# Charged Compact Stars in $f(\mathcal{G})$ Gravity

M. Ilyas
Centre for High Energy Physics, University of the Punjab,
Quaid-i-Azam Campus, Lahore-54590, Pakistan<EMAIL_ADDRESS>
###### Abstract

This work is devoted to investigate some of the interior configuration of
static anisotropic spherical stellar charged structures in the regime of
$f(\mathcal{G})$ gravity, where $\mathcal{G}$ is the Gauss Bonnet invariant.
The structure of particular charged stars is analyzed with the help of
solution obtained by Krori and Barua under different viable models in
$f(\mathcal{G})$ gravity theory. The behavior of some physical aspects is
investigated with the help of plots and the viability of our modeling is
analyzed through different energy conditions. We have also studied some
behavior of these realistic charged compact stars and discuss some aspects
like density variation, evolution of stresses, different forces, stability of
these stars, measure of anisotropy, equation of state parameters and the
distribution of charges.

##  1 Introduction

Despite of the great, well established and successful theory, the general
theory of relativity, in the past century, numerous valuable modifications are
being suggested by researchers. In these modification, the Ricci scalar is
replace by some arbitrary function, like $f(\mathcal{R})$ in which
$\mathcal{R}$ is ricci scalar, $f(\mathcal{G})$, where $\mathcal{G}$ in Gauss-
Bonnet invariant and many others as discussed in ref [1, 2, 3, 4, 5, 6, 7, 8,
9].
The expansion of the universe is the remarkable phenomenon which is being
addressed by these modified theories [10]. The well-established fact is, the
accelerated expansion of universe cannot be explaining by GR alone in its
regular arrangement without adding extra term in the gravitational Lagrangian
or exotic matter [11, 12]. The simplest modification was given by Buchdal in
1970 [13] with the help of replacement of $\mathcal{R}$ by $f(\mathcal{R})$
arbitrary function of $\mathcal{R}$ in the Hilbert-Einstein gravitational
action. In literature, lot of information regarding modified theories of
gravity is available [14, 15, 16, 17, 18, 19, 20]. One of the well-known
modified theory is, Gauss Bonnet gravity, which has been studied many times in
the recent past years [21, 22]. In this modified gravity, the Hilbert-Einstein
action consist of a function $f(\mathcal{R},\mathcal{G})$ instead of
$\mathcal{R}$. This is one of the fact that the additional Gauss-Bonnet term
resolve the shortcomings of $f(\mathcal{R})$ gravity theory in the background
of large expansion of universe [23, 21, 22, 24, 25]. The simplest form of
$f(\mathcal{R},\mathcal{G})$ modified gravity is the $f(\mathcal{G})$ gravity
which is widely addressed and can reproduce any kind of cosmological
solutions.
Like, it could help out in the possible study of an acceleration regimes, and
their transition to decelerated regimes, inflationary epoch and passes all
tests evoked by solar system experiments and crossing phantom divide line [26,
27]. The $f(\mathcal{G})$ gravity is less constrained than $f(\mathcal{R})$
gravity as discussed in [28]. In addition, the $f(\mathcal{G})$ gravity offer
an efficient platform to analyze several cosmic issues as an alternate to dark
energy [29]. similarly, the $f(\mathcal{G})$ gravity is very supportive to
study the behavior of finite time future singularities along with late time
eras of an accelerating universe [30, 31]. Furthermore, in the background of
some viable models in $f(\mathcal{G})$ gravity, the cosmic accelerating nature
followed by matter era is also studied [28, 29]. Several viable
$f(\mathcal{G})$ gravity models were suggested for the purpose to pass some
certain solar system constraints [28, 29] which are studied in [32] and
further bounds on $f(\mathcal{G})$ gravity models may develop from the
behavior of energy conditions [33, 34, 35].
Observations of compact objects like pulsars, neutron stars and black holes
have attracted the researchers towards the useful physical modeling stuff
based on highly precise observational data instead of just finding the
mathematical expeditions [36].
Recently, some of the physical properties of different strange compact stars
were studied in the framework of different modified gravity theories and it
was concluded that all these strange stars under consideration are stable,
matter content is realistic and obeys all the energy conditions [37, 38, 39].
In favor of modeling static objects, supposition of spherical symmetry
geometry is very useful and natural while there are more options in the choice
of matter content. In past, many researchers focused their attention on
perfect fluid matter content. While fluids with viscosity and pressure
anisotropic fluids have also been studied and concluded that the anisotropy
disturb the stability of the configuration relative to local isotropic case.
Furthermore, the effects of local anisotropy have been elaborated with the
help of equation of state [40]. Therefore it looks suitable to deal the
anisotropic pressure with modified gravity models. Some of the physical
properties of compact stars have been studied in the presence of pressure
anisotropy and charge [41, 42, 43, 44].
The aim of this research work is to investigate the role of $f(\mathcal{G})$
gravity models in modeling of realistic charged compact stellar structures. We
investigate the different structural properties, like evolution of charged
matter density and anisotropic pressure, the Tolman-Oppenheimer-Volkoff
equation, the stability, the equation of state parameters as well as the
different energy conditions, for different observational data of compact
stars. This paper is design as, in a very next section, we discuss the
modified $f(\mathcal{G})$ gravity with charged anisotropic matter distribution
of the static spherically symmetric geometry. In section 3, we demonstrate
some of viable $f(\mathcal{G})$ gravity models. Section 4 is dedicated to
check the physical analysis and viability of different well known compact
stars through plots. And finally, we summarize the main results in last
section.

