#  The Tolman VII Space-time in the Presence of a Charge and a Cosmological
Constant

James Ripple<EMAIL_ADDRESS>Anish Agashe Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

The Tolman VII space-time is one of the few physically acceptable exact
solutions in general relativity. In this paper, we derive a generalised Tolman
VII solution which includes a charge and a cosmological constant. We analyse
the spatial geometry of the solution and present conditions for zero and non-
zero spatial curvature. We show that for a particular value of the boundary,
the Tolman VII space-time can be matched to the charged Nariai space-time.
This represents a new class of solutions interior to Nariai. Matching with the
Reissner-Nordström-de Sitter space-time, we derive analytic expressions for
the metric functions and the pressure. Using this, we show that the solution
allows for trapped null geodesics for a broad range of values for the charge
and the cosmological constant. We investigate a few physical properties and
derive an equation of state for the fluid. Performing numerical fitting, we
show that the fluid can be considered a polytrope with, $\Gamma\sim 2.4$.
Finally, we analyse the energy conditions and conclude that the solution
follows the basic physical acceptability criteria.

Keywords: Tolman VII, charge, cosmological constant, Nariai solution, trapped
geodesics

## 1 Introduction

Static spherically symmetric fluid solutions of the Einstein field equations
have been of great interest ever since the advent of general relativity (GR).
These solutions commonly denote the interior space-time of a finite sized
fluid sphere and hence can be used to model compact objects such as stars [1].
The earliest such solution can be traced back to Schwarzschild who assumed a
fluid with a uniform density [2]. Many other solutions including those with a
varying density have been found since. The Tolman VII solution is one such
example [3]. It assumes a density that is quadratic in the radial coordinate.
It is one of the very few physically acceptable spherically symmetric fluid
solutions [4] and has been shown to be a viable candidate to model neutron
stars [5, 6, 7, 8]. Therefore, it is of interest to analyse the properties of
this solution and its generalisations.

In this paper, we derive a generalised form of the Tolman VII solution that
includes a charge and a cosmological constant. Charged fluid spheres have been
amply discussed in the literature [9, 10, 11] and so have been solutions with
a cosmological constant [12, 13, 14]. However, solutions with both charge and
a cosmological constant have been given relatively less attention. In [15],
uniform density charged fluids in the presence of a cosmological constant were
considered. The aim of this paper is to present a similar but much more
extended analysis for the Tolman VII space-time111Tolman, in his paper [3],
did initially include a cosmological constant but ignored it mentioning that
it is “too small to produce appreciable effects”. Also, see [16] for Tolman
VII solution with a charge..

We start our analysis with describing the Einstein-Maxwell field equations
with a cosmological constant. Then, we choose an ‘appropriate’ form for the
charge distribution and take the Tolman VII density profile. Using this, we
solve the field equation and derive the generalised form of the Tolman VII
space-time. Both the geometric and physical properties of the space-time now
depend on a parameter space that contains five parameters: central density,
boundary radius, total charge, cosmological constant, and the self-boundedness
parameter. We analyse the spatial curvature and obtain conditions for which
the spatial hypersurfaces are curved (positive or negative) and flat. For a
particular choice of the central density and the total charge, the spatial
curvature becomes identical to the case of an uncharged uniform density
sphere. For physically reasonable values of the five parameters mentioned
above, the fluid sphere exhibits a positively curved spatial geometry.

Then, we consider matching the interior and the exterior space-times. The
presence of a charge and a cosmological constant means that the exterior
space-time can be described by either the Reissner-Nordström-de Sitter
(charged Kottler) metric or the charged Nariai metric. We show that the
generalised Tolman VII space-time can be matched to the charged Nariai space-
time if the matching is done at a specific value of the boundary radius. This
has been previously shown for uniform density spheres [15]. Here we extend
this to the Tolman VII solution, thus deriving a new class of solutions
interior to Nariai.

By matching with the Reissner-Nordström-de Sitter metric, we solve for all the
unknown constants that arise from integrating the field equations. This gives
us complete expressions for the metric functions and the pressure. Then, we
solve the geodesic equations and derive a generalised condition for the
trapping of null geodesics. We use the Tolman-Oppenheimer-Volkoff equation to
convert this into a condition on the pressure and density of the fluid. We
show that our solution allows for trapped null orbits for specific values of
the central density that correspond to a tenuity between 2.5 and 3.2. This is
consistent with previous studies such as [17, 18]. However, we expect it to
change with larger values for the charge and the cosmological constant.

Further, we solve for the radial profile of the pressure obtain an analytic
expression for the equation of state. Assuming the fluid to be a polytrope, we
perform numerical fitting to find the values for the polytropic index that
correspond the best to the analytic expression. We do this for different
values of the self-boundedness parameter Finally, we investigate the energy
conditions and the sound speed. We show that the weak energy condition ensures
a positive spatial curvature. From this analysis, we conclude that the
generalised Tolman VII solution is physically acceptable.

The paper is arranged in the following manner: in section 2, we present the
mathematical setup necessary to solve for the metric functions and the
pressure. We solve these equations for the Tolman VII density ansatz and
present geometric properties of this solution in section 3. In section 4, we
show that the solution allows of trapping of null geodesics. In section 5, we
present the equation of state for the fluid and investigate the energy
conditions. Finally, in section 6, we summarise the results in this paper and
provide some concluding remarks.

We work with geometric units, i.e., $G=1=c$ and follow the Landau-Lifshitz
spacelike convention (LLSC) [1].

## 2 Charged Perfect Fluid Spheres with a Cosmological Constant

The line element for a static spherically symmetric space-time in spherical
coordinates $(t,r,\theta,\phi)$ is given by [19, 20],

$\textrm{d}s^{2}=-\textrm{e}^{2\Phi(r)}\textrm{d}t^{2}+\textrm{e}^{2\Psi(r)}\textrm{d}r^{2}+r^{2}\left(\textrm{d}\theta^{2}+\sin^{2}\theta\textrm{d}\phi^{2}\right)$
(2.1)

The time-like unit 4-vector field admitted by such space-time is given by,
$u^{\alpha}=\left[\textrm{e}^{-\Phi},0,0,0\right]$. The energy-momentum tensor
for a perfect fluid is given by,

$^{(\rm
matter)}{T^{\alpha}}_{\beta}=(\rho+p)u^{\alpha}u_{\beta}+p\delta^{\alpha}_{\beta}$
(2.2)

where, $\rho\equiv\rho(r)$ and $p\equiv p(r)$ are, respectively, the energy
density and pressure of the fluid. The fluid is taken to be charged such that
it generates a static radial electric field, $E(r)$. The energy-momentum
tensor for the electromagnetic field is given by,

$^{(\rm
emfield)}{T^{\alpha}}_{\beta}=\frac{1}{4\pi}\left(F^{\alpha\sigma}F_{\beta\sigma}-\frac{1}{4}\delta^{\alpha}_{\beta}F^{\epsilon\sigma}F_{\epsilon\sigma}\right)$
(2.3)

where, due to spherical symmetry and staticity, we have,

$F_{\epsilon\sigma}=2\delta^{1}_{[\epsilon}\delta^{0}_{\sigma]}E(r)$ (2.4)

The charge density can be found using,

$J^{0}=\frac{1}{4\pi}\nabla_{\sigma}F^{0\sigma}$ (2.5)

Using this, the charge distribution is given by,

$q(r)=\int\sqrt{-g}\ J^{0}\textrm{d}^{3}x=r^{2}\textrm{e}^{-(\Phi+\Psi)}E(r)$
(2.6)

where, $g=\det(g_{\alpha\beta})$.

The Einstein-Maxwell field equations with a cosmological constant are given
by,

${G^{\alpha}}_{\beta}:={R^{\alpha}}_{\beta}-{\frac{1}{2}}R\delta^{\alpha}_{\beta}=8\pi\left({}^{(\rm
matter)}{T^{\alpha}}_{\beta}+^{(\rm
emfield)}{T^{\alpha}}_{\beta}\right)+\Lambda\delta^{\alpha}_{\beta}$ (2.7)

Using equations (2.2) - (2.6), the field equations for the metric in (2.1)
take the form,

${G^{0}}_{0}:=\frac{\textrm{e}^{-2\Psi}}{r^{2}}\left(1-2r\Psi^{\prime}\right)-\frac{1}{r^{2}}=-8\pi\rho-\frac{q^{2}}{r^{4}}+\Lambda$
(2.8a)

${G^{1}}_{1}:=\frac{\textrm{e}^{-2\Psi}}{r^{2}}\left(1+2r\Phi^{\prime}\right)-\frac{1}{r^{2}}=8\pi
p-\frac{q^{2}}{r^{4}}+\Lambda$ (2.8b)
${G^{2}}_{2}={G^{3}}_{3}:=\frac{\textrm{e}^{-2\Psi}}{r^{2}}\left[r\left(\Phi^{\prime}-\Psi^{\prime}\right)+r^{2}\left(\Phi^{\prime\prime}+{\Phi^{\prime}}^{2}-\Phi^{\prime}\Psi^{\prime}\right)\right]=8\pi
p+\frac{q^{2}}{r^{4}}+\Lambda$ (2.8c)

where, ‘dot’ and ‘prime’ represent differentiation with respect to the
coordinates $t$ and $r$, respectively. The variables, $\Phi,\Psi,\rho,p,q$ are
functions of $r$, while $\Lambda$ is a constant.

