Anisotropic stars via embedding approach in Brans-Dicke gravity
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Eur. Phys. J. C (2021) 81:729
https://doi.org/10.1140/epjc/s10052-021-09519-5
Regular Article - Theoretical Physics
Anisotropic stars via embedding approach in Brans–Dicke gravity
S. K. Maurya1,a , Ksh. Newton Singh2,b , M. Govender3,c , Abdelghani Errehymy4,d ,
Francisco Tello-Ortiz5,e
1 Department of Mathematical and Physical Sciences, College of Arts and Science, University of Nizwa, Nizwa, Sultanate of Oman
2 Department of Physics, National Defence Academy, Khadakwasla, Pune 411023, India
3 Department of Mathematics, Durban University of Technology, Durban 4000, South Africa
4 Laboratory of High Energy Physics and Condensed Matter (LPHEMaC), Department of Physics, Faculty of Sciences Aïn Chock, Hassan II
University of Casablanca, B.P. 5366, Maarif, Casablanca 20100, Morocco
5 Departamento de Física, Facultad de Ciencias Básicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile
Received: 19 May 2021 / Accepted: 30 July 2021
© The Author(s) 2021
Abstract In the present article, the solution of the Einstein– 1 Introduction
Maxwell field equations in the presence of a massive scalar
field under the Brans-Dicke (BD) gravity is obtained via Einstein’s general relativity (GR) has been fruitful in describ-
embedding approach, which describes a charged anisotropic ing gravitational phenomena on both cosmological and astro-
strange star model. The interior spacetime is described by physical scales. The predictions of GR has gone beyond the
a spherically symmetric static metric of embedding class I. realms of theory and has been successfully confirmed through
This reduces the problem to a single-generating function of a plethora of experiments. With the advancement of tech-
the metric potential which is chosen by appealing to physics nology these predictions have been refined. The perihelion
based on regularity at each interior point of the stellar inte- precession of Mercury, one of the first solar system tests
rior. The resulting model is subjected to rigorous physical of GR has been drastically improved by the collection of
checks based on stability, causality and regularity for partic- data from Mercury MESSENGER which orbited Mercury in
ular object PSR J1903+327. We also show that our solutions 2011 [1]. The joint European-Japanese Mercury spacecraft
describe compact objects such as PSR J1903+327; Cen X-3; BepiColombo project which launched in 2018 is expected to
EXO 1785-248 and LMC X-4 to an excellent approximation. reveal more precise measurements of the peculiarities of Mer-
Novel results of our investigation reveal that the scalar field cury’s orbit. The first gravitational wave events were detected
leads to higher surface charge densities which in turn affects in September 2015 by the LIGO and Virgo collaborations
the compactness and upper and lower values imposed by the thus reinforcing the prediction of classical GR. There is no
modified Buchdahl limit for charged stars. Our results also more greater signalling of the Golden Age of astrophysical
show that the electric field and scalar field which originate observations than the 2019 capturing of the image of the
from entirely different sources couple to alter physical char- black hole at the center of galaxy M87 by the Event Horizon
acteristics such as mass-radius relation and surface redshift Telescope [2].
of compact objects. This superposition of the electric and Despite these confirmations of GR there are still many
scalar fields is enhanced by an increase in the BD coupling observations that leave Einstein’s classical gravity theory
constant, ω B D . short. In cosmology researchers are still faced with vari-
ous problems including the late-time acceleration of the Uni-
verse, dark matter and dark energy conundrums, baryon sym-
metry and the horizon problem, just to name a few [3]. On
the other hand there are outstanding problems in astrophysics
some of which include the origin of large surface redshifts
a e-mail: sunil@unizwa.edu.om (corresponding author) in compact objects, the behaviour of matter at extreme den-
b e-mail: ntnphy@gmail.com sities such as in the core of neutron stars and the end-states
c e-mail: megandhreng@dut.ac.za of continued gravitational collapse, amongst others.