##  2 $f(\mathcal{G})$ gravity

This section is to provide the extended version of Gauss-Bonnet gravity with
its equations of motion. For $f(\mathcal{G})$ gravity, the usual Einstein-
Hilbert action is modified as follows

$S=\int{{d^{4}}x\sqrt{-g}\left[{\frac{\mathcal{R}}{2}+f(\mathcal{G})}\right]+{S_{m}}\left({{g^{\mu\nu}},\psi}\right)+{S_{e}}\left({{g^{\mu\nu}},\psi}\right)},$
(1)

where ${\kappa^{2}}=8\pi G\equiv 1$,
$\mathcal{R},~{}f,~{}{S_{m}}({{g^{\mu\nu}},\psi}),~{}{S_{e}}({{g^{\mu\nu}},\psi})$
are the Ricci scalar, arbitrary function of Gauss-Bonnet invariant, the matter
action and the charged action ,respectively. The Gauss-Bonnet invariant
quantity is

$\mathcal{G}=\mathcal{R}^{2}-4{R_{\mu\nu}}{R^{\mu\nu}}+{R_{\mu\nu\alpha\beta}}{R^{\mu\nu\alpha\beta}},$
(2)

where $R_{\mu\nu}$, ${R_{\mu\nu\alpha\beta}}$ are the Ricci and the Riemannian
tensors. Upon varying the above action with respect to $g_{\mu\nu}$, we get
the modified field equations for $f(\mathcal{G})$ gravity as

${R_{\mu\nu}}-\frac{1}{2}\mathcal{R}{g_{\mu\nu}}=T_{\mu\nu}^{\textrm{eff}},$
(3)

where $T_{\mu\nu}^{\textrm{eff}}$ is named as effective stress-energy tensor
with its expression as follows

$\displaystyle T_{\mu\nu}^{{\rm{eff}}}$
$\displaystyle={\kappa^{2}}({T_{\mu\nu}}+{E_{\mu\nu}})-8\left[{{R_{\mu\rho\nu\sigma}}}\right.+{R_{\rho\nu}}{g_{\sigma\mu}}-{R_{\rho\sigma}}{g_{\mu\nu}}-{R_{\mu\nu}}{g_{\rho\sigma}}+{R_{\mu\sigma}}{g_{\nu\rho}}$
$\displaystyle+\frac{1}{2}\mathcal{R}({g_{\mu\nu}}{g_{\rho\sigma}}-{g_{\mu\sigma}}\left.{{g_{\nu\rho}})}\right]{\nabla^{\rho}}{\nabla^{\sigma}}{f_{G}}+\left({\mathcal{G}{f_{G}}-f}\right){g_{\mu\nu}},$
(4)

where subscript $G$ defines the derivation of the corresponding term with the
GB term, while ${T_{\mu\nu}}$ is the usual stress energy momentum tensor and

${E_{\mu\nu}}=\frac{{{g_{\mu\mu}}}}{2}\left[{-{F^{\mu\alpha}}{F_{\alpha\nu}}+\frac{1}{4}\delta_{\nu}^{\mu}{F^{\alpha\beta}}{F_{\alpha\beta}}}\right].$
(5)

###  2.1 Anisotropic matter distribution in $f(\mathcal{G})$ gravity

Here, we wish to examine the effects of anisotropic stresses over the
stability of compact charged stars. For this purpose, we take the distribution
of matter content source to be anisotropic having the following mathematical
formulation

$\displaystyle{T_{\alpha\gamma}}=(\rho+{P_{r}}){V_{\alpha}}{V_{\gamma}}-{P_{t}}{g_{\alpha\gamma}}+\Pi{U_{\alpha}}{U_{\gamma}},$
(6)

where $\rho$ is fluid energy density, $P_{t}$ is tangential pressure
component, $P_{r}$ is radial pressure component and $\Pi$ is equal to
$P_{r}-P_{t}$. Furthermore, ${V_{\gamma}}$ and $U_{\gamma}$ are four velocity
and four vector of the fluid, respectively. These quantities obey
${V^{\gamma}}{V_{\gamma}}=1$ and ${U^{\alpha}}{U_{\alpha}}=-1$ relation under
the comoving coordinate system,.
Now, we suppose the interior relativistic structure to be static and spherical
symmetric everywhere. In this direction, we take the general line element
static spherical symmetric geometry as following

$d{s^{2}}={e^{a(r)}}d{t^{2}}-{e^{b(r)}}d{r^{2}}-{r^{2}}\left({d{\theta^{2}}+{{\sin}^{2}}\theta
d{\phi^{2}}}\right),$ (7)