The conservation of energy-momentum now becomes,

$\nabla_{\alpha}\left({}^{(\rm matter)}{T^{\alpha}}_{\beta}+^{(\rm
emfield)}{T^{\alpha}}_{\beta}\right)=0$ (2.9)

This leads to only one equation (for $\beta~{}=~{}1$), while other components
are identically zero,

$p^{\prime}+\Phi^{\prime}\left(\rho+p\right)=\frac{qq^{\prime}}{4\pi r^{4}}$
(2.10)

### 2.1 Solving for the Metric Functions

To specify the space-time completely, one needs to find the explicit
functional form the metric coefficients. This is done by solving Einstein
field equations. The first field equation (2.8a) can be simplified as,

$\left[r\left(\textrm{e}^{-2\Psi}-1\right)\right]^{\prime}=-8\pi\rho
r^{2}-\frac{q^{2}}{r^{2}}+\Lambda r^{2}$ (2.11)

This can be integrated to give,

$\textrm{e}^{2\Psi(r)}=\left[1-\frac{2m(r)}{r}-\frac{\epsilon(r)}{r}+\frac{\Lambda
r^{2}}{3}\right]^{-1}$ (2.12)

where, we have defined,

$m(r):=4\pi\int\rho(r)r^{2}\ \textrm{d}r$ (2.13a)
$\epsilon(r):=\int\frac{q^{2}(r)}{r^{2}}\ \textrm{d}r$ (2.13b)

This remains true for any given density profile or charge distribution.
Therefore, a general spherically symmetric charged fluid space-time has the
following line element,

$\textrm{d}s^{2}=-\textrm{e}^{2\Phi(r)}\textrm{d}t^{2}+\frac{\textrm{d}r^{2}}{1-\frac{2m(r)}{r}-\frac{\epsilon(r)}{r}+\frac{\Lambda
r^{2}}{3}}+r^{2}\left(\textrm{d}\theta^{2}+\sin^{2}\theta\textrm{d}\phi^{2}\right)$
(2.14)

Solving for the other metric coefficient is much more complicated than this.
To do this, we first use the isotropy of the pressure to get,
${G^{1}}_{1}-{G^{2}}_{2}+\frac{2q^{2}}{r^{4}}=0$. This gives the following
equation,

$\textrm{e}^{-2\Psi}\left[\left(1+r\Phi^{\prime}\right)\left(1+r\Psi^{\prime}\right)-r^{2}\left(\Phi^{\prime\prime}+{\Phi^{\prime}}^{2}\right)\right]-1+\frac{2q^{2}}{r^{2}}=0$
(2.15)

Using this, we get,

$\textrm{e}^{-2\Psi}r^{2}\left(\Phi^{\prime\prime}+{\Phi^{\prime}}^{2}\right)-\textrm{e}^{-2\Psi}\left(1+r\Psi^{\prime}\right)r\Phi^{\prime}-\left\\{\textrm{e}^{-2\Psi}\left(1+r\Psi^{\prime}\right)-1+\frac{2q^{2}}{r^{2}}\right\\}=0$
(2.16)

To simplify this further, we use the substitution, $U=\textrm{e}^{\Phi}$, such
that, $\Phi^{\prime}=\frac{U^{\prime}}{U}$ and
$\Phi^{\prime\prime}+{\Phi^{\prime}}^{2}=\frac{U^{\prime\prime}}{U}$. This
leads to,

$\textrm{e}^{-2\Psi}r^{2}U^{\prime\prime}-\textrm{e}^{-2\Psi}\left(1+r\Psi^{\prime}\right)rU^{\prime}-\left\\{\textrm{e}^{-2\Psi}\left(1+r\Psi^{\prime}\right)-1+\frac{2q^{2}}{r^{2}}\right\\}U=0$
(2.17)

From equation (2.12), one can get,

$\textrm{e}^{-2\Psi}\left(1+r\Psi^{\prime}\right)=1+m^{\prime}-\frac{3m}{r}+\frac{\epsilon^{\prime}}{2}-\frac{3\epsilon}{2r}$
(2.18)

Using equations (2.18) and (2.12) in equation (2.17), one gets a second order
differential equation for $\Phi$. To solve this equation, one needs a specific
form of the mass function, $m(r)$, and the charge function, $\epsilon(r)$. We
will do this using the Tolman VII density profile and assuming a constant
charge density. However, before that, let us derive the equations used to
solve for pressure.

### 2.2 Solving for the Pressure

Besides the metric coefficients, one also needs the radial dependency of the
pressure to analyse the properties of the fluid distribution. A differential
equation for pressure can be derived by combining the field equations and the
conservation equation. This equation is known as the Tolman-Oppenheimer-
Volkoff (TOV) equation [3, 21]. Here, we derive a generalised form of this
equation with a charge and a cosmological constant. Using equation (2.12) in
equation (2.8b), and separating out $\Phi^{\prime}$, we get,

$\Phi^{\prime}=\frac{24\pi pr^{4}+6mr+3\epsilon
r-3r^{2}\epsilon^{\prime}+2\Lambda r^{4}}{2r^{2}\left(3r-6m-3\epsilon+\Lambda
r^{3}\right)}$ (2.19)

Using this in equation (2.10), we get the generalised TOV equation,

$p^{\prime}=-(\rho+p)\frac{24\pi pr^{4}+6mr+3\epsilon
r-3r^{2}\epsilon^{\prime}+2\Lambda r^{4}}{2r^{2}\left(3r-6m-3\epsilon+\Lambda
r^{3}\right)}+\frac{\epsilon^{\prime\prime}}{8\pi
r^{2}}+\frac{\epsilon^{\prime}}{4\pi r^{3}}$ (2.20)

There is another equation for pressure which may prove more useful in some
cases than the TOV equation. This is derived by using,
${G^{1}}_{1}-{G^{0}}_{0}=8\pi(\rho+p)$, which leads to,

$p=\frac{\textrm{e}^{-2\Psi}}{4\pi
r}\left(\Phi^{\prime}+\Psi^{\prime}\right)-\rho$ (2.21)

The above equations reinforces the well-known fact that for fluid
distributions in GR, given an ansatz for the density profile, the pressure
(and hence the equation of state) cannot be arbitrarily chosen but rather must
be fixed through the field equations.

### 2.3 The Exterior Space-times

We are analysing fluid distributions that have a finite radial extent, $r\leq
r_{b}$. Beyond this ($r\geq r_{b}$), one can take the density and pressure to
vanish $(\rho=0=p)$. In the presence of a charge and a cosmological constant,
this exterior region can be described by either of the two well-known
solutions – the Reissner-Nordström de Sitter space-time [19] and the charged
Nariai space-time [22, 23, 15]. The line elements for these two space-times
are, respectively, given by,

$\textrm{d}s^{2}=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+\frac{\Lambda
r^{2}}{3}\right)\textrm{d}t^{2}+\frac{\textrm{d}r^{2}}{\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+\frac{\Lambda
r^{2}}{3}\right)}+r^{2}\left(\textrm{d}\theta^{2}+\sin^{2}\theta\textrm{d}\phi^{2}\right)$
(2.22a)
$\textrm{d}s^{2}=\frac{1}{\Lambda-\frac{Q^{2}}{r_{b}^{4}}}\left[-\left\\{X\cos(u)+Y\sin(u)\right\\}^{2}\textrm{d}t^{2}+\textrm{d}u^{2}\right]+\frac{1}{\Lambda+\frac{Q^{2}}{r_{b}^{4}}}\left(\textrm{d}\theta^{2}+\sin^{2}\theta\textrm{d}\phi^{2}\right)$
(2.22b) where, $M=m(r_{b})$ and $Q=q(r_{b})$ are, respectively, the total mass
and the total charge contained within the region, $r\leq r_{b}$, for some
boundary radius, $r_{b}$; $X$ and $Y$ are arbitrary constants.

## 3 The Generalised Tolman VII Solution

The Tolman VII solution is characterised by the following ansatz on the energy
density [3, 5],

$\rho=\rho_{0}\left[1-\beta\left(\frac{r}{r_{b}}\right)^{2}\right]$ (3.1)

where, $\rho_{0}$ is the central density ($\rho_{0}\equiv\rho(r=0)$), $r_{b}$
represents a boundary radius beyond which the space-time can be considered to
be electro-vacuum, and $\beta$ is the so-called self-boundedness parameter
that takes values between 0 and 1 [5]. Using this ansatz, one can solve
equation (2.17) in an analogous manner as done for the usual Tolman VII
solution [24, 25, 5]. In figure 1, we plot the Tolman VII density ansatz. The
density is monotonically decreasing ($\rho^{\prime}<0$) for $0<r<r_{b}$. The
central density and the boundary radius are taken to be, $\rho_{0}=10^{-9}$
m-2 and $r_{b}=10^{4}$ m, which correspond to their typical values for a
neutron star [5].