d e-mail: abdelghani.errehymy@gmail.com To this end researchers in gravitational physics have
e e-mail: francisco.tello@ua.cl (corresponding author) appealed to modified theories of gravitation in the hope of
0123456789().: V,-vol 123729 Page 2 of 19 Eur. Phys. J. C (2021) 81:729
finding mechanisms that will account for the observations and Starobinsky f (R; T )-function can be seen in the follow-
which cannot be resolved by GR. These modified theories ing Refs. [9–13]. The Brans–Dicke (BD) gravitational theory
must have as their weak field limit Einstein’s general rela- was the first of many scalar-tensor theories of gravitation in
tivity. A simple modification to the standard 4D theory is to which the non-minimally coupled scalar field represents the
accommodate more than just the linear forms of the Rie- spacetime-varying gravitational “constant”. The BD gravity
mann tensor, the Ricci tensor and the Ricci scalar in the theory successfully incorporated Mach’s principle. The BD
action principle. It is well-known that incorporation of just gravity theory of gravitation is also called the Jordan–Brans–
linear tensorial quantities produces second order equations Dicke gravity theory which is a theoretical framework that
of motion which are compatible with the 4D equations [4]. can be represented in Jordan–Brans–Dicke gravity as well
The so-called Einstein–Gauss–Bonnet (EGB) gravity arises as Einstein’s frame. It continues to be one of the more pop-
from the more general class of theories called the Lovelock ular theories of modified classical GR and has been widely
polynomial Lagrangians which incorporates tensorial quan- utilised in cosmological models. BD gravity has elegantly
tities to be of quadratic order. The beauty associated with explained the inflationary epoch of the universe and the cur-
the EGB Lagrangian is that the equations of motion continue rent accelerated phase of the universe without invoking any
to be second order quasi-linear. There has been widespread exotic matter fields or dissipative processes [14]. An emer-
interest in modeling compact objects within the framework gent universe via quantum tunneling within a Jordan–Brans–
of EGB gravity. Several exact solutions of the modified pres- Dicke framework has been recently proposed by Labrana
sure isotropy condition have been derived and these models and Cossi [15]. The initial static universe is supported by a
were shown to obey the stability, regularity and causality scalar field contained within a false vacuum. The staticity
conditions required for stellar configurations. More ever, it of the model is broken via quantum tunneling in which the
was shown that higher order corrections alter physical prop- scalar field decays into a true vacuum and the universe begins
erties such as compactness, stability and surface redshifts of to evolve dynamically. In a recent study within the Brans–
stellar models. Recent work by Chakraborty and Dadhich on Dicke framework motivated by a f (R) = R + α R n − δ R 2−n
charged compact objects showed that for a given spacetime modified Starobinsky model inflation and a nonzero resid-
dimension, 4D stellar models are more compact than their ual value for the Ricci scalar was obtained. More impor-
pure Lovelock counterparts [5]. They further showed that an tantly, it was shown in the high energy limit (BD theory
increase in the intensity of the electromagnetic field results with a Jordan framework) predictions are consistent with data
in a greater compactification of the stellar object. They also obtained by PLANCK or BICEP2 [16]. On the astrophysical
demonstrated that within the context of 4D EGB gravity an front recent work by several authors using the BD formalism
increase in the strength of the Gauss-Bonnet coupling (behav- have successfully generated models of anisotropic compact
ing as an effective electric charge), leads to an increase in the objects [17,18]. Sharif and Majid [19] obtained models of
compactness of the stellar object. anisotropic bounded configurations via gravitational decou-
Besides the Lovelock gravity and the popular EGB gravity pling through MGD approach. They show that the stability
theory there are a wide spectrum of other modified gravity of the model is affected by the anisotropy parameter which
theories which are being frequently utilised within both cos- is inherently linked to the decoupling constant.
mological and astrophysical contexts. Unimodular gravity We have seen a virtual explosion of exact solutions
which is based on the trace-free Einstein equations was con- describing compact objects in classical GR and modified
jured to solve the magnitude of the vacuum energy density gravity theories. The past decade in particular has seen a pro-
conundrum. A hugely popular modified theory of gravitation liferation of realistic stellar models which have catapulted
is the so-called f (R; T ) theory proposed by Harko et al. [6] the search for solutions of the field equations into main-
in which the action is the Ricci scalar R and the the trace of stream astrophysics. Solution-generating methods have been
the energy-momentum tensor T . The Rastall theory is cen- inherently linked to physical viability tests which are backed
tered on the notion that divergence of the energy-momentum by observational data. The Karmarkar condition has been
tensor is proportional to the divergence of the Ricci scalar. extensively utilised to generate compact objects in which the
This has implications for the conservation of energy momen- radial and transverse stresses are unequal at each interior
tum [7]. Starobinsky in his attempt to explain the accelerated point of the stellar fluid [20–22]. In classical GR the Kar-
expansion of the universe put forth a theory of gravitation markar condition is immediately integrated to give a relation
whose action is quadratic in the Ricci scalar. This modified between the two metric potentials. This reduces the problem
theory has come to be known as the f (R) theory of grav- of finding exact solutions of the field equations to a singe-
ity. Stellar modeling in f (R) gravity has received interest in generating function [23–29]. In a recent paper, Hansraj and
the recent past which produced models which are compatible Moodly demonstrated that nonexistence of conformally flat
with observational data, see [8] and references therein. How- charged isotropic fluid sphere of embedding Class one [30].