Where $a$ and $b$ are arbitrary constant. Now by solving the field equations
(3), we get

$\displaystyle\rho+E^{2}=$
$\displaystyle\frac{1}{{2{r^{2}}}}{{\rm{e}}^{-2b}}\left[{-2{{\rm{e}}^{b}}}\right.+2{{\rm{e}}^{2b}}-{{\rm{e}}^{2b}}f{r^{2}}+{{\rm{e}}^{2b}}{r^{2}}{f_{G}}\mathcal{G}$
$\displaystyle+2b^{\prime}({{\rm{e}}^{b}}r-2({{\rm{e}}^{b}}-3){f_{G}}^{\prime})-8{f_{G}}^{\prime\prime}+8\left.{{{\rm{e}}^{b}}{f_{G}}^{\prime\prime}}\right],$
(8) $\displaystyle{p_{r}}-E^{2}=$
$\displaystyle\frac{{{{\rm{e}}^{-2b}}}}{{2{r^{2}}}}\left[{{{\rm{e}}^{b}}(2}\right.+{{\rm{e}}^{b}}({r^{2}}f-2))-{{\rm{e}}^{2b}}{r^{2}}{f_{G}}\mathcal{G}$
$\displaystyle+2a^{\prime}({{\rm{e}}^{b}}r-2({{\rm{e}}^{b}}-3)\left.{{f_{G}}^{\prime})}\right],$
(9) $\displaystyle{p_{t}}+E^{2}=$
$\displaystyle\frac{1}{{4r}}{{\rm{e}}^{-2b}}\left[{-2{{\rm{e}}^{2b}}}\right.r{f_{G}}\mathcal{G}+{{a^{\prime}}^{2}}({{\rm{e}}^{b}}r+4{f_{G}}^{\prime})+2({{\rm{e}}^{2b}}rf-{{\rm{e}}^{b}}b^{\prime}$
$\displaystyle+({{\rm{e}}^{b}}r+4{f_{G}}^{\prime}){a^{\prime\prime}})+a^{\prime}(-b^{\prime}({{\rm{e}}^{b}}r+12{f_{G}}^{\prime})+2({{\rm{e}}^{b}}+4\left.{{f_{G}}^{\prime\prime}))}\right].$
(10)

Here, $E^{2}=\frac{Q^{2}}{8\pi r^{4}}$. We suppose $a=r^{2}B+C$ and $b=r^{2}A$
as suggested by Krori and Barua [45], here $A$, $B$ and $C$ are the arbitrary
constant. Using these definitions, we reach at

$\displaystyle\rho=$ $\displaystyle
2A{{\rm{e}}^{-A{r^{2}}}}-\frac{f}{2}+\frac{1}{{{r^{2}}}}-\frac{{{{\rm{e}}^{-A{r^{2}}}}}}{{{r^{2}}}}+\frac{1}{2}{f_{G}}\mathcal{G}-\frac{{{Q^{2}}}}{{8\pi{r^{4}}}}+\frac{{12A{{\rm{e}}^{-2A{r^{2}}}}{f_{G}}^{\prime}}}{r}$
$\displaystyle-\frac{{4A{{\rm{e}}^{-A{r^{2}}}}{f_{G}}^{\prime}}}{r}-\frac{{4{{\rm{e}}^{-2A{r^{2}}}}{f_{G}}^{\prime\prime}}}{{{r^{2}}}}+\frac{{4{{\rm{e}}^{-A{r^{2}}}}{f_{G}}^{\prime\prime}}}{{{r^{2}}}},$
(11) $\displaystyle{p_{r}}=$ $\displaystyle
2B{{\rm{e}}^{-A{r^{2}}}}+\frac{f}{2}-\frac{1}{{{r^{2}}}}+\frac{{{{\rm{e}}^{-A{r^{2}}}}}}{{{r^{2}}}}-\frac{1}{2}{f_{G}}\mathcal{G}+\frac{{{Q^{2}}}}{{8\pi{r^{4}}}}$
$\displaystyle+\frac{{12B{{\rm{e}}^{-2A{r^{2}}}}{f_{G}}^{\prime}}}{r}-\frac{{4B{{\rm{e}}^{-A{r^{2}}}}{f_{G}}^{\prime}}}{r},$
(12) $\displaystyle p_{t}=$ $\displaystyle
2B{{\rm{e}}^{-A{r^{2}}}}-A{{\rm{e}}^{-A{r^{2}}}}+\frac{f}{2}-AB{{\rm{e}}^{-A{r^{2}}}}{r^{2}}+{B^{2}}{{\rm{e}}^{-A{r^{2}}}}{r^{2}}-\frac{1}{2}{f_{G}}\mathcal{G}-\frac{{{Q^{2}}}}{{8\pi{r^{4}}}}$
$\displaystyle+\frac{{4B{{\rm{e}}^{-2A{r^{2}}}}{f_{G}}^{\prime}}}{r}-12AB{{\rm{e}}^{-2A{r^{2}}}}r{f_{G}}^{\prime}+4{B^{2}}{{\rm{e}}^{-2A{r^{2}}}}r{f_{G}}^{\prime}+4B{{\rm{e}}^{-2A{r^{2}}}}{f_{G}}^{\prime\prime}$
(13)