Figure 1: The Tolman VII ansatz for the density profile, $\rho(r)$, as a
function of the radius, $r$, for different values of the parameter, $\beta$.
The values of various constants are: $\rho_{0}=10^{-9}$ m-2, $r_{b}=10^{4}$ m.
The density is well behaved and is monotonically decreasing throughout the
extent of the fluid distribution.

Using equation (3.1), the mass function (equation (2.13a)) inside the boundary
becomes,

$m(r)=4\pi\rho_{0}\left(\frac{r^{3}}{3}-\frac{\beta r^{5}}{5r_{b}^{2}}\right)$
(3.2)

Further, we assume a uniform charge density, such that,

$q(r)=\frac{Qr^{3}}{r_{b}^{3}}\quad\Rightarrow\
\epsilon(r)=\frac{Q^{2}r^{5}}{5r_{b}^{6}}$ (3.3)

Using equations (3.2) and (3.3) in equation (2.12), we get222Originally, in
[3], an ansatz of the form (3.4) was taken on the metric coefficient and the
density profile in equation (3.1) was obtained by solving the field
equations.,

$\textrm{e}^{2\Psi(r)}=\left(1-\frac{Ar^{2}}{3}+\frac{Br^{4}}{5}\right)^{-1}$
(3.4)

where, $A:=8\pi\rho_{0}-\Lambda$ and
$B:=\frac{8\pi\rho_{0}\beta}{r_{b}^{2}}-\frac{Q^{2}}{r_{b}^{6}}$. In figure 2,
we plot the metric function above as a function of the radial coordinate.
Again, we take, $\rho_{0}=10^{-9}$ m-2 and $r_{b}=10^{4}$ m. Further, the
cosmological constant is taken to be, $\Lambda=10^{-52}$ m-2 [26] and the
total charge, $Q=0.02$ m [27]. For these values, the metric function remains
regular inside the fluid sphere.

Figure 2: The metric function, $\textrm{e}^{2\Psi(r)}$, as a function of the
radius, $r$, for different values of the parameter, $\beta$. The values of
various constants are: $\rho_{0}=10^{-9}$ m-2, $Q=0.02$ m, $\Lambda=10^{-52}$
m-2, $r_{b}=10^{4}$ m. The metric function remains regular throughout the
extent of the fluid distribution.

Using the above expression for the metric function, equation (2.17) becomes,

$\left(1-\frac{Ar^{2}}{3}+\frac{Br^{4}}{5}\right)r^{2}U^{\prime\prime}+\left(\frac{Br^{4}}{5}-1\right)rU^{\prime}+\left(\frac{B}{5}-\frac{2Q^{2}}{r_{b}^{6}}\right)r^{4}U=0$
(3.5)

Multiplying the above equation by $\frac{5}{B}$; defining a new variable,
$x:=r^{2}$, and writing the derivatives with respect to $x$, the above
equation becomes,

$\left(\frac{5}{B}-\frac{5Ax}{3B}+x^{2}\right)\widetilde{\widetilde{U}}+\left(x-\frac{5A}{6B}\right)\widetilde{U}+\left(\frac{1}{4}-\frac{5Q^{2}}{2Br_{b}^{6}}\right)U=0$
(3.6)

where, ‘tilde’ represents differentiation with respect to $x$. To simplify
this further, we define,
$M:=\frac{5}{B}-\frac{5Ax}{3B}+x^{2}=\frac{5}{B}\textrm{e}^{-2\Psi}$;
$N:=x-\frac{5A}{6B}=\frac{\widetilde{M}}{2}$; and
$D^{2}:=\frac{1}{4}-\frac{5Q^{2}}{2Br_{b}^{6}}$; such that the above equation
takes a simpler form,

$M\widetilde{\widetilde{U}}+N\widetilde{U}+D^{2}U=0$ (3.7)

To solve this equation, we yet again change all the derivatives to be with
respect to a new variable, $w:=\ln\left(\sqrt{M}+N\right)$. This gives,
$\widetilde{w}=\frac{1}{\sqrt{M}}$, and,
$\widetilde{\widetilde{w}}=-\frac{N}{M\sqrt{M}}$. Using this, we get,
$\widetilde{U}=\frac{1}{\sqrt{M}}\frac{\textrm{d}U}{\textrm{d}w}$, and,
$\widetilde{\widetilde{U}}=\frac{1}{M}\frac{\textrm{d}^{2}U}{\textrm{d}w^{2}}-\frac{N}{M\sqrt{M}}\frac{\textrm{d}U}{\textrm{d}w}$.
Substituting these in equation (3.7), we get,

$\frac{\textrm{d}^{2}U}{\textrm{d}w^{2}}+D^{2}U=0$ (3.8)

This can be easily solved to get333One can, in fact, work with a less general
solution comprising of either a cosine term or a sine term as done in [18].
However, here, we choose to work with the more general form of the solution.,

$U=C_{1}\cos(Dw)+C_{2}\sin(Dw)=\textrm{e}^{\Phi}$ (3.9)

where, $C_{1}$ and $C_{2}$ are arbitrary constants. Using this, we also get,

$\Phi^{\prime}=\frac{U^{\prime}}{U}=2rD\sqrt{\frac{B}{5}}\textrm{e}^{\Psi-\Phi}\
\left[C_{2}\cos(Dw)-C_{1}\sin(Dw)\right]$ (3.10)

Using the above equation in (2.21), we get,

$p=\frac{D}{2\pi}\sqrt{\frac{B}{5}}\left[\textrm{e}^{-(\Phi+\Psi)}\left(C_{2}\cos(Dw)-C_{1}\sin(Dw)\right)\right]-\frac{1}{2\pi}\left(\frac{B}{5}r^{2}-\frac{A}{6}\right)-\rho$
(3.11)

To solve for both the metric function and the pressure, one needs to fix the
constants, $C_{1}$ and $C_{2}$. Before doing that, we present some properties
of the solution that do not involve the integration constants.

### 3.1 Spatial Geometry

The line element in equation (2.14) can be written as,

$\textrm{d}s^{2}=-\textrm{e}^{2\Phi}\textrm{d}t^{2}+\frac{\textrm{d}r^{2}}{1-k(r)r^{2}}+r^{2}\textrm{d}\theta^{2}+r^{2}\sin^{2}\theta\textrm{d}\phi^{2}$
(3.12)

where,

$k(r):=\frac{2m(r)}{r^{3}}+\frac{\epsilon(r)}{r^{3}}-\frac{\Lambda}{3}=\frac{{}^{(3)}R}{6}$
(3.13)

where, ${}^{(3)}R$ is the Ricci scalar for the $t={\rm const.}$ spatial
hypersurfaces. The function, $k(r)$, determines the spatial curvature.
Depending on whether this function is positive, zero, or negative, we will
have a fluid sphere with spherical, flat, and hyperbolic spatial geometry,
respectively. Using equations (3.2) and (3.3), for the generalised Tolman VII
space-time, we get,

$k(r)=\frac{A}{3}-\frac{B}{5}r^{2}=\frac{1}{3}\left(8\pi\rho_{0}-\Lambda\right)-\left(\frac{8\pi\rho_{0}\beta}{r_{b}^{2}}-\frac{Q^{2}}{r_{b}^{6}}\right)\frac{r^{2}}{5}$
(3.14)

As expected, the spatial curvature depends on the values of the central
density, the total charge, the cosmological constant, and the boundary radius.
In Figure 3, we plot the spatial curvature as a function of radius for
realistic values of these variables. It turns out that the generalised Tolman
VII solution has a positively curved spatial geometry between $0\leq r\leq
r_{b}$.

Figure 3: The spatial curvature, $k(r)$, as a function of the radius, $r$, for
different values of the parameter, $\beta$. The values of constants are:
$\rho_{0}=10^{-9}$ m-2, $Q=0.02$ m, $\Lambda=10^{-52}$ m-2, $r_{b}=10^{4}$ m.
We see that for these values, the solution exhibits a spherical spatial
geometry throughout the extent of the fluid distribution.

For $Q^{2}=8\pi\rho_{0}\beta r_{b}^{4}$, we have a constant spatial curvature
that depends only on the variable, $A$. Then, positive, negative, or zero
curvature corresponds to, $\Lambda<$, $>$, or $=8\pi\rho_{0}$, respectively.
Curiously, these conditions are the same as the ones for a uniform density
fluid sphere [28]. Below, we derive general conditions on these variables for
the three types of the spatial curvature.

