. Middle panel: The compactness factor versus the dimensionless radial coordinate $r/R$. Right panel: the gravitational surface red–shift versus $r/R$. These plots were built for different values mentioned in tables 1.  Table 6: Lower and upper bounds for the mass-radius ratio and the mass-radius relation for different values of $\chi$ and values mentioned in table 1 Strange | $\chi$ | Lower bound |  | Upper bound
---|---|---|---|---
Star |  | $\frac{Q^{2}\left(18R^{2}+Q^{2}\right)}{2R^{2}\left(12R^{2}+Q^{2}\right)}$ |  | $\frac{2R^{2}+3Q^{2}+2R\sqrt{R^{2}+3Q^{2}}}{9R^{2}}$
SMC X–1 r47  | 0.2 | 0.002798 |  | 0.44693
SMC X–1 r47  | 1.2 | 0.000546 |  | 0.44493
LMC X–4 r47  | 0.2 | 0.003797 |  | 0.44781
LMC X–4 r47  | 1.2 | 0.000741 |  | 0.44510

##  VII Concluding Remarks

In this article we have obtained a well–behaved interior solution describing
compact objects such as neutron and quark stars. The main ingredient of this
model is a charged anisotropic matter distribution in the stellar interior. To
obtain this toy model we have closed the Einstein–Maxwell set of equations by
using the class I approach and a link between the anisotropy factor and the
electric field. The class I methodology provides the geometry of inner space-
time once one of the metric potentials is imposed. In this opportunity we have
selected the $g_{rr}$ potential corresponding to the isotropic Buchdhal
solution. This choice is well motivated since it is free from physical and
mathematical singularities. Placing this metric potential into Eq. (26) the
temporal metric component is determined, completing the geometrical
description of the problem. On the other hand, to obtain the full
energy–momentum tensor (3), instead of impose an equation of state, we have
established a relation between the anisotropy factor ($\Delta>0$) and the
electric field (32). This relation is mediated by a dimensionless parameter,
namely $\chi$. As $\chi$ increases in magnitude the electric field also
increases, while the anisotropy factor decreases and vice versa. The structure
of this link discards negative values for $\chi$ in order to assure $\Delta>0$
everywhere. The main features characterizing the model ı.e, the density
$\rho$, radial pressure $p_{r}$ and transverse pressure $p_{\perp}$ satisfy
all the requirement to represent a well behaved stellar interior solution.
This is corroborated in Fig. 1, where it is clear that these thermodynamic
quantities are positive defined and decreasing functions with increasing
radial coordinate $r$. In the same figure, in upper left panel, the metric
potentials are exhibited. It is shown that both potentials coincide at the
boundary of the object indicating that the junction condition procedure with
the exterior Reissner–Nordström space-time is correct. It is worth mentioning
that the energy–momentum tensor satisfies all energy conditions (see Fig. 2
for more details), thus the matter distribution is well behaved and admissible
from the physical point of view. Respect to the electric properties, they
behaved as expected ı.e, null at the center of the structure and positive
defined and increasing function within the stellar interior. It should be
noted that the order of magnitude of the central density, the electric field
and electric charge evaluated at the boundary of the compact star are in
complete agreement with the values reported for charged quark stars (see
tables 2 and 3). In addition, we have obtained in table 4 the values for the
electric field and electric charge for different values of the parameter
$\chi$, and as expected for increasing $\chi$ the electric properties
decrease.