We will use these equations with different models. Here, we see charge
contribute in $\rho$, $p_{r}$ and $p_{t}$. Now consider the quark matter EoS

${p_{r}}=\frac{1}{3}\left[{\rho-4{B_{g}}}\right]$ (14)

where $B_{g}$ is bag constant. Using this equation, we find the expression for
charge, read as

$\displaystyle Q=$ $\displaystyle
2{{\rm{e}}^{-A{r^{2}}}}\sqrt{\pi}r\left[{{{\rm{e}}^{2A{r^{2}}}}}\right.(2-2Bg{r^{2}}+{r^{2}}(-f+\mathcal{G}f^{\prime}))+2{f^{\prime\prime}}(3(A-3B)r\mathcal{G}^{\prime}-{\mathcal{G}^{\prime\prime}})$
$\displaystyle-2{{\mathcal{G}^{\prime}}^{2}}{f^{\left(3\right)}}+{{\rm{e}}^{A{r^{2}}}}(-2+(A-3B){r^{2}}+2{f^{\prime\prime}}(-\left({A-3B}\right)r\mathcal{G}^{\prime}+{\mathcal{G}^{\prime\prime}})+2{{\mathcal{G}^{\prime}}^{2}}{\left.{{f^{\left(3\right)}})}\right]^{\frac{1}{2}}}$
(15)

The expression for $Q$ contain a square-root which means both sign for charge
are acceptable but we will consider the positive sign of charge for further
investigation.

##  3 Matching condition and Different Models

In this section, we consider a hypersurface $\Sigma$ that is a boundary of
both exterior and interior regions. Furthermore, we suppose Reissner-Nordstr m
metric for the description of exterior geometry, written as

$d{s^{2}}=\left[{1-\frac{{2m}}{r}+\frac{{{Q^{2}}}}{{{r^{2}}}}}\right]d{t^{2}}-{\left[{1-\frac{{2m}}{r}+\frac{{{Q^{2}}}}{{{r^{2}}}}}\right]^{-1}}d{r^{2}}-{r^{2}}\left({d{\theta^{2}}+\sin{\theta^{2}}d{\varphi^{2}}}\right)$
(16)

Where $m$, $r$ and $Q$ is the mass, radius and charge, respectively. The
interior of given metric in Eq. (7) for the charged fluid distribution join
smoothly with the above exterior of Reissner-Nordstr m metric. By matching the
these two geometries at $r=R$ and $m(R)=M$, we get

$A=-\frac{{1}}{{{R^{2}}}}\ln\left[{1+\frac{{{Q^{2}}}}{{{R^{2}}}}-\frac{{2M}}{R}}\right],$
(17)
$B=\left({\frac{M}{R^{3}}+\frac{{{Q^{2}}}}{{{R^{4}}}}}\right){\left[{1+\frac{{{Q^{2}}}}{{{R^{2}}}}-\frac{{2M}}{R}}\right]^{-1}},$
(18)
$C=\ln\left[{1+\frac{{{Q^{2}}}}{{{R^{2}}}}-\frac{{2M}}{R}}\right]-\left({\frac{M}{R}-\frac{{{Q^{2}}}}{{{R^{2}}}}}\right){\left[{1+\frac{{{Q^{2}}}}{{{R^{2}}}}-\frac{{2M}}{R}}\right]^{-1}}.$
(19)

Compact Stars | $M$ | $R(km)$ | $\mu_{M}=\frac{M}{R}$ | $A$ | $B$ | $\mu_{C}=\frac{Q^{2}}{R^{2}}$
---|---|---|---|---|---|---
Vela X - 1 (CS1) | $1.77M_{\odot}$ | $9.56$ | $0.273091$ | $0.00832706$ | $0.00608302$ | $0.0133624$
SAXJ1808.4-3658 (CS2) | $1.435M_{\odot}$ | $7.07$ | $0.299$ | $0.0169456$ | $0.0127081$ | $0.0266898$
4U1820-30 (CS3) | $2.25M_{\odot}$ | $10$ | $0.332$ | $0.00760739$ | $0.00555676$ | $0.0133208$
Table 1: The approximate values of the masses, radii and compactness for
charged compact stars, Vela X - 1, SAXJ 1808.4-3658, and 4U 1820-30 and their
numerical values of the constants $A$, and $B$.

We find the numerical values of these constat for three different strange
compact physical stars as shown in Table: 1. Furthermore, we will take some of
the viable models for the study of different compact star properties like the
stability analysis energy conditions etc As

$f(\mathcal{G})=f_{i}(\mathcal{G})$

where we will take three different models $i=1,2,3$

###  3.1 Model 1

First, we assume the power-law model with the additional logarithmic
correction term [46]

$f_{1}=\alpha_{1}{\mathcal{G}^{n_{1}}}+\beta_{1}\mathcal{G}\log(\mathcal{G}),$
(20)

where $\alpha_{1},~{}n_{1}$ and $\beta_{1}$ are arbitrary constants. This
model could provide observationally well-consistent cosmic results because of
its extra degrees of freedom allowed in the dynamics.