#### Spherical Spatial Geometry

For the spatial geometry to be spherical, we need the spatial curvature,
$k(r)$, to be positive. This gives,

$\displaystyle\frac{1}{3}\left(8\pi\rho_{0}-\Lambda\right)$
$\displaystyle>\left(\frac{8\pi\rho_{0}\beta}{r_{b}^{2}}-\frac{Q^{2}}{r_{b}^{6}}\right)\frac{r^{2}}{5}$
(3.15a) $\displaystyle\Rightarrow r^{2}$
$\displaystyle<\frac{5r_{b}^{2}}{3}\left(\frac{8\pi\rho_{0}-\Lambda}{8\pi\rho_{0}\beta-\frac{Q^{2}}{r_{b}^{4}}}\right)$
(3.15b)

Therefore, depending on the relationship between the constants, $\rho_{0}$,
$\Lambda$, and $Q^{2}$, we will get the range for the radial coordinate up to
which, the fluid sphere will exhibit a spherical spatial geometry. If the
right hand side of equation (3.15b) is greater than unity, then we get a
space-time that has positively curved spatial hypersurfaces for the whole
extent of the fluid distribution. This condition takes the form,

$\Lambda>\frac{1}{5}\left[8\pi\rho_{0}\left(5-3\beta\right)+\frac{Q^{2}}{r_{b}^{4}}\right]$
(3.16)

#### Hyperbolic Spatial Geometry

For the spatial curvature to be negative, the condition in equation (3.15b)
will be reversed, and we get,

$r^{2}>\frac{5r_{b}^{2}}{3}\left(\frac{8\pi\rho_{0}-\Lambda}{8\pi\rho_{0}\beta-\frac{Q^{2}}{r_{b}^{4}}}\right)$
(3.17)

If one has, $\Lambda=8\pi\rho_{0}$ and
$8\pi\rho_{0}>\frac{1}{\beta}\frac{Q^{2}}{r_{b}^{4}}$, then the above
condition becomes, $r>0$. That is, for all values of the radial coordinate, we
will have negatively curved spatial hypersurfaces.

#### Flat Spatial Geometry

We get a condition for flat spatial geometry by replacing the inequality sign
in equation (3.15a) by an equality. Then, for the condition,
$\Lambda<\frac{1}{5}\left[8\pi\rho_{0}\left(5-3\beta\right)+\frac{Q^{2}}{r_{b}^{4}}\right]$,
we will have a value of $r$, say $r_{\rm flat}<r_{b}$ at which the spatial
geometry would be flat. That is, there will exist a point within the fluid
distribution, at which the spatial curvature is zero while at other points, it
can be either positive or negative depending on the relationship between the
constants defining the space-time.

### 3.2 Matching with the Charged Nariai Space-time

Using equation (3.9), the line element for the generalised Tolman VII solution
becomes,

$\textrm{d}s^{2}=-\left[C_{1}\cos(Dw)+C_{2}\sin(Dw)\right]^{2}\textrm{d}t^{2}+\textrm{e}^{2\Psi}\textrm{d}r^{2}+r^{2}\textrm{d}\theta^{2}+r^{2}\sin^{2}\theta\textrm{d}\phi^{2}$
(3.18)

Now, using the fact that, $\frac{\textrm{d}w}{\textrm{d}x}=\frac{1}{\sqrt{M}}$
and $\frac{1}{M}=\frac{B\textrm{e}^{2\Psi}}{5}$, we get,

$\textrm{e}^{2\Psi}\textrm{d}r^{2}=\frac{5}{4B}\frac{\textrm{d}w^{2}}{r^{2}}$
(3.19)

Using this, the line element becomes,

$\textrm{d}s^{2}=-\left[C_{1}\cos(Dw)+C_{2}\sin(Dw)\right]^{2}\textrm{d}t^{2}+\frac{5}{4B}\frac{\textrm{d}w^{2}}{r^{2}}+r^{2}\textrm{d}\theta^{2}+r^{2}\sin^{2}\theta\textrm{d}\phi^{2}$
(3.20)

Introducing a new coordinate, $u=Dw$, we get,

$\textrm{d}s^{2}=-\left[C_{1}\cos(u)+C_{2}\sin(u)\right]^{2}\textrm{d}t^{2}+\frac{5}{4BD^{2}}\frac{\textrm{d}u^{2}}{r^{2}}+r^{2}\textrm{d}\theta^{2}+r^{2}\sin^{2}\theta\textrm{d}\phi^{2}$
(3.21)

Comparing the above line element to that in equation (2.22b), we see that the
metric functions corresponding to the $\textrm{d}t^{2}$ term are identical for
both the line elements as long as the following conditions are satisfied,

$C_{1}=\frac{X}{\sqrt{\Lambda-\frac{Q^{2}}{r_{b}^{4}}}}\quad{\rm and}\quad
C_{2}=\frac{Y}{\sqrt{\Lambda-\frac{Q^{2}}{r_{b}^{4}}}}$ (3.22)

Since $C_{1},C_{2},X,Y$ are arbitrary constants, the above conditions can
always be taken to be satisfied without making any assumptions on the values
of $\Lambda$, $Q$, and $r_{b}$.

From this, it is clear that Tolman VII space-time is a natural candidate to be
the interior of the charged Nariai space-time. To match the two space-times,
in addition to the above two conditions, we require that at the boundary
radius, $r_{b}$, we have,

$\frac{5}{4BD^{2}r_{b}^{2}}=\frac{1}{\Lambda-\frac{Q^{2}}{r_{b}^{4}}}\quad{\rm
and}\quad r_{b}^{2}=\frac{1}{\Lambda+\frac{Q^{2}}{r_{b}^{4}}}$ (3.23)

Using, $4BD^{2}r_{b}^{2}=8\pi\rho_{0}\beta-\frac{11Q^{2}}{r_{b}^{4}}$, the
conditions above translate to the following,

$\frac{Q^{2}}{r_{b}^{4}}=\frac{8\pi\rho_{0}\beta-5\Lambda}{6}\quad{\rm
and}\quad\frac{Q^{2}}{r_{b}^{4}}=\frac{1}{r_{b}^{2}}-\Lambda$ (3.24)

Combining these two together, we get,

$r_{b}=\pm\sqrt{\frac{6}{8\pi\rho_{0}\beta+\Lambda}}$ (3.25)

The expression above remains valid for arbitrary values of the constants,
$X,Y,C_{1},C_{2}$, as well as the physical quantities, $\rho_{0}$ and
$\Lambda$. Therefore, this represents a class of solutions interior to Nariai.

Requiring the boundary radius to be a positive quantity, we can ignore the
negative solution. In the case of a negative cosmological constant, we also
require, $8\pi\rho_{0}\beta>|\Lambda|$, so that $r_{b}$ remains real. For
realistic values, $\Lambda=10^{-52}$ m-2 and $\rho_{0}=10^{-9}$ m-2, the
boundary radius turns out to be, $r_{b}\sim 15000$ m, which is a reasonable
size for a neutron star. In the case of vanishing charge, we get the
condition, $8\pi\rho_{0}\beta=5\Lambda$, and the boundary radius becomes,
$r_{b}=\frac{1}{\sqrt{\Lambda}}$ which is the cosmological horizon. Obviously,
this does not correspond to typical size of neutron stars.

### 3.3 Matching with the Reissner-Nordström-de Sitter Space-time

In the previous section, we found a new class of solutions interior to the
Nariai space-time. However, any further analysis with these solutions may
prove difficult since the constants, $C_{1},C_{2}$ still remain to be
determined. To get the full expression for the metric coefficients and the
pressure, we need to find the values of these constants. This can be done by
matching with the Reissner-Nordström-de Sitter space-time. Additionally, we
will also demand that at the boundary radius, $r_{b}$, the pressure vanishes
($p(r_{b})=0$). This, along with,
$\textrm{e}^{-\Psi(r_{b})}=\textrm{e}^{\Phi(r_{b})}$, used in equations (3.9)
and (3.11), leads to,

$\displaystyle C_{1}\cos(Dw_{b})+C_{2}\sin(Dw_{b})$
$\displaystyle=\left(1-\frac{Ar_{b}^{2}}{3}+\frac{Br_{b}^{4}}{5}\right)^{{\frac{1}{2}}}$
(3.26a) $\displaystyle C_{2}\cos(Dw_{b})-C_{1}\sin(Dw_{b})$
$\displaystyle=\frac{1}{D}\sqrt{\frac{5}{B}}\left\\{\frac{B}{5}r_{b}^{2}-\frac{A}{6}+2\pi\rho_{0}(1-\beta)\right\\}$
(3.26b)

where, $w_{b}=w(r_{b})$. The above two equations can be solved easily to get,

$\displaystyle C_{1}$
$\displaystyle=\left(1-\frac{Ar_{b}^{2}}{3}+\frac{Br_{b}^{4}}{5}\right)^{{\frac{1}{2}}}\cos(Dw_{b})-\frac{1}{D}\sqrt{\frac{5}{B}}\left\\{\frac{B}{5}r_{b}^{2}-\frac{A}{6}+2\pi\rho_{0}(1-\beta)\right\\}\sin(Dw_{b})$
(3.27a) $\displaystyle C_{2}$
$\displaystyle=\left(1-\frac{Ar_{b}^{2}}{3}+\frac{Br_{b}^{4}}{5}\right)^{{\frac{1}{2}}}\sin(Dw_{b})+\frac{1}{D}\sqrt{\frac{5}{B}}\left\\{\frac{B}{5}r_{b}^{2}-\frac{A}{6}+2\pi\rho_{0}(1-\beta)\right\\}\cos(Dw_{b})$
(3.27b)

Using these in equation (3.9), we get the following expression for the metric
coefficient,

$\textrm{e}^{2\Phi}=\scalebox{0.9}{\mbox{$\displaystyle\left[\left(1-\frac{Ar_{b}^{2}}{3}+\frac{Br_{b}^{4}}{5}\right)^{{\frac{1}{2}}}\cos[D(w-w_{b})]+\frac{1}{D}\sqrt{\frac{5}{B}}\left\\{\frac{B}{5}r_{b}^{2}-\frac{A}{6}+2\pi\rho_{0}(1-\beta)\right\\}\sin[D(w-w_{b})]\right]^{2}$}}$
(3.28)

In figure 4, we plot the metric function above as a function of the radial
coordinate.