The hydrostatic balance and stability of the system, under the action of the
hydrostatic $F_{h}$, gravitational $F_{g}$, anisotropic $F_{a}$ and electric
$F_{e}$ gradients was analyzed. The configuration remains in hydrostatic
balances under the mentioned gradients. In this regard the electric gradient
play an important role to counteract the gravitational one, avoiding a
collapse into a point singularity (refer to Fig. 4). As mentioned before, if
$\chi$ decreases in magnitude then the anisotropic gradient $F_{a}$ increases,
being more relevant than $F_{e}$ in this process. However, regardless of which
gradient dominates, they all help to avoid collapse against the gravitational
gradient. We studied the stability of the hydrostatic balance from three
different schemes, namely: i) relativistic adiabatic index, ii)
Harrison–Zeldovich–Novikov and iii) Abreu’s et. al criterion. As can be seen
in Figs. 5 and 6 the system is stable in all frames. In table 2 it is
appreciated that the central relativistic adiabatic index overcome the
critical value, and the left panel in Fig. 6 shows that the matter
distribution meets causality condition along the principal directions of the
fluid sphere. Finally, we have studied the impact of electric properties on
the macro physical observables ı.e, total gravitational mass, mass–radius
ratio and surface gravitational red–shift. The trend of this quantities are
depicted in Fig. 7. Besides, in tables 4 and 6 is shown the impact of
parameter $\chi$ on these important observables and also on the electric
properties. So, taking into account all these things, we can conclude that the
obtained model could represent charged anisotropic compact objects.

## Acknowledgements

Y. Gomez-Leyton thanks the financial support by the CONICYT PFCHA/DOCTORADO-
NACIONAL/$2020$\- $21202056$. F. Tello-Ortiz thanks the financial support by
the CONICYT PFCHA/DOCTORADO-NACIONAL/$2019$-$21190856$ and projects ANT-$1756$
and SEM $18-02$ at the Universidad de Antofagasta, Chile. F. Tello-Ortiz
thanks the PhD program Doctorado en Física mención en Física Matemática de la
Universidad de Antofagasta for continuous support and encouragement.

##  Appendix A The full energy–momentum tensor

In this appendix we show how the get the energy–momentum tensor given by Eq.
(3). The derivation provided here is the classical one, that is, we start from
the Einstein–Hilbert action minimally coupled to the Maxwell electromagnetic
field including both the interaction and particle contributions weinberg . So
we have

$S_{\text{Total}}=S_{\text{E--
H}}+S_{\text{EM}}+S_{\text{Int}}+S_{\text{Part}},$ (73)

where each action is given by

$\displaystyle S_{\text{E--H}}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\int d^{4}x\sqrt{-g}R,$ (74) $\displaystyle
S_{\text{EM}}$ $\displaystyle=$ $\displaystyle-\frac{1}{16\pi}\int
d^{4}x\sqrt{-g}F_{\mu\nu}F^{\mu\nu},$ (75) $\displaystyle S_{\text{Int}}$
$\displaystyle=$ $\displaystyle-\sum_{n}e_{n}\int
dp\frac{x^{\mu}_{n}}{dp}A_{\mu},$ (76) $\displaystyle S_{\text{Part}}$
$\displaystyle=$ $\displaystyle-\sum_{n}m_{n}\int
dp\left[-g_{\mu\nu}\frac{dx^{\mu}}{dp}\frac{dx^{\nu}}{dp}\right]^{1/2}.$ (77)

It worth mentioning that $g_{\mu\nu}$ and $A_{\mu}$ are functions of
$x^{\mu}(p)$ being $p$ some quantity parametrizing the particle trajectories.
Furthermore, $m_{n}$ and $e_{n}$ are the mass and electric charge for each
point particles. Now, variations with respect to $A_{\mu}$ and $x^{\mu}$ lead
to the following field equations for he electromagnetic field and the point
particles

$\displaystyle\nabla_{[\alpha}F_{\beta\gamma]}$ $\displaystyle=$
$\displaystyle 0,$ (78)
$\displaystyle\partial_{\mu}\left[\sqrt{-g}F^{\mu\nu}\right]$ $\displaystyle=$
$\displaystyle
4\pi\sum_{n}e_{n}\int\delta^{4}\left(x-x_{n}\right)\frac{dx^{\nu}_{n}}{d\tau_{n}}d\tau_{n},$
(79)

where it is clear from (79) that

$\sqrt{-g}J^{\nu}\equiv\sum_{n}e_{n}\int\delta^{4}\left(x-x_{n}\right)\frac{dx^{\nu}_{n}}{d\tau_{n}}d\tau_{n}.$
(80)