###  3.2 Model 2

Next, we take another model having the form [47]

$f_{2}=\alpha_{2}\mathcal{G}^{n_{2}}\left(\beta_{2}\mathcal{G}^{m}+1\right),$
(21)

where $\alpha_{2},~{}\beta_{2}$ and $m$ are any constant number, while
$n_{2}>0$. This model is very helpful for the treatment of finite time future
singularities.

###  3.3 Model 3

Further, we assume another viable model of the form

$f_{3}=\frac{{{a_{1}}{\mathcal{G}^{n_{3}}}+{b_{1}}}}{{{a_{2}}{\mathcal{G}^{n_{3}}}+{b_{2}}}},$
(22)

here $a_{1}$, $a_{2}$, $b_{1}$, $b_{2}$ and $n_{3}$ are arbitrary constants
with $n_{3}>0$.
From the condition $p_{r}(R)=0$, We find the value of $B_{g}$ as shown in
Table: 2.

Models |  $B_{g}$ for CS1 |  $B_{g}$ for CS2 |  $B_{g}$ for CS3
---|---|---|---
Model 1 | 0.00336605 | 0.00635613 | 0.00307594
Model 2 | 0.00336605 | 0.006356117 | 0.003075942
Model 3 | 0.00336603 | 0.006355635 | 0.003075926
Table 2: The approximate values of the the constant $B_{g}$ for the three
different stars under three different models.

Using these model with eq. (11-13), we get $\rho$, $p_{r}$ and $p_{t}$ from
which we check the different aspect of compact stars as shown in Table 1. We
will discuss these aspect one by one in the following section.

##  4 Aspects of $f(\mathcal{G})$ Gravity Models

In this section, we discuss some of physical aspects of the above charged
stars from the interior solution. We present the anisotropic behavior and
stability of these charged stars under consideration of three different
$f(\mathcal{G)}$ viable models. We discuss these aspects one by one in
following

###  4.1 Variation of Energy Density and anisotropic stresses

We study the influence of quark matter EoS with the anisotropic stresses at
the center with modified $f(\mathcal{G})$ gravity models. The corresponding
variations in the vicinity of energy density along with anisotropic stresses
are shown in Figs. 1, 2 and 3, respectively.
The evolution of the density for the strange star candidate $VelaX-1$,
$SAXJ1808.43658$, and $4U1820-30$ are shown in Fig. 1. Here, for $r\rightarrow
0$, the density goes to its maximum value. In fact, this indicates the high
compactness of the core of these stars and validating our models in
$f(\mathcal{G})$ gravity under investigation for the outer region of the core.
In case of model 1: The density at core of Vela X - 1 is $1.33881\times
10^{15}gcm^{-3}$, the density at core of SAX J 1808.43658 is $2.72436\times
10^{15}gcm^{-3}$, while 4U 1820-30 having density at core is $1.223106\times
10^{15}gcm^{-3}$.
Furthermore, the surface density of Vela X - 1 under model 1 is
$7.215861\times 10^{14}gcm^{-3}$, SAX J 1808.43658 having $1.362573\times
10^{15}gcm^{-3}$, while 4U 1820-30 having $6.593945\times 10^{14}gcm^{-3}$.
The Central and surface density of these stars for different models are shown
in Table 3.
Similarly, the variation of the radial and traverser pressure,are shown in
Fig. 2-3.
The behavior of radial pressure is, for $r\rightarrow 0$, the radial pressure
$1.849116\times 10^{35}gcm^{-1}sec^{-2}$ for Vela X - 1 under model 1. SAX J
1808.43658 having $4.0796995\times 10^{35}gcm^{-1}sec^{-2}$ at core and in
case of 4U 1820-30, the radial pressure at core is $1.688794\times
10^{35}gcm^{-1}sec^{-2}$.
The transverse pressure in consideration of model 1, For Vela X - 1, the
transverse pressure is $1.84911386\times 10^{35}gcm^{-1}sec^{-2}$, for SAX J
1808.43658 is $4.0795867\times 10^{35}gcm^{-1}sec^{-2}$ while for 4U 1820-30
is $1.6887926204\times 10^{35}gcm^{-1}sec^{-2}$.

Figure 1: Variation of density profile for charged stars, Vela X - 1, SAX J
1808.4-3658, and 4U 1820-30, under different viable $f(\mathcal{G})$ models.
Figure 2: Evolution of radial pressure for charged stars, Vela X - 1, SAX J
1808.4-3658, and 4U 1820-30, under different viable $f(\mathcal{G})$ models.
Figure 3: Evolution of transverse pressure for charged stars, Vela X - 1, SAX
J 1808.4-3658, and 4U 1820-30, under different viable $f(\mathcal{G})$ models.