ever, some recent works on the f (R; T ) theory, Rastall theory The embedding of the generalized Vaidya (GV) solution via
123Eur. Phys. J. C (2021) 81:729 Page 3 of 19 729
the Karmarkar solution shows that embedding does not allow within the core of the compact stellar structure. In this respect,
the interpretation of the generalized Vaidya spacetime as a equilibrium is achieved after a certain amount of electrons
diffusive medium. In other words, the Karmarkar condition escape and the net electric charge approaches to about 100
prohibits the GV solution to be interpreted as an atmosphere Coulombs per solar mass. As shown by the authors [51], this
composed of radiation and diffusive strings of a star undergo- result applies to any bounded system whose size is smaller
ing dissipative collapse in the form of a radial heat flux [31]. than the Debye length of the surrounding media. In his semi-
The Karmarkar condition has been extended to incorporate nal paper addressing hydrostatic equilibrium and the gravita-
time-dependent systems which include modelling shear-free, tional collapse of charged bodies, Bekenstein argues that the
dissipative collapse [32–35]. The Karmarkar condition has densities within neutron star cores may lead to electric fields
been successfully used in modified gravity theories to investi- whose magnitudes may exceed the critical field required for
gate contributions from the inclusion of quadratic terms of the pair creation, 1016 V/cm, and hence annihilate themselves.
tensorial quantities and higher dimensional effects on com- On the other hand, the possibility of a collapsing charged stel-
pactness, stability and surface redshifts of compact objects lar structure to a point singularity might be avoided by the
residing in these exotic spacetimes. presence of charge. Ghezzi’s work on charged neutron stars
The role of an equation of state (EoS) in modeling com- showed that the outcome of gravitational
√ collapse is sensitive
pact objects has been highlighted in several recent studies. to the charge to mass ratio, Q/ Gμ. Simulations based on
The simple linear EoS which expresses the pressure as a various values of this ratio predict several possible outcomes
linear function of the fluid density ( pr = αρ) where α ≥ of collapse which include the formation of black holes, naked
is a constant has been extended to include α < 0 used in singularities or exploding remnants. An interesting outcome
modeling so-called dark stars and phantom fields. The MIT of this study is that extremal black holes cannot form from the
Bag model EoS has gained popularity amongst researchers collapse of a charged fluid sphere. This huge body of work on
and has been successfully utilised to model compact objects static, charged stellar objects has provided us with more ques-
in classical GR and modified gravity theories [36,37]. The tions than answers and has prompted an invigorated interest
quadratic EoS, polytropic EoS and Chaplygin gas EoS have in studying the effect of charge on the hyrdostatic equilibrium
also led to physically reasonable models of static stars [38– or dynamical collapse of such configurations. Herein lies the
40]. By appealing to results in quantum chromodynamics motivation for our study of charged compact objects within
and quark interactions within the stellar core, the so-called the BD framework with a massive scalar field. It is impor-
colour-flavoured-Locked (CFL) EoS has been recently used tant to note that these toy models play an important role in
in obtaining models of compact objects which approximate checking numerical results and simulations. In this way, it is
realistic neutron stars, pulsars and strange stars to a very clear that stellar objects can be endowed with nonzero charge
good degree [41,42]. The CFL EoS has also been employed which can give rise to high intensity electric fields [52–54].
to study the surface tension of neutron stars. This study shows The Einstein-Maxwell system can be interpreted as repre-
that the surface tension is sensitive to the magnitude of the senting an anisotropic fluid with the pressure isotropy con-
Bag constant [43]. It was observed that larger values of the dition becoming the definition of the electric field. A recent
Bag constant led to stellar models with lower tangential pres- paper by Maurya and Tello-Ortiz [55] highlighted an inter-
sures and surface tensions. esting interplay between the anisotropy parameter and the
The role of the electromagnetic field in the (in)stability electric field intensity which provides a mechanism for main-
of compact objects has occupied the interest of researchers taining stability of the stellar configuration. They found that
since the discovery of the Schwarzschild solution. The study the force due to the pressure anisotropy initially dominates
of charged objects in general relativity has taken a differ- the Coulombic repulsion closer to the center of the star with
ent and refreshing trajectory compared to the early attempts the anisotropic force out-growing the Coulombic repulsion
at just finding exact solutions to the Einstein-Maxwell sys- towards the surface layers of the star. A similar phenomenon
tem. The theoretical possibility of celestial models endowed was discovered in a charged compact star of embedding Class
with an electric field has been studied by several researchers one [56]. The effect of charge on stellar characteristics within
[44–48]. As early as 1924 Rosseland [49] (see also Edding- the framework of Einstein–Gauss–Bonnet (EGB) gravity has
ton [50]), investigated the possibility that a self-gravitating been investigated. It is found that mass-radius relation and
stellar structure within the framework of Eddington’s theory, surface redshifts are modified by the presence of the EGB
where the stellar structure is treated as a ball of hot ionized coupling constant [57].