Moreover, to satisfy (78)–(79) the tensor $F_{\mu\nu}$ is defined as
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Next, for the point
particles one has

$\frac{d^{2}x^{\mu}_{n}}{d\tau^{2}_{n}}+\Gamma^{\mu}_{\nu\lambda}\frac{dx^{\lambda}_{n}}{d\tau_{n}}\frac{dx^{\nu}_{n}}{d\tau_{n}}=\left(\frac{e_{n}}{m_{n}}\right)F^{\mu}_{\
\nu}\frac{dx^{\nu}_{n}}{d\tau_{n}},$ (81)

and $d\tau_{n}$ is given by

$d\tau_{n}\equiv\left[-g_{\mu\nu}dx^{\mu}dx^{\nu}\right]^{1/2}.$ (82)

Variation with respect to $g_{\mu\nu}$ leads to the Einstein tensor

$G_{\mu\nu}\equiv R_{\mu\nu}-\frac{R}{2}g_{\mu\nu},$ (83)

whilst the remaining terms conform the energy–momentum tensor

$T^{\mu\nu}=\frac{1}{\sqrt{-g}}\sum_{n}m_{n}\int
d\tau_{n}\frac{dx^{\mu}_{n}}{d\tau_{n}}\frac{dx^{\nu}_{n}}{d\tau_{n}}\delta^{4}\left(x-x_{n}\right)+\frac{1}{4\pi}\left(\frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}-F^{\mu\alpha}F^{\nu}_{\
\alpha}\right).$ (84)

The first term in (84), under certain assumptions could represent any type of
matter distribution sh . In this case this term is describing an imperfect or
anisotropic fluid, expressed by

$\frac{1}{\sqrt{-g}}\sum_{n}m_{n}\int
d\tau_{n}\frac{dx^{\mu}_{n}}{d\tau_{n}}\frac{dx^{\nu}_{n}}{d\tau_{n}}\delta^{4}\left(x-x_{n}\right)=\left({\rho}+{\bar{p}}_{\perp}\right)\,\chi^{\mu}\,\chi^{\nu}-g^{\mu\nu}\,{\bar{p}}_{\perp}+\left({\bar{p}}_{r}-{\bar{p}}_{\perp}\right)\,u^{\mu}\,u^{\nu}.$
(85)

Then, the field equations for the gravitational sector are

$G_{\mu\nu}=8\pi T_{\mu\nu}.$ (86)

By virtue of Bianchi’s identities one has

$\nabla_{\mu}G^{\mu\nu}=0\Rightarrow\nabla_{\mu}T^{\mu\nu}=0.$ (87)

So, from Eqs. (84)–(85) it is not hard to obtain

$\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}\left[\left({\rho}+{\bar{p}}_{\perp}\right)\,\chi^{\mu}\,\chi^{\nu}-g^{\mu\nu}\,{\bar{p}}_{\perp}+\left({\bar{p}}_{r}-{\bar{p}}_{\perp}\right)\,u^{\mu}\,u^{\nu}\right]-F^{\nu}_{\
\alpha}J^{\alpha}.$ (88)

In obtaining the second member in the right hand side of (88) we have employed
the Eqs. (78)–(79). Now, combining Eqs. (80), (81) and (85) one arrives to

$\nabla_{\mu}\left[\left({\rho}+{\bar{p}}_{\perp}\right)\,\chi^{\mu}\,\chi^{\nu}-g^{\mu\nu}\,{\bar{p}}_{\perp}+\left({\bar{p}}_{r}-{\bar{p}}_{\perp}\right)\,u^{\mu}\,u^{\nu}\right]=F^{\nu}_{\
\alpha}J^{\alpha}.$ (89)

Thus, inserting (89) into (88) one gets $\nabla_{\mu}T^{\mu\nu}=0$ as desired.
As can be seen, the whole energy–momentum tensor is conserved ı.e,
electromagnetic field (containing the interaction) plus point particles
contribution.

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