The variation of radial derivative of density, $\frac{d\rho}{{dr}}$, radial
derivative of radial pressure, $\frac{dp_{r}}{{dr}}$, and radial derivative of
transverse pressure, $\frac{dp_{t}}{{dr}}$ are shown in Fig. 4, 5 and 6
respectively. We see that these variations are negative and for $r=0$, we get

${\left.{\frac{{d\rho}}{{dr}}}\right|_{r=0}}=0$
${\left.{\frac{{d{p_{r}}}}{{dr}}}\right|_{r=0}}=0$

which is expected. e.g. central density of stars $\rho(r=0)=\rho_{c}$.

Figure 4: Evolution of $d\rho/dr$ for charged stars, Vela X - 1, SAX J 1808.4-3658, and 4U 1820-30, under different viable $f(\mathcal{G})$ models. Figure 5: Evolution of $dp_{r}/dr$ for charged stars, Vela X - 1, SAX J 1808.4-3658, and 4U 1820-30, under different viable $f(\mathcal{G})$ models. Figure 6: Evolution of $dp_{t}/dr$ for charged stars, Vela X - 1, SAX J 1808.4-3658, and 4U 1820-30, under different viable $f(\mathcal{G})$ models. Density of Stars | Model 1 | Model 2 | Model 3
---|---|---|---
Vela X - 1 $\rho_{c}(g/cm^{3})$ | $1.33881\times 10^{15}$ | $1.33881\times 10^{15}$ | $1.33904\times 10^{15}$
Vela X - 1 $\rho_{R}(g/cm^{3})$ | $7.21586\times 10^{14}$ | $7.21586\times 10^{14}$ | $7.21581\times 10^{14}$
SAX J 1808.43658 $\rho_{c}(g/cm^{3})$ | $2.72436\times 10^{15}$ | $2.72449\times 10^{15}$ | $2.72857\times 10^{15}$
SAX J 1808.43658 $\rho_{R}(g/cm^{3})$ | $1.36257\times 10^{15}$ | $1.36257\times 10^{15}$ | $1.36247\times 10^{15}$
4U 1820-30 $\rho_{c}(g/cm^{3})$ | $1.22311\times 10^{15}$ | $1.22311\times 10^{15}$ | $1.22327\times 10^{15}$
4U 1820-30 $\rho_{R}(g/cm^{3})$ | $6.59394\times 10^{14}$ | $6.59394\times 10^{14}$ | $6.59391\times 10^{14}$
Table 3: The approximate values of Central density $\rho_{c}$ and Surface
density $\rho_{R}$ for three different models.

###  4.2 Energy conditions

To deal with a physically viable and acceptable matter field, there are some
mathematical constraints which should be obeyed by stress-energy tensor, these
constraints are known as energy conditions. These energy conditions are
coordinate invariant and can be written as following.

  * •

NEC: $\rho+{p_{i}}\geq 0$ .

  * •

WEC: $\rho\geq 0$, $\rho+{p_{i}}\geq 0$ .

  * •

SEC: $\rho+{p_{i}}\geq 0$, $\rho+{p_{i}}+{p_{t}}\geq 0$ .

  * •

DEC: $\rho\geq|{p_{i}}|$ .

Here $i=r,t$ and $\rho$, $p_{r}$ and $p_{t}$ include electric charge
contributions as well.
All these above energy conditions for three different charged compact
relativistic structures are well satisfied under consideration of different
viable $f(\mathcal{G})$ gravity models. The evolution of these energy
conditions are shown graphically in Fig. 7, 8 and 9.

Figure 7: Different Energy conditions for Model 1

Figure 8: Different Energy conditions for Model 2

Figure 9: Different Energy conditions for Model 3

###  4.3 Equilibrium condition

To investigate the equilibrium of inner structure of these charged compact
stars, we use the generalized Tolman-Oppenheimer-Volko (TOV) equation. For
charged spherical anisotropic stellar interior geometry, this equation is
written

$\frac{{d{p_{r}}}}{{dr}}+\frac{{\nu^{\prime}(\rho+{p_{r}})}}{2}+\frac{{2({p_{r}}-{p_{t}})}}{r}+\frac{{\sigma
Q}}{{{r^{2}}}}{e^{\lambda/2}}=0$ (23)

Where $\sigma$ is charge density. Furthermore, the above Eq. (23) may be
written as a sum of different forces e.g. gravitational, hydrostatic,
anisotropic and electric forces

$F_{g}+F_{h}+F_{a}+F_{e}=0,$ (24)

which yields

$F_{g}=-rB(\rho+p_{r}),F_{h}=\frac{{-d{p_{r}}}}{{dr}},F_{a}=2\frac{{({p_{r}}-{p_{t}})}}{r},F_{e}=\frac{{\sigma
Q}}{{{r^{2}}}}{e^{\lambda/2}}$

Figure 10: The Variation of hydrostatic force ($F_{h}$), gravitational force
($F_{g}$), anisotropic force $(F_{a})$ and electric force $F_{e}$ under
consideration of different viable $f(\mathcal{G})$ models.