Figure 4: The metric function, $\textrm{e}^{2\Phi(r)}$, as a function of the
radius, $r$, for different values of the parameter, $\beta$. The values of
various constants are: $\rho_{0}=10^{-9}$ m-2, $Q=0.02$ m, $\Lambda=10^{-52}$
m-2, $r_{b}=10^{4}$ m. The metric function remains regular throughout the
extent of the fluid distribution.

Using equations (3.28) and (3.4), we finally get the line element for the
generalised Tolman VII space-time,

$\textrm{d}s^{2}=-\scalebox{0.84}{\mbox{$\displaystyle\left[\left(1-\frac{Ar_{b}^{2}}{3}+\frac{Br_{b}^{4}}{5}\right)^{{\frac{1}{2}}}\cos[D(w-w_{b})]+\frac{1}{D}\sqrt{\frac{5}{B}}\left\\{\frac{B}{5}r_{b}^{2}-\frac{A}{6}+2\pi\rho_{0}(1-\beta)\right\\}\sin[D(w-w_{b})]\right]^{2}$}}\textrm{d}t^{2}\\\
+\frac{\textrm{d}r^{2}}{1-\frac{Ar^{2}}{3}+\frac{Br^{4}}{5}}+r^{2}\textrm{d}\theta^{2}+r^{2}\sin^{2}\theta\textrm{d}\phi^{2}$
(3.29)

The analysis that follows will be done for the space-time described the above
line element.

## 4 Trapping of Null Geodesics

The Tolman VII solution has been shown to exhibit internal trapping of null
geodesics [17, 29, 18]. Study of geodesic trapping provides an insight into
the geodesic structure of the space-time and gives a criterion to classify
exact solutions (Tolman VII solution is the only physically acceptable
solution to exhibit internally trapped circular geodesics [29, 17]). This is
also relevant in astrophysics due to the trapping of gravitational waves and
neutrinos [18]. Here, we investigate the effects of the cosmological constant
and charge on trapping of null geodesics.

### 4.1 Effective Potential

Due to spherical symmetry, we can fix, $\theta=\frac{\pi}{2}$ (equatorial
plane). Then, the geodesic equations for the line element (2.1) take the form,

$\displaystyle\ddot{t}+2\dot{r}\dot{t}\Phi^{\prime}$ $\displaystyle=0$ (4.1)
$\displaystyle\ddot{r}+\dot{r}^{2}{\Psi^{\prime}}+e^{2\Phi-2\Psi}\dot{t}^{2}{\Phi^{\prime}}-\textrm{e}^{-2\Psi}r\dot{\phi}^{2}$
$\displaystyle=0$ (4.2)
$\displaystyle\ddot{\phi}+\frac{2\dot{r}\dot{\phi}}{r}$ $\displaystyle=0$
(4.3)

Additionally, for null geodesics, we also have,

$-\textrm{e}^{2\Phi}\dot{t}^{2}+\textrm{e}^{2\Psi}\dot{r}^{2}+r^{2}\dot{\phi}^{2}=0$
(4.4)

Here, ‘dot’ represents a derivative with respect of some affine parameter. The
first integrals of equations (4.1) and (4.3) lead to the familiar conserved
quantities,

$\textrm{e}^{2\Phi}\dot{t}=E\quad{\rm and}\quad r^{2}\dot{\phi}=\ell$ (4.5)

Using this in equation (4.4) leads to,

$r^{2}\dot{r}^{2}=\textrm{e}^{-2\Psi}E^{2}\left(r^{2}\textrm{e}^{-2\Phi}-\frac{\ell^{2}}{E^{2}}\right)$
(4.6)

We define the impact parameter as, $b=\frac{\ell}{E}$ [1, 17]. Then, one can
get the turning points of the geodesic motion by introducing an effective
potential, $V_{\rm eff}(r)=r^{2}\textrm{e}^{-2\Phi}$, such that, the above
equation leads to the constraint, $b^{2}\leq V_{\rm eff}$.

For the generalised Tolman VII space-time, the effective potential is given
by,

$V_{\rm
eff}(r)=\left[\frac{r\sec[D(w-w_{b})]}{\left(1-\frac{Ar_{b}^{2}}{3}+\frac{Br_{b}^{4}}{5}\right)^{{\frac{1}{2}}}+\frac{1}{D}\sqrt{\frac{5}{B}}\left\\{\frac{B}{5}r_{b}^{2}-\frac{A}{6}+2\pi\rho_{0}(1-\beta)\right\\}\tan[D(w-w_{b})]}\right]^{2}$
(4.7)

We will have a circular null orbits of radius, $r_{c}$, if,
$\left.\frac{\textrm{d}V_{\rm eff}}{\textrm{d}r}\right|_{r_{c}}=0$. If the
radius is such that we have, $r_{c}<r_{b}$, then the circular orbits will be
trapped inside the fluid sphere. In figure 5, we plot the effective potential
for different values of the central density and $\beta=1$. We observe
essentially the same features as in [18]. Therefore, realistic values of $Q$
and $\Lambda$ (which are quite small) do not have any substantial effect on
the trapping of null geodesics in the Tolman VII space-time.

Figure 5: The effective potential, $V_{\rm eff}(r)$, plotted for different
values of the central density, $\rho_{0}$. The values of other constants are:
$\beta=1$, $Q=0.02$ m, $\Lambda=10^{-52}$ m-2, $r_{b}=10^{4}$ m. The green
(maxima) and red (minima) dots mark where one will get circular orbits. For
$1.94\times 10^{-9}$ m${}^{-2}\geq\rho_{0}\geq 1.87\times 10^{-9}$ m-2
($2.98\leq\frac{r_{b}}{M}\leq 3.19$), both stable and unstable orbits lie
inside the boundary. For $\rho_{0}>1.94\times 10^{-9}$ m-2
($\frac{r_{b}}{M}<2.98$) we only have the stable orbits trapped inside the
boundary. The effective potential diverges as $\rho_{0}\to 2.36\times
10^{-19}$ m-2 ($\frac{r_{b}}{M}\to 2.53$).

### 4.2 Conditions for Trapped Null Orbits

For having trapped null orbits, we require that for some $r_{c}<r_{b}$, we
have,

$\left.\frac{\textrm{d}V_{\rm eff}}{\textrm{d}r}\right|_{r_{c}}=0$ (4.8)

Using, $V_{\rm eff}(r)=r^{2}\textrm{e}^{-2\Phi}$ (which is an invariant
definition for arbitrary static spherically symmetric space-times [17]), this
reduces to,

$r_{c}\Phi^{\prime}(r_{c})=1$ (4.9)

This is a necessary and sufficient condition for the existence of trapped null
orbits444An alternate way to derive this condition (without using $V_{\rm
eff}$) is through equation (4.2). Using equation (4.4) in (4.2), we get,
$\ddot{r}+\dot{r}^{2}\left(\Psi^{\prime}+\Phi^{\prime}\right)+\textrm{e}^{-2\Psi}\dot{\phi}^{2}r\left(r\Phi^{\prime}-1\right)=0$
(4.10) The space-time will have a trapped circular orbit if at some,
$r_{c}<r_{b}$, we have, $\dot{r}=0=\ddot{r}$. This reduces the above equation
to, $r\Phi^{\prime}=1$, which is the same condition as equation (4.9). .

This is essentially a geometric condition. However, it can be converted into a
condition on the physical properties of the fluid. To do this, we use equation
(2.19), which gives us,

$4\pi
pr^{2}+\frac{3m}{r}+\frac{3}{2}\frac{\epsilon}{r}-{\frac{1}{2}}\epsilon^{\prime}=1$
(4.11)

Further, using equation (2.10) or the TOV equation (2.20), we can also
write555Similarly, conditions involving both geometric and physical quantities
can also be derived. For example, combining equation (4.11) with (2.18), we
get,  $\textrm{e}^{-2\Psi}(1+r\Psi^{\prime})=m^{\prime}+4\pi pr^{2}=4\pi
r^{2}(\rho+p)$ (4.12) When combined with the null energy condition, this takes
the form, $r\Psi^{\prime}\geq-1$.,

$rp^{\prime}+p+\rho=\frac{qq^{\prime}}{4\pi r^{3}}\quad{\rm or}\quad
rp^{\prime}+p+\rho=\frac{\epsilon^{\prime\prime}}{8\pi
r}+\frac{\epsilon^{\prime}}{4\pi r^{2}}$ (4.13)

Using equation (4.7), the condition for existence of trapped null orbits in
the generalised Tolman VII space-time becomes,

$\frac{\left(1-\frac{Ar_{b}^{2}}{3}+\frac{Br_{b}^{4}}{5}\right)^{{\frac{1}{2}}}}{\frac{1}{D}\sqrt{\frac{5}{B}}\left\\{\frac{B}{5}r_{b}^{2}-\frac{A}{6}+2\pi\rho_{0}(1-\beta)\right\\}}\left[\frac{1+rDw^{\prime}\tan\left[D(w-w_{b})\right]}{rDw^{\prime}-\tan\left[D(w-w_{b})\right]}\right]=1$
(4.14)

In figure 6, we plot $r\Phi^{\prime}$ for the generalised Tolman VII solution
as a function of the radial coordinate.