gases containing a considerable quantity of charge. In such This paper is structured as follows: In Sect. 2. we provide
a system, the large number of lighter particles (electrons) as the necessary equations within the BD formalism necessary
compared to positive heavier particles (ions) escape from its to model a charged compact object. The Class one embed-
surface due to their higher kinetic energy and this migration ding condition is derived for the BD framework in Sect. 3.
of electrons continues resulting in an induced electric field The junction conditions required for the smooth matching
123729 Page 4 of 19 Eur. Phys. J. C (2021) 81:729
of the interior spacetime to the Reissner-Nordström-de Sitter Here, denotes the d’Alembert operator, then Φ can be
(RNdS) exterior is presented in Sect. 4. In Sect. 5 we discuss given as,
the regularity of the metric functions and thermodynamical
quantities at the center of the stellar configuration and we T (m) 1 dL (Φ)
Φ = + Φ − 2L (Φ) , (7)
derive the modified TOV equation in the presence of a mas- 3 + 2 ωB D 3 + 2 ωB D dΦ
sive scalar field. The nonzero charge density and pressure
anisotropy together with mass-radius relation and moment of Here, ρ, pr and pt denote the energy density, radial pres-
inertia thorough M−R and M−I curves have been discussed sure and transverse pressure, respectively with T (m) being
(m)
in Sect. 5. A detailed discussion of the physical attributes the trace of the energy tensor Ti j . Since we are interested
together with the conclusion of our model follows in Sect. 6. in spherically symmetric stellar structure, then we assume a
static spherically symmetric line element which can be cast
as,
2 The background of Brans–Dicke gravity theory and ds 2 = eξ(r ) dt 2 − eη(r ) dr 2 − r 2 (dθ 2 + sin2 θ dφ 2 ), (8)
field equations
where ξ(r ) and η(r ) are metric potentials and rely just upon
The action of scalar-tensor theory in Brans–Dicke frame in the radial distance r that ensures the static nature of the space-
relativistic units G = c = 1 is defined as, time. Also, u i = e−ξ/2 δ0i and v i = e−η/2 δ1i , given in Eq. (5),
are denoting the four-velocity and the unit space like vector
√ respectively, which are specified as, u i u i = 1 and v i vi = −1,
S = 16π1
RΦ − ωΦB D ∇ i ∇i Φ − L (Φ) −g d 4 x
and Φ = Φ;i,i = (−g)−1/2 [(−g)−1/2 Φ ,i ],i .
√ √
+ Lm −g d 4 x + Le −g d 4 x, (1) In addition, the anti-symmetric electromagnetic field ten-
sor Fi j given in Eq. (5) is characterized as
where R, g, Lm , and Le describe the Ricci scalar, deter-
minant of metric tensor, matter Lagrangian density, and Fi j = ∇i A j − ∇ j Ai (9)
Lagrangian electromagnetic field respectively, while ω B D is for which Maxwell’s equations have been satisfied,
a dimensionless Dicke coupling constant and Φ is a Brans–
Dicke scalar field. Here the function L (Φ) depends com- Fi j,k + F jk,i + Fki, j = 0 (10)
pletely on the scalar field Φ. In the present case we define
this scalar field function L (Φ) as, with
1 2 2 F ik ; k = 4π J i (11)
L (Φ) = m Φ (2)
2 φ
Now by varying of the action (1) with respect to the metric where, J i is the electromagnetic 4-current vector. This can
tensor g i j and scalar field Φ provides the following field be expressed as
equations and evaluation equation, respectively, which can
σ dxi
be written as, Ji = √ = σ ui , (12)
g00 d x 0
1 (m) (Φ)
Gi j = − 8π Ti j + 8π E i j + Ti j , (3) where σ = eξ/2 J 0 (r ) representing the charge density. It
Φ
turns out that for a static matter distribution with spherical
(m) symmetry, there is only one non-zero component of the elec-
where, Ti j and E i j denote the energy-momentum tensor for
matter distribution and electromagnetic field tensor, respec- tromagnetic 4-current J i which is J 0 , a function of the radial
(Φ)
tively while Ti j represents a scalar tensor appearing in the distance, r . The F 01 and F 10 components are the only non-
system due to the scalar field Φ. All the field tensors can be zero components of electromagnetic field tensor expressed
written as, in (5) and they are connected by the formula F 01 = −F 10 ,
which characterizes the radial constituent of the electric field.