By using these definitions with the values of different parameters from Table
1, we check the variations of these forces and their hydrostatic equilibrium,
as shown in Fig. 10.
The left plot shows the evolution of these forces in background of first
model, the middle one is for second model and the right plot describe the
variation of these forces because of third model. It is clear from Fig. 10,
that the electric force has a very negligible effect in this balancing
mechanism.

###  4.4 Stability Analysis

In this section, we investigate the stability of the interior of stars under
modified $f(\mathcal{G})$ theory. For the mathematical modeling of compact
stellar structures, it is to be noted that only those stellar models are
significant which are stable against the variations. Hence, the role of
stability is very crucial and burning issue in the modeling of compact
objects. The stability of stellar structure has been studied by many
researches. Here we adopt the techniques which is based on the concept of
overturning (or cracking) [48]. According to this, the radial speed of sound
$v_{sr}^{2}$ as well as transverse speed of sound $v_{st}^{2}$ must be in the
range of a closed interval [0, 1] to preserve the causality condition and for
stability the necessary condition $0\leq v_{sr}^{2}-v_{sr}^{2}\leq 1$ should
be obeyed.
The the radial and transverse speeds is defined as

$\frac{{d{p_{r}}}}{{d\rho}}=v_{sr}^{2}$

and

$\frac{{d{p_{t}}}}{{d\rho}}=v_{st}^{2}$

In our case, $v_{sr}^{2}\sim 1/3$ and $v_{sr}^{2}$ is plotted in fig. 11,
which obey the condition $0\leq v_{sr}^{2}\leq 1$ and $0\leq v_{st}^{2}\leq 1$
which is the indication of causality preservation within these charged compact
stars.

Figure 11: Variations of $v_{st}^{2}$ for different viable $f(\mathcal{G})$
gravity models. Figure 12: Variations of $v_{st}^{2}-v_{sr}^{2}$ for different
viable $f(\mathcal{G})$ gravity models.

Similarly, for stability, we plot $v_{st}^{2}-v_{sr}^{2}$ as shown in Fig. 12.
It is to be noted that all of our charged stellar structures under
consideration of different viable $f(\mathcal{G})$ models obey the constraint:

$0<|v_{st}^{2}-v_{sr}^{2}|<1$

We concluded that the stability is attained in $f(\mathcal{G})$ gravity models
for three considered strange candidate stars, $VelaX-1$, $SAXJ1808.43658$, and
$4U1820-30$.

###  4.5 EoS Parameter

Now for anisotropic stresses, there are two equation of state parameters,
written as

$w_{r}=\frac{p_{r}}{\rho}$

and

$w_{t}=\frac{p_{t}}{\rho}$

For a radiation dominant era, equation of state parameters must lie between 0
and 1. More precisely, $0<w_{r}<1$ and $0<w_{t}<1$. Here, we check the
evolution of EoS parameters for three different charged stars and their
behavior are shown graphically in Fig. 13 and 14.
We can see that both $w_{r}$ and $w_{t}$ lies in given range.

Figure 13: Variations of radial EoS parameter for different viable
$f(\mathcal{G})$ gravity models. Figure 14: Variations of the transverse EoS
parameter for different viable $f(\mathcal{G})$ gravity models.

###  4.6  Mass Radius Relationship, Compactness and Redshift Analysis

The mass of charged compact stars can be written as

$m(r)=\int\limits_{0}^{r}{4\pi{{r^{\prime}}^{2}}\rho dr^{\prime}}$ (25)

Here we know that mass $m$ is function of $r$ and $m(r=0)=0$ while $m(r=R)=M$.
The variation in masses of charged compact stars are shown in Fig. 15.

Figure 15: Variations of the mass function for different charged compact
stars.

We see that the mass is regular at core because it is directly proportional to
radial distance e.g. $m(r)\rightarrow 0$ for $r\rightarrow 0$. The maximum
mass is attained at $r=R$, as shown in fig. 15. The mass radius relation is
also compatible with the study of neutron stars under $f(G)$ gravity [49].
Furthermore, the compactness, $\mu$ can be define as

$\mu(r)=\frac{1}{r}\int\limits_{0}^{r}{4\pi{{r^{\prime}}^{2}}\rho
dr^{\prime}}$ (26)

The compactness for three different strange stars are shown in Fig. 16.
similarly, the Redshift, $Z_{s}$ can be define for compact object

$Z_{s}={\left({1-2\mu}\right)^{-\frac{1}{2}}}-1$

The bound over $Z_{s}\leq 2$.

Figure 16: Variations of the compactness for different charged compact stars.
Figure 17: Variations of the Redshift for different charged compact stars.

In our case, wee check the variation in redshift from the core to surface of
stars. These evolution are shown with the help of plots, as given in Fig. 17.

###  4.7 The Measurement of Anisotropy

In modeling of relativistic stellar interior structures, it is important to
discuss the anisotropicity or anisotropy which is defined as

$\Delta=\frac{2}{r}({p_{t}}-{p_{r}})$ (27)

We check the anisotropy for three different charged strange stars under
consideration of three viable models in $f(\mathcal{G})$ gravity. After
plugging the constant values with these models, we plot the anisotropy and get
that $\Delta>0$ e.g. $p_{t}>p_{r}$. This implies that the anisotropy is
directed outward for all three stars. These plots are shown in Fig. 18. It is
important to note that $\Delta\rightarrow 0$ at $r\rightarrow 0$ and becomes
monotonically increasing outwards with the increase of $r$ near the surface of
the star.