Figure 6: The quantity, $r\Phi^{\prime}(r)$, plotted for different values of
the central density, $\rho_{0}$. The values of other constants are the same as
figure 5. The points at which $r\Phi^{\prime}=1$ are marked with green and red
dots. These points correspond exactly to the extremum points of the effective
potential, thus showing that the condition in equation (4.9) is indeed
correct.

## 5 Physical Properties of the Fluid

### 5.1 Radial Profile of the Pressure

Using equation (3.27) in (3.11), we get the full expression for the radial
profile of the pressure,

$p(r)=\frac{D}{2\pi}\sqrt{\frac{B}{5}}\left(1-\frac{Ar^{2}}{3}+\frac{Br^{4}}{5}\right)^{\frac{1}{2}}\scalebox{0.76}{\mbox{$\displaystyle\left[\frac{\frac{1}{D}\sqrt{\frac{5}{B}}\left\\{\frac{B}{5}r_{b}^{2}-\frac{A}{6}+2\pi\rho_{0}(1-\beta)\right\\}-\left(1-\frac{Ar_{b}^{2}}{3}+\frac{Br_{b}^{4}}{5}\right)^{{\frac{1}{2}}}\tan[D(w-w_{b})]}{\left(1-\frac{Ar_{b}^{2}}{3}+\frac{Br_{b}^{4}}{5}\right)^{{\frac{1}{2}}}+\frac{1}{D}\sqrt{\frac{5}{B}}\left\\{\frac{B}{5}r_{b}^{2}-\frac{A}{6}+2\pi\rho_{0}(1-\beta)\right\\}\tan[D(w-w_{b})]}\right]$}}\\\
-\frac{1}{2\pi}\left(\frac{B}{5}r^{2}-\frac{A}{6}\right)-\rho_{0}\left[1-\beta\left(\frac{r}{r_{b}}\right)^{2}\right]$
(5.1)

This is quite a complicated expression, even when we have used the shorthands,
$A$ and $B$, for various constants defining the system. To see the radial
dependence of the pressure more explicitly, we present the complete expression
for the pressure in equation (A.9). In figure 7, we plot the pressure as a
function of the radius. The pressure profile is well behaved within the bounds
of the fluid sphere and is monotonically decreasing.

Figure 7: The radial profile of the pressure, $p(r)$, plotted for different
values of the parameter, $\beta$. The values of various constants are:
$\rho_{0}=10^{-9}$ m-2, $Q=0.02$ m, $\Lambda=10^{-52}$ m-2, $r_{b}=10^{4}$ m.
The pressure is well behaved and monotonically decreasing for $0<r<r_{b}$.

### 5.2 Equation of State

To find the equation of state, one can invert equation (3.1) and replace,
$r^{2}=\frac{r_{b}^{2}}{\rho_{0}\beta}\left(\rho_{0}-\rho\right)$ in the above
equation. We do this, and after performing some rather cumbersome algebra, we
find the equation of state to be,

$p(\rho)=\frac{D}{2\pi}\sqrt{\frac{B}{5}}\left\\{1-\frac{Ar_{b}^{2}}{3\rho_{0}\beta}\left(\rho_{0}-\rho\right)+\frac{Br_{b}^{4}}{5\beta^{2}\rho_{0}^{2}}(\rho_{0}-\rho)^{2}\right\\}^{\frac{1}{2}}\\\
\times\scalebox{1.0}{\mbox{$\displaystyle\left[\frac{\frac{1}{D}\sqrt{\frac{5}{B}}\left\\{\frac{B}{5}r_{b}^{2}-\frac{A}{6}+2\pi\rho_{0}(1-\beta)\right\\}-\left(1-\frac{Ar_{b}^{2}}{3}+\frac{Br_{b}^{4}}{5}\right)^{{\frac{1}{2}}}\tan[D(w-w_{b})]}{\left(1-\frac{Ar_{b}^{2}}{3}+\frac{Br_{b}^{4}}{5}\right)^{{\frac{1}{2}}}+\frac{1}{D}\sqrt{\frac{5}{B}}\left\\{\frac{B}{5}r_{b}^{2}-\frac{A}{6}+2\pi\rho_{0}(1-\beta)\right\\}\tan[D(w-w_{b})]}\right]$}}\\\
-\frac{1}{2\pi}\left(\frac{Br_{b}^{2}}{5\rho_{0}\beta}(\rho_{0}-\rho)-\frac{A}{6}\right)-\rho$
(5.2)

where, we have,

$w(\rho)=\ln\left[\left\\{\frac{15\rho_{0}\beta-5Ar_{b}^{2}(\rho_{0}-\rho)}{3B\rho_{0}\beta}+\frac{r_{b}^{4}}{\rho_{0}^{2}\beta^{2}}(\rho_{0}-\rho)^{2}\right\\}^{\frac{1}{2}}+\frac{r_{b}^{2}}{\rho_{0}\beta}(\rho_{0}-\rho)-\frac{5A}{6B}\right]$
(5.3)

We present the complete analytic expression for the equation of state in the
appendix (equation (A.10)). It is interesting to note that the equation of
state changes with the value of the central density and the boundary radius.
In figure 8, we plot the equation of state.

Figure 8: The equation of state, $p(\rho)$, plotted for different values of
the parameter, $\beta$. The values of various constants are:
$\rho_{0}=10^{-9}$ m-2, $Q=0.02$ m, $\Lambda=10^{-52}$ m-2, $r_{b}=10^{4}$ m.
The equation of state is a well behaved function for $0<r<r_{b}$.

Even though we are able to obtain an analytic expression for the equation of
state, it is quite abstruse. Therefore, it is useful to fit the equation of
state to a simpler form, e.g., that of a polytrope,
$p(\rho)=\kappa\rho^{\Gamma(r)}$. In figure 9, we plot the polytropic index,
$\Gamma(r)=\frac{\partial\log p}{\partial\log\rho}$, for the Tolman VII fluid
as a function of the radius.

Figure 9: The polytropic index, $\Gamma(r)$, plotted against the radial
coordinate for different values of $\beta$. The other constants are:
$\rho_{0}=10^{-9}$ m-2, $Q=0.02$ m, $\Lambda=10^{-52}$ m-2, $r_{b}=10^{4}$ m.
The polytropic index does not vary for most of the fluid distribution, thus,
the fluid is well described as a polytrope. The steep change near the boundary
for $\beta<1$ is due to the discontinuity in the density [5].

In the case of a constant polytropic index, we list the best fit values for
the constants, $\kappa$ and $\Gamma$, for different values of the parameter,
$\beta$, in table 1.

$\beta$ | $\kappa$ | $\Gamma$
---|---|---
$1.0$ | $1.04\times 10^{12}$ | $2.43$
$0.9$ | $2.76\times 10^{12}$ | $2.47$
$0.8$ | $3.97\times 10^{13}$ | $2.58$
$0.7$ | $4.93\times 10^{15}$ | $2.81$
$0.6$ | $1.09\times 10^{19}$ | $3.17$
Table 1: The best fit values for the constants, $\kappa$ and $\Gamma$, in the
polytropic equation fo state. The parameter, $\beta$, changes the value of the
polytropic index, however, it is still centred around $\Gamma\sim 2.5$, while
$\kappa$ changes by several orders of magnitude.

### 5.3 Sound Speed

Once, we have the equation of state, it is straightforward to calculate the
sound speed, $c_{s}^{2}=\frac{\partial p}{\partial\rho}$. The sound speed is
plotted against the radial coordinate in figure 10. We see that for
$0<\beta<0.55$, the sound speed becomes greater than the speed of light, thus
violating causality. This puts a constraint on the values the self-boundedness
parameter, $\beta$, can take.

Figure 10: The variation in the speed of sound squared, $c_{s}^{2}$, with the
radius, $r$, plotted for different values of the parameter, $\beta$. The
values of various constants are: $\rho_{0}=10^{-9}$ m-2, $Q=0.02$ m,
$\Lambda=10^{-52}$ m-2, $r_{b}=10^{4}$ m. The values of $\beta$ are chosen
such that, $c_{s}^{2}<1$, thus following the causality condition.

### 5.4 Energy Conditions

The energy conditions are constraints on the total energy-momentum tensor
which, in our case, includes matter fields, electromagnetic fields, and the
cosmological constant. Therefore, we have,

$T_{\alpha\beta}=\ ^{(\rm matter)}T_{\alpha\beta}\ +\ ^{(\rm
emfield)}T_{\alpha\beta}+\Lambda g_{\alpha\beta}$ (5.4)

  1. 1.

The null energy condition, $T_{\alpha\beta}k^{\alpha}k^{\beta}\geq 0$, reduces
to,

$\rho+p\geq 0$ (5.5)

where, $k^{\alpha}=\left[\textrm{e}^{-\Phi},\textrm{e}^{-\Psi},0,0\right]$ is
a null vector. We see that the presence of a charge and a cosmological
constant does not change the form of the null energy condition.