(m)
Ti j = (ρ + pt )u i u j − pt gi j + ( pr − pt )vi v j , (4) The constituent of the electric field is determined through
Eqs. (11) and (12) as follows
1 1
Ei j = −Fin F j n + gi j Fγ n F γ n , (5) q −(ξ +η)/2
4π 4 F 01 = −F 10 = e (13)
r2
(Φ) ωB D gi j Φ,δ Φ ,δ
Ti j = Φ,i; j − gi j Φ + Φ,i Φ, j − The quantity q(r ) represents the effective charge of a spheri-
Φ 2
L (Φ) gi j cal system of radial coordinate, r , subsequently, this electric
− . (6) charge can be characterized by the relativistic Gauss law and
2
123Eur. Phys. J. C (2021) 81:729 Page 5 of 19 729
corresponding electric field E explicitly as, pressure pr can be cast as
r pr = p f − B, f = u, d, s (23)
q(r ) = 4 π σ r 2 eη/2 dr = r 2 −F10 F 10 (14)
0 f
q2
E 2 = −F10 F 10 = . (15) where p f describes the individual pressures due to each
r4 quark flavor which is balanced by the Bag constant (or total
It is noted that Doneva et al. [58] and Yazadjiev et al. [59] external Bag pressure) B. The deconfined quarks inside the
have already discussed both slowly and rapidly rotating neu- MIT Bag model have the accompanying total energy density
tron stars by using the above potential function (2). By using
Eqs. (2)–(8), we obtain the following field equations, ρ= ρ f + B, (24)
f
η 11 1 q2
e−η
0(Φ)
− 2= 8πρ +
+ + T , (16) where ρ f indicates the matter density due to each flavor
r r2
r Φ r4 0
which is connected to the corresponding pressure by the for-
ξ 1 1 1 q2 mula given as ρ f = 3 p f . Consequently, Eqs. (23) and (24)
e−η
1(Φ)
+ 2 − 2 = 8π pr − 4 − T1 , (17)
r r r Φ r are consolidated to express the following simplified MIT Bag
e−η ξ 2 ξ − η ξ η model,
ξ +
+ −
2 2 r 2 1
pr = (ρ − 4B), (25)
1 q2 3
= 8π pt + 4 − T22(Φ) , (18)
Φ r It should be mentioned here that this specific linear form of
the MIT Bag model EoS has been applied for portraying the
where prime denotes differentiation with respect to r . On the stellar systems made of the strange quark matter distribution
other hand, the scalar tensor components T00(Φ) , T11(Φ) , and in pure GR and modified gravity theories.
2(Φ)
T2 in terms of ξ and η are given as, Now using the Eqs. (16) and (17) along with EoS (25), we
obtain the expression for the electric field as,
2 η ω B D 2
T00(Φ) = e−η Φ + − Φ + Φ 4 (1 − e−η )
r 2 2Φ 4q 2 −η η − 3ξ
=Φ e +
L (Φ) r4 r r2
−eη , (19) 1(Φ) 0(Φ)
2 − 3T1 + T0 − 32π B. (26)
2 ξ ω B D 2 L (Φ)
= e−η Φ − Φ − eη
1(Φ)
T1 + ,
r 2 2Φ 2
(20) 3 Basic formulation of Class one condition and its
solution in Brans–Dicke gravity
1 η ξ ω B D 2
T22(Φ) = e−η Φ + − + Φ + Φ
r 2 2 2Φ
It is well-known that the embedding of n-dimensional space
L (Φ)
−eη . (21) V n in a pseudo-Euclidean space E n attracted much consid-
2 eration as inferred by Eisland [60] and Eisenhart [61]. In the
However, from Eqs. (7) and (8) we obtain, case where a n-dimensional space V n can be isometrically
immersed in (n + m)-dimensional space, where m is a min-
2 η ξ imum number of supplementary dimensions, at this stage
Φ = −e−η − + Φ (r ) + Φ (r )
r 2 2 V n is said to be m-Class embedding. Habitually, the metric
expressed in (8) provides the four-dimensional spherically
1 dL (Φ)
= T (m) + Φ − 2 L (Φ) . (22) symmetric space-time which describes a space-time of Class
(3 + 2 ω B D ) dΦ
two i.e, when m = 2, which shows that it is embedded in a
It has been argued that extreme temperatures and pressures six-dimensional pseudo-Euclidean space. On the other hand,
at the core of massive neutron stars can transform into quark it should be noted that one can reveal a possible parametriza-
stars with up (u), down (d) and strange (s) quark flavors. tion in order to incorporate the space-time expressed in (8)
In this regard, we suppose that the MIT Bag model rules into a five-dimensional pseudo-Euclidean space which leads
the matter variables (density and pressure) in the interior of to Class m = 1 known as embedding Class one [60–62].