Figure 18: Variations of anisotropic measure $\Delta$ with respect to the
radial.

###  4.8 Electric field and Charge

We observed that the electric charge on the boundary for star 1 is
$6.459234\times 10^{20}C$, for star 2 $6.7510838\times 10^{20}C$ and for star
3 $6.7460087\times 10^{20}C$ and zero at the core of these stars under
consideration of model 1. The charge profile is monotonically increasing away
from the center, as shown in Fig. 19.

Figure 19: Variations of electric charge $Q$ with respect to the radial.

Furthermore, the electric charge density is monotonically decreasing outward
and is maximum at the center of these stars as shown in Fig. 20.

Figure 20: Variations of surface charge density $\sigma$ with respect to the
radial.

Similarly, the behavior of electric field intensity $E^{2}$ is also discuss
and their variation for different stars are shown in Fig. 21.

Figure 21: Variations of electric field square $E^{2}$ with respect to the
radial.

From these plots, we conclude that the core of these stars contain $Q(r=0)=0$,
$\sigma(r=0)=\sigma_{0}$ and $E^{2}(r=0)=0$ while at surface of these stars
$Q(r=R)=Q_{m}$ $\sigma(r=R)=0$ and $E^{2}(r=R)=E^{2}_{m}$ where $\sigma_{0}$,
$Q_{m}$ and $E^{2}_{m}$is the maximum charge density, charge and electric
field intensity.
The stellar structure formation in the background of modified gravity are
comparatively higher contraction in the collapsing rate of spherical systems
at its initial stages unlike GR. The extra curvature (non-gravitational fluid)
on the existence of compact structure could lead arena of having relatively
more compact stars than in GR.
Similarly, the influences of these additional dark source terms on mass radius
relationships for compact stars predict more massive relativistic systems with
comparatively smaller radii than in GR. Perhaps, the calculated apparent
masses of neutron star models in modified gravity are more massive star with
smaller radii than in GR. Such type of investigations could provide
theoretical well-consistent way to handle and study classes of massive and
super massive structures at large scales.

##  5 Summary

It has been attracting challenge to find the correct model for charged
realistic geometry of interior compact objects not only in general relativity
but also in extended theories of gravity like $f(\mathcal{G})$ gravity. For
this purpose, we have considered the three-different observed compact stars,
labeled as Vela X - 1, SAX J 1808.4-3658, and 4U 1820-30. Our desire is to
study the real composition of these compact objects in their central regions
under consideration of three different viable models.
We have investigated several aspects of compact stars in the regime of
$f(\mathcal{G})$ gravity with the anisotropic matter content under Einstein
Maxwell spacetime. We have utilized the solutions for the metric function
suggested by Krori-Barua for a spherical compact object whose arbitrary
constants are calculated across the boundary of interior and exterior
geometry. The values of these arbitrary constants are determined with the help
of charge, mass, and radius of any compact object. We have used three
different strange candidate stars with their experimental observational data
to study the effects of additional degree of freedom coming from modified
gravity theories. For this purpose, we used three different viable models in
$f(\mathcal{G})$ gravity. By using these models along with calculated values
for three different stars, we have plotted the relevant quantities like
variation of anisotropic stress and energy density against radial distance. It
is found that the energy density is very high at core of these stars and
gradually decreases with the increasing radius, thereby indicating the high
compactness structures of these stellar interiors.
We concluded our discussion as:

  * •

The variation in energy density and both radial as well as transverse stress
are positive throughout these charged stars configurations.

  * •

The radial derivative of density and anisotropic pressures ($d\rho/dr$,
$dP_{r}/dr$, $dP_{t}/dr$) remains negative and for $r=0$, these values vanish
which confirm the density and anisotropic stress maximum value at core.

  * •

All the energy conditions are well satisfied which show the realistic matter
content.

  * •

Both the radial and transverse sound speed remains within the bounds, which
mean the causality condition is obeyed.

  * •

All these stars are stable.

  * •

Both radial and transverse EoS parameters lie in the range of 0 and 1.

  * •

The isotropy remains positive throughout these charged stars.

  * •

The distribution of charges increase from central to surface of stars.

  * •

Electric field intensity is maximum at the surface of these stars.

We see that $f(\mathcal{G})$ gravity is much attractive in the study of
compact stars. In this sense, to our knowledge, the existence and study of
different charged stars and particle physics inside their highly dense cores
compelled the researchers for more genuine solutions of field equations.
Similarly, the study of compact charged stellar configuration and better
observational data on the mass radius relation have the potential to exclude a
larger region of the parameter space of alternative theories. There is of
course the possibility that theoretical and observational work may give us a
direction on how to modify general relativity to make it compatible with the
standard model of particle physics, which would be even more exciting.

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