  2. 2.

The weak energy condition, $T_{\alpha\beta}u^{\alpha}u^{\beta}\geq 0$, reduces
to,

$8\pi\rho+\frac{q^{2}}{r^{4}}-\Lambda\geq 0$ (5.6)

where, $u^{\alpha}=\left[\textrm{e}^{-\Phi},0,0,0\right]$ is a timelike
vector. Using the ansatzes (3.1) and (3.3), the weak energy condition reduces
to,

$A-Br^{2}\geq 0$ (5.7)

Comparing this with equation (3.15a), we can conclude that the weak energy
condition ensures positivity of the spatial curvature in the generalised
Tolman VII solution666In general, for any static spherically symmetric perfect
fluid solution, the weak energy condition only ensures the non-negativity of
the spatial curvature..

  3. 3.

The dominant energy condition,
$T_{\alpha\beta}v^{\beta}{T^{\alpha}}_{\sigma}v^{\sigma}\leq 0$, reduces to,

$\displaystyle 8\pi\rho+\frac{q^{2}}{r^{4}}-\Lambda$ $\displaystyle\geq
0\quad{\rm when}\ v^{\alpha}\ {\rm is\ timelike}$ (5.8) $\displaystyle
4\pi(\rho-p)+\frac{q^{2}}{r^{4}}-\Lambda$ $\displaystyle\geq 0\quad{\rm when}\
v^{\alpha}\ {\rm is\ null}$ (5.9)

The timelike case is the same as the weak energy condition, therefore, the
null case is the only independent condition.

  4. 4.

The strong energy condition,
$\left(T_{\alpha\beta}-{\frac{1}{2}}{T^{\sigma}}_{\sigma}g_{\alpha\beta}\right)u^{\alpha}u^{\beta}\geq
0$, reduces to,

$4\pi(\rho+3p)+\frac{q^{2}}{r^{4}}+\Lambda\geq 0$ (5.10)

In figure 11, we plot the four energy conditions to show that solution follows
all of them.

Figure 11: The four energy conditions plotted for different values of the
parameter, $\beta$. The values of various constants are the same as other
figures.

The analysis in this section along with the regularity and finiteness of the
metric functions (figures 4 and 2) and density (figure 1) tells us that the
solution agrees with all the basic criteria [4] for physical acceptability.
Additionally, existence of an analytic, albeit complicated, expression for the
equation of state makes the solution even more physically relevant.

## 6 Summary and Discussion

In this paper, we presented a comprehensive analysis of the Tolman VII
solution in the presence of a charge and a cosmological constant. We started
with using the Einstein-Maxwell field equations for a perfect fluid with a
cosmological constant to arrive upon a generalised TOV equation. Then, with
the Tolman VII ansatz on the density profile, we solved the field equations to
derive the generalised Tolman VII solution. All the features of the solution
were characterised by a parameter space containing five components: the
central density($\rho_{0}$), the self-boundedness parameter ($\beta$), the
boundary radius ($r_{b}$), the total charge ($Q$), and the cosmological
constant ($\Lambda$).

Using the metric functions, we analysed the spatial geometry of the solution
and showed that for realistic values of the parameters, it remains positively
curved for $0\leq r\leq r_{b}$. Then, we investigated the issue of matching
the interior and exterior space-times. Due to the presence of a cosmological
constant, there were two candidates to describe the vacuum space-time
surrounding the charged fluid sphere: the Reissner-Nordström (anti-)de Sitter
solution and the charged Nariai solution. We showed that the Tolman VII
solution can be matched to the Nariai solution if the matching is done at a
specific boundary radius given in terms of $\beta$, $\rho_{0}$, $\Lambda$, and
$Q$. This constituted a new class of solutions interior to the charged Nariai
space-time. Such solutions have been used to construct exact two-mass
solutions [15, 30]. It will be interesting to see if the results in this paper
can be used to do the same. This will be particularly significant for
astrophysics since the Tolman VII solution is a reasonable model for a neutron
star.

Next, we matched the interior solution with the Reissner-Nordström de Sitter
space-time. This led to the generalised form of the Tolman VII solution in its
usual form. Using the geodesic equations, we derived the expression for the
effective potential for the fluid sphere. We also presented the general
conditions for the existence of trapped null geodesics. Both the $Q$ and
$\Lambda$ enter these conditions. However, for their realistic values (which
are small), these parameters do not have any effect on geodesic trapping. We
found that the generalised solution exhibits trapped null geodesics when the
tenuity ($\frac{r_{b}}{M}$) is between $2.5$ and $3.2$. This is similar to
what has been found for the usual Tolman VII solution [17, 29, 18]. However,
one should expect this to change with larger values of $Q$ and $\Lambda$.
Since the issue of geodesic trapping is of certain astrophysical relevance, it
will be interesting to introduce rotation within our solution similar to [31,
32].

Finally, we derived an analytic expression for the pressure and the equation
of state of the fluid. Both contained $\Lambda$ and $Q$ terms. The pressure
was positive definite, monotonically decreasing, and vanished at the boundary.
We showed that when $\beta=1$, the fluid can be best described as a polytrope
with an index, $\Gamma\sim 2.4$. This was consistent with previous findings
[5]. We also showed that the solution follows the causality criterion
($c_{s}^{2}<1$) and all the energy conditions. Therefore, the cosmological
constant and charge (at least for reasonable values) do not have any effect on
the physical acceptability of the solution. It is worth noting here that the
form of the charge distribution, $q(r)=\frac{Qr^{3}}{r_{b}^{3}}$ was chosen
since it allowed for analytic solutions. Other forms could be analysed using
numerical methods. One could further generalise the solution by including
anisotropic pressure [33, 34, 35] or use it in alternate/modified gravity
theories [36, 37, 35]. We keep these and other issues discussed above for
future work.

## Acknowledgements

We thank Sai Madhav Modumudi and Shalini Ganguly for useful discussions. AA
acknowledges partial financial support from an internal grant at St. Mary’s
College of Maryland.

##  Appendix A Complete Expressions for the Metric Coefficients, Pressure, and
the Equation of State

Let us recall the definitions of the constants, $A$, $B$, and $D$,

$\displaystyle A$ $\displaystyle:=8\pi\rho_{0}-\Lambda$ (A.1) $\displaystyle
B$ $\displaystyle:=\frac{1}{r_{b}^{6}}\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)$ (A.2) $\displaystyle D$
$\displaystyle:=\frac{8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}}{4\left(8\pi\rho_{0}\beta r_{b}^{4}-Q^{2}\right)}$ (A.3)

Using the definitions of these constants, the variable, $w$, becomes,

$w(r)=\ln\left[\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r^{4}\right\\}^{{\frac{1}{2}}}+r^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}\right]$ (A.4)

and, $w_{b}=w(r_{b})$, is given by,

$w(r_{b})=\ln\left[\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r_{b}^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r_{b}^{4}\right\\}^{\frac{1}{2}}+r_{b}^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}\right]$ (A.5)

Using this, we present the metric coefficients solely in terms of the radial
coordinate, $r$, and the constants, $\rho_{0}$, $\beta$, $Q$, $\Lambda$, and
$r_{b}$,

$\textrm{e}^{2\Psi(r)}=\left\\{1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r^{2}}{3}+\left(\frac{8\pi\rho_{0}\beta}{r_{b}^{2}}-\frac{Q^{2}}{r_{b}^{6}}\right)\frac{r^{4}}{5}\right\\}^{-1}$
(A.6)

$\textrm{e}^{2\Phi(r)}=\scriptscriptstyle\left[\left\\{1-8\pi\rho_{0}r_{b}^{2}\left(\frac{1}{3}-\frac{\beta}{5}\right)+\frac{\Lambda
r_{b}^{2}}{3}-\frac{Q^{2}}{5r_{b}^{2}}\right\\}^{{\frac{1}{2}}}\cos\left[\left(\frac{8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}}{32\pi\rho_{0}\beta
r_{b}^{4}-4Q^{2}}\right)^{\frac{1}{2}}\ln\left[{\frac{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r^{4}\right\\}^{{\frac{1}{2}}}+r^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r_{b}^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r_{b}^{4}\right\\}^{\frac{1}{2}}+r_{b}^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}}\right]\right]\right.\\\
\left.\scriptscriptstyle+\left\\{\frac{4\pi\rho_{0}r_{b}^{4}(5-3\beta)+5\Lambda
r_{b}^{4}-6Q^{2}}{3\sqrt{5}r_{b}\left(8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}\right)^{\frac{1}{2}}}\right\\}\sin\left[\left(\frac{8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}}{32\pi\rho_{0}\beta
r_{b}^{4}-4Q^{2}}\right)^{\frac{1}{2}}\ln\left[{\frac{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r^{4}\right\\}^{{\frac{1}{2}}}+r^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r_{b}^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r_{b}^{4}\right\\}^{\frac{1}{2}}+r_{b}^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}}\right]\right]\right]^{2}$ (A.7)

Using these expressions, the line element for the generalised Tolman VII
space-time is given by,