these relativistic massive stars. In addition, we assume that For a spherically symmetric space-time in both cases static
non-interacting and massless quarks occupy the inside of the or non-static to be Class-one, the system has to be consistent
stellar structures. According to the MIT Bag model the quark with the following necessary and suitable conditions:
123729 Page 6 of 19 Eur. Phys. J. C (2021) 81:729
– For a stellar system, the symmetric tensor bi j should be where (b01 )2 = sin2 θ (R1202 )2 /R2323 . In this situation, the
determined under the associated Gauss conditions: embedding Class-one condition known as Karmarkar condi-
tion [60,62] takes the following form
Ri j hk = bi h b jk − bik b j h , (27)
R0202 R1313 = R0101 R2323 + R1202 R1303 . (34)
where = ±1 is everywhere normal to the manifold is
Although in our circumstance, the relation (32) between Rie-
time-like (+1) or space-like (-1).
mann components under the static spherically symmetric line
– The symmetric tensor bi j must fulfill the following dif-
element (8) will be equivalent to relation (34). The condition
ferential equation, known as Codazzi equation, as:
(34) plays a fundamental role for describing the space-time
(8) to be a Class-one, which is also well-known as a neces-
∇h bi j − ∇i bh j = 0. (28) sary and sufficient condition. At this point, by incorporating
the Riemann components in expression (34), we obtain the
Now the non-vanishing Riemann components for the line following differential equation,
element (8) can be expressed as follows,
2ξ η eη
ξ η ξ ξ 2 + ξ = , (35)
R0101 = − eξ − + ; ξ eη − 1
2 4 4
r with eη = 1. By integrating equation (35), we find the rela-
R1313 = − η sin2 θ ;
2 tionship amongst the gravitational potentials in the following
r 2 sin2 θ η
r form
R2323 = η
1 − e ; R0303 = − ξ eξ −η sin2 θ ;
e 2
r ξ −η r √
R0202 = − ξ e ; R1212 = − η . (29) ξ(r ) = 2 ln A + B η
e − 1 dr . (36)
2 2
Substituting these Riemann components into Gauss’s equa-
Here A and B are integration constants. Moreover, the solu-
tion (27) leads to
tion determined by the relation (36) is called a embedding
b01 b33 = R1303 = 0; b01 b22 = R1212 = 0; Class one solution for the line element (8). It is mentioned
that the above method has been utilized to model compact
b00 b33 = R0303 ;
stellar objects in various contexts. In order to find the solu-
b00 b22 = R0202 ; b11 b33 = R1313 ; b22 b33 = R2323 ; tion of the field Eqs. (16)–(18) in Brans–Dicke gravity under
b11 b22 = R1212 ; b00 b11 = R0101 . (30) Class I condition (36), we need to find the metric potentials
admitting Karmarkar condition with Pandey–Sharma con-
These relations expressed in (30) lead immediately to the
dition [63] (R2323 = 0). As it is well-known in general, the
following expressions
invariance of the Ricci tensor necessitates that the matter vari-
(R0202 )2 (R1212 )2 ables viz., energy density ρ, radial pressure pr and transverse
(b00 )2 = sin2 θ ; (b11 )2 = sin2 θ ; pressure pt ought to be finite at the center. The regularity of
R2323 R2323
R2323 the Weyl invariant necessitates that the following two quan-
(b22 )2 = ; (b33 )2 = sin2 θ R2323 . (31) tities: mass m(r ) and electric charge q(r ) must satisfy the
sin2 θ
following conditions: m(0) = q(0) = 0 and m(0) = 0,
By combining the last term of the relation (30) together with m (r ) > 0 and q(0) = 0, q (r ) > 0 i.e., both said quanti-
the components of Eq. (31), we found the following relation- ties reach their minimum and maximum values at the center
ship in terms of the Riemann components as well as at the surface of the celestial body, respectively.
In this regard, Maurya and collaborators [84] have already
R0202 R1313 = R0101 R2323 , (32) demonstrated that the gravitational potential ξ(0) is equal to
a finite constant value, q(0) = 0, ξ (0) = 0 and ξ (0) > 0
subject to R2323 = 0 (Pandey and Sharma condition [63]). according to the modelling of charged anisotropic compact
It ought to be noticed that all the components are given in celestial bodies. Since both physical quantities viz., energy
(31) fulfill the Codazzi equation (28). On the other hand, in density ρ and radial pressure pr are positive finite and contin-
the case of a general non-static spherically symmetric space- uous and also pursue the condition r > 2m(r ) [85,86]. From
time, the relation between components for symmetric tensor pr (r ) ≥ 0 with the help of the condition r > 2m(r ), we can
bi j and Riemann tensor Ri j hk can be given as follows obtain ξ = 0, which implies that the general function ξ(r )
is regular minimum at the center and a monotonic increasing
b01 b22 = R1212 and b00 b11 − (b01 )2 = R0101 , (33) function of the radial coordinate r . Consequently, the general
123Eur. Phys. J. C (2021) 81:729 Page 7 of 19 729
+6bΦ − 6aebr Φ − m 2φ Φ 2 + 32 B π + aΦ ebr r 2 + 4ab
2 2
function ξ(r ) should conserve the said physical characteris-
tics. Moreover, the gravitational potential function eη should 2 2 2
×ebr Φr 2 − 4a 2 e2br Φr 2 − 2aebr m 2φ Φ 2 r 2 + 64a B ebr
2
fulfill the accompanying form eη = 1 + O(r 2 ) to ensure
×π r 2 − a 2 e2br m 2φ Φ 2 r 4 + 32a 2 B e2br πr 4 ) − Φ Φ(8 +
2 2
the regularity and stability of the compact stellar object. In
×7aebr r 2 + br 2 (3 + 2aebr r 2 )) + Φ 2 r (1 + aebr r 2 )ω B D
2 2 2
this way by keeping all the attributes in our mind, we have
assumed the eη as follow:
+A Φ Φ(−8 + aebr r 2 (−7 + br 2 )) + Φ r m 2φ Φ 2 − 32
2
×B π − Φ (1 + aebr r 2 ) + a 2 e2br r 2 (4Φ + m 2φ Φ 2 r 2 − 32
2 2
eη(r ) = 1 + ar 2 eb r .
2
(37)
2
×B πr 2 ) + 2aebr (3Φ + bΦr 2 + m 2φ Φ 2 r 2 − 32Bgπr 2 )
Using (37) in (36), we get
+Φ 2 r (1 + ae2br r 2 )ω B D .
2
(42)
ξ(r ) = 2 ln A + C e br /2 .
2
(38)
√
where, C = ab B . Furthermore, the explicit expression for 4 Junction conditions
energy density and pressure can be given as,
To describe the complete structure of the self gravitat-
1 ing anisotropic compact object, the interior spacetime must
8πρ = be matched smoothly with the exterior spacetime at the
4Φ (A + Cebr /2 ) (1 + aebr r 2 )2
2 2
pressure-free boundary . The exterior spacetime is consid-
2
× C ebr /2 6aebr Φ 2
2
ered to be the Reissner–Nordström–de Sitter (RNdS) space-
time given by,
+32B Φπ + 3aΦ ebr Φr + 2 + ar 2 ebr a B ebr Φ
2 2 2
−1
2M Q2
×πr 2 − 3Φ Φ(1 + aebr r 2 ) + 3bΦ(2Φ + Φ r )(1 + 2a
2
ds+2
= − 1− + 2 − r2 dr 2 − r 2 dθ 2
r r 3
×ebr r 2 ) − 3Φ 2 ω B D − 3aΦ 2 ebr r 2 ω B D + A [32B
2 2
2M Q2
×Φ π + 32 a 2 B e2br Φ π r 4 − 3Φ Φ(1 + aebr r 2 )
2 2 − sin2 θ dφ 2 + 1 − + 2 − r 2 dt 2 ,
r r 3
−3Φ 2 ω B D + aebr {6Φ 2 (1 + br 2 ) + Φr (3Φ + 64B
2
(43)
×πr + 3bΦ r 2 ) − 3Φ 2 r 2 ω B D }] , (39) where, M is the total mass, Q the total charge and the
cosmological constant. In order to satisfy the smoothness
8π
8π pr = (ρ − 4 B ), (40) and continuity of internal spacetime metric ds− 2 and the
3 external spacetime ds+ at the boundary surface , the fol-
2
1 2
8π pt = C ebr /2 Φ Φ lowing conditions must be fulfilled at the hypersurface
4Φ r (A + Cebr /2 ) (1 + aebr r 2 )2
2 2
( f = r − R = 0, where R is a radius).
×(12 + 7 b r 2 + 7 a ebr r 2 + 2 a b ebr r 4 ) + Φ r 5Φ
2 2
[ds 2 − ] = [ds 2 + ] , [K i j − ] = [K i j + ] , (44)
+14 b Φ − 10 a ebr Φ − 2m 2φ Φ 2 + 32B π + 5 a Φ
2
2 2 2 [Φ(r )− ] = [Φ(r )+ ] , [Φ (r )− ] = [Φ (r )+ ] .
×ebr r 2 + 4b2 Φr 2 + 4 a b ebr Φ r 2 − 4 a 2 e2br Φ r 2
2 2 2 (45)
−4 a ebr m 2φ Φ 2 r 2 + 64 a B ebr π r 2 − 2a 2 e2br
Here, as usual − and + denote the interior and exterior space-
×m 2φ Φ 2 r 4 + 32 a 2 B e2br π r 4 + Φ 2 r (1 + aebr r 2 )
2 2
times, respectively while K i j represents the curvature. By
×ω B D + A − Φ Φ[−12 + aebr r 2 (−7 + 5br 2 )] + Φ r
2
using the continuity of the first fundamental form ([ds 2 ] =0),
we will always get
× 5Φ (1 + aebr r 2 ) − 2(m 2φ Φ 2 − 16B π + aebr (5Φ
2 2
[F] ≡ F(r −→ R + ) − F
2
+3bΦr 2 + 2m 2φ Φ 2 r 2 − 32B πr 2 ) + a 2 e2br r 2 (2Φ
(r −→ R − ) ≡ F + (R) − F − (R).
+m 2φ Φ 2 r 2 − 16B πr 2 )) + Φ 2 r (1 + a ebr r 2 )ω B D .
2
(46)
(41)
− (R) =
for any function F(r ). This condition provides us grr
+ − +
Consequently, the expression for the electric field intensity grr (R) and gtt (R) = gtt (R). On the other hand, the space-
can be obtained from the Eq. (26) as, time (8) must satisfy the second fundamental form (K i j ) at
the surface which is equivalent to the O Brien and Synge
1 2
E2 = 2 /2
C ebr /2 − Φ r (Φ [64] junction condition. This condition says that radial pres-
4Φ r (A + Cebr ) (1 + aebr r 2 )2
2
123729 Page 8 of 19 Eur. Phys. J. C (2021) 81:729
− +
sure must be zero at boundary r = r , which leads to Then the junction condition [K 22 ] = [K 22 ] together with
[r ] provides,
pr (R) = 0. (47)
2M Q2 2
e−η(R) = 1 − + 2 − R . (53)
In addition to above, the BD scalar field Φ corresponding R R 3
to the vacuum Schwarzschild solution is derived using the Plugging the above expression into the matching condition
technique in [72]. We consider τ − and τ + as the inner and −
[K 00 +
] = [K 00 ] yields,
outer space, respectively. Then the hypersurface is defined
by the following line element 2M 2Q 2 2
ξ (R) = − 3 − R
R2 R 3
ds 2 = dγ 2 − R 2 (dθ 2 + sin2 θ dφ 2 ). (48)
2M Q2 2 −1
× 1− + 2 − R . (54)
where γ defines the proper time boundary. The extrinsic cur- R R 3
vature of this boundary can be written as, Therefore, the matching conditions given by Eqs. (52)–(54)
μ leads to the following relations at the hypersurface,
∂ 2 y±
k
k ∂ y± ∂ y±
l
K i±j = −m ± − m±
k Γμ l (49) 2M Q2 2 1/2
k
∂n n
i j ∂n ∂n j
i
A+Ce b R 2 /2
= 1− + 2 − R , (55)
R R 3
Here the symbol n i defines the coordinates in the boundary
2M Q2 2 −1
, while m ±
2
k denotes the four-velocity normal to . The 1 + a R 2 eb R = 1 − + 2 − R , (56)
components of this four-velocity are given in the coordinates R R 3
ν ) of τ ± as
(y± b R 2 /2 M Q2
2C b R e = − 3 − R ×
R2 R 3
μ l d f d f −1/2
m±
df g 2M Q2 2 −1/2
k =± with m k m k = 1. (50)
dy k dy μ dy l × 1−
R
+ 2 −
R 3
R (57)
Then the unit normal vectors for inner and outer regions can After solving the relations (55)–(57), we obtain expressions
be cast as for the constants,
e−b R 6M R − 3Q 2 + R 4
2
m−k = 0, e η/2
, 0, 0 , and a =− 2 , (58)
R R 6M + R 3 − 3R − 3Q 2
2M Q2 2 −1/2
m− = 0, 1 − + 2 − r , 0, 0 . 1 9Q 2 18M √ b R2
k
r r 3 A= b − − 3R + 9 − 3 a C e
2 2 ,
3b R2 R
(51)
(59)
In view of line elements (8) and (48) together with the C=
Reisner–Nordström spacetime (43), we can write 3Q 2 +R 4 − 3M R R 6M +R 3 − 3R − 3Q 2
.
−1/2 18b R 6 eb R
2 /2
dt 2M Q2
= e−ξ/2 = 1 − + 2 − r2 . (60)
dγ r r 3
(52) As per the current observational evidences the value of the
cosmological constant 0 ≈ 10−46 /km2 at the present uni-
where [r ] = R. The non-zero components of the curvature verse. Hence, its effects on the local structures will be negli-
(K i j ) can be obtained from Eq. (51) as, gibly small, and then its numerical value does not affect the