$\textrm{d}s^{2}=\scriptscriptstyle-\left[\left\\{1-8\pi\rho_{0}r_{b}^{2}\left(\frac{1}{3}-\frac{\beta}{5}\right)+\frac{\Lambda
r_{b}^{2}}{3}-\frac{Q^{2}}{5r_{b}^{2}}\right\\}^{{\frac{1}{2}}}\cos\left[\left(\frac{8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}}{32\pi\rho_{0}\beta
r_{b}^{4}-4Q^{2}}\right)^{\frac{1}{2}}\ln\left[{\frac{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r^{4}\right\\}^{{\frac{1}{2}}}+r^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r_{b}^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r_{b}^{4}\right\\}^{\frac{1}{2}}+r_{b}^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}}\right]\right]\right.\\\
\left.\scriptscriptstyle+\left\\{\frac{4\pi\rho_{0}r_{b}^{4}(5-3\beta)+5\Lambda
r_{b}^{4}-6Q^{2}}{3\sqrt{5}r_{b}\left(8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}\right)^{\frac{1}{2}}}\right\\}\sin\left[\left(\frac{8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}}{32\pi\rho_{0}\beta
r_{b}^{4}-4Q^{2}}\right)^{\frac{1}{2}}\ln\left[{\frac{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r^{4}\right\\}^{{\frac{1}{2}}}+r^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r_{b}^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r_{b}^{4}\right\\}^{\frac{1}{2}}+r_{b}^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}}\right]\right]\right]^{2}\textrm{d}t^{2}\\\
+\frac{\textrm{d}r^{2}}{\left\\{1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r^{2}}{3}+\left(\frac{8\pi\rho_{0}\beta}{r_{b}^{2}}-\frac{Q^{2}}{r_{b}^{6}}\right)\frac{r^{4}}{5}\right\\}}+r^{2}\textrm{d}\theta^{2}+r^{2}\sin^{2}\theta\textrm{d}\phi^{2}$
(A.8)

To see the radial dependence of the pressure in its complete form, we replace
the expressions for the constants, $A$, $B$, and $D$ in equation (5.1) to get,

$p(r)=\frac{\left(8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}\right)^{\frac{1}{2}}}{10\sqrt{2}\pi
r_{b}^{3}}\left\\{1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r^{2}}{3}+\left(\frac{8\pi\rho_{0}\beta}{r_{b}^{2}}-\frac{Q^{2}}{r_{b}^{6}}\right)\frac{r^{4}}{5}\right\\}^{\frac{1}{2}}\\\
\times\scriptstyle\left[\frac{1-\frac{\left\\{1-8\pi\rho_{0}r_{b}^{2}\left(\frac{1}{3}-\frac{\beta}{5}\right)+\frac{\Lambda
r_{b}^{2}}{3}-\frac{Q^{2}}{5r_{b}^{2}}\right\\}^{{\frac{1}{2}}}}{\left\\{\frac{4\pi\rho_{0}r_{b}^{4}(5-3\beta)+5\Lambda
r_{b}^{4}-6Q^{2}}{3\sqrt{5}r_{b}\left(8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}\right)^{\frac{1}{2}}}\right\\}}\tan\left[\left(\frac{8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}}{32\pi\rho_{0}\beta
r_{b}^{4}-4Q^{2}}\right)^{\frac{1}{2}}\ln\left[{\frac{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r^{4}\right\\}^{{\frac{1}{2}}}+r^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r_{b}^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r_{b}^{4}\right\\}^{\frac{1}{2}}+r_{b}^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}}\right]\right]}{\frac{\left\\{1-8\pi\rho_{0}r_{b}^{2}\left(\frac{1}{3}-\frac{\beta}{5}\right)+\frac{\Lambda
r_{b}^{2}}{3}-\frac{Q^{2}}{5r_{b}^{2}}\right\\}^{{\frac{1}{2}}}}{\left\\{\frac{4\pi\rho_{0}r_{b}^{4}(5-3\beta)+5\Lambda
r_{b}^{4}-6Q^{2}}{3\sqrt{5}r_{b}\left(8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}\right)^{\frac{1}{2}}}\right\\}}+\tan\left[\left(\frac{8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}}{32\pi\rho_{0}\beta
r_{b}^{4}-4Q^{2}}\right)^{\frac{1}{2}}\ln\left[{\frac{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r^{4}\right\\}^{{\frac{1}{2}}}+r^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r_{b}^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r_{b}^{4}\right\\}^{\frac{1}{2}}+r_{b}^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}}\right]\right]}\right]\\\
-\frac{1}{2\pi}\left\\{\left(\frac{8\pi\rho_{0}\beta}{r_{b}^{2}}-\frac{Q^{2}}{r_{b}^{6}}\right)\frac{r^{2}}{5}-\frac{1}{6}\left(8\pi\rho_{0}-\Lambda\right)\right\\}-\rho_{0}\left[1-\beta\left(\frac{r}{r_{b}}\right)^{2}\right]$
(A.9)

Using this, the equation of state is given by,

$p(\rho)=\scalebox{0.9}{\mbox{$\displaystyle\frac{\left(8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}\right)^{\frac{1}{2}}}{10\sqrt{2}\pi
r_{b}^{3}}\left\\{1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r_{b}^{2}}{3\beta}\left(1-\frac{\rho}{\rho_{0}}\right)+\left(\frac{8\pi\rho_{0}\beta}{r_{b}^{2}}-\frac{Q^{2}}{r_{b}^{6}}\right)\frac{r_{b}^{4}}{5\beta^{2}}\left(1-\frac{\rho}{\rho_{0}}\right)^{2}\right\\}^{\frac{1}{2}}$}}\\\
\times\scalebox{0.7}{\mbox{$\displaystyle\left[\frac{1-\frac{\left\\{1-8\pi\rho_{0}r_{b}^{2}\left(\frac{1}{3}-\frac{\beta}{5}\right)+\frac{\Lambda
r_{b}^{2}}{3}-\frac{Q^{2}}{5r_{b}^{2}}\right\\}^{{\frac{1}{2}}}}{\left\\{\frac{4\pi\rho_{0}r_{b}^{4}(5-3\beta)+5\Lambda
r_{b}^{4}-6Q^{2}}{3\sqrt{5}r_{b}\left(8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}\right)^{\frac{1}{2}}}\right\\}}\tan\left[\left(\frac{8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}}{32\pi\rho_{0}\beta
r_{b}^{4}-4Q^{2}}\right)^{\frac{1}{2}}\ln\left[{\frac{\left\\{\frac{5r_{b}^{6}\left\\{1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r_{b}^{2}}{3\beta}\left(1-\frac{\rho}{\rho_{0}}\right)\right\\}}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+\frac{r_{b}^{4}}{\beta^{2}}\left(1-\frac{\rho}{\rho_{0}}\right)^{2}\right\\}^{{\frac{1}{2}}}+\frac{r_{b}^{2}}{\beta}\left(1-\frac{\rho}{\rho_{0}}\right)-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r_{b}^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r_{b}^{4}\right\\}^{\frac{1}{2}}+r_{b}^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}}\right]\right]}{\frac{\left\\{1-8\pi\rho_{0}r_{b}^{2}\left(\frac{1}{3}-\frac{\beta}{5}\right)+\frac{\Lambda
r_{b}^{2}}{3}-\frac{Q^{2}}{5r_{b}^{2}}\right\\}^{{\frac{1}{2}}}}{\left\\{\frac{4\pi\rho_{0}r_{b}^{4}(5-3\beta)+5\Lambda
r_{b}^{4}-6Q^{2}}{3\sqrt{5}r_{b}\left(8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}\right)^{\frac{1}{2}}}\right\\}}+\tan\left[\left(\frac{8\pi\rho_{0}\beta
r_{b}^{4}-11Q^{2}}{32\pi\rho_{0}\beta
r_{b}^{4}-4Q^{2}}\right)^{\frac{1}{2}}\ln\left[{\frac{\left\\{\frac{5r_{b}^{6}\left\\{1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r_{b}^{2}}{3\beta}\left(1-\frac{\rho}{\rho_{0}}\right)\right\\}}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+\frac{r_{b}^{4}}{\beta^{2}}\left(1-\frac{\rho}{\rho_{0}}\right)^{2}\right\\}^{{\frac{1}{2}}}+\frac{r_{b}^{2}}{\beta}\left(1-\frac{\rho}{\rho_{0}}\right)-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}{\left\\{\frac{5r_{b}^{6}\left(1-\left(8\pi\rho_{0}-\Lambda\right)\frac{r_{b}^{2}}{3}\right)}{8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}}+r_{b}^{4}\right\\}^{\frac{1}{2}}+r_{b}^{2}-\frac{5\left(8\pi\rho_{0}-\Lambda\right)r_{b}^{6}}{6\left(8\pi\rho_{0}\beta
r_{b}^{4}-Q^{2}\right)}}}\right]\right]}\right]$}}\\\
-\frac{1}{2\pi}\left\\{\left(\frac{8\pi\rho_{0}\beta}{r_{b}^{2}}-\frac{Q^{2}}{r_{b}^{6}}\right)\frac{r_{b}^{2}}{5\beta}\left(1-\frac{\rho}{\rho_{0}}\right)-\frac{1}{6}\left(8\pi\rho_{0}-\Lambda\right)\right\\}-\rho$
(A.10)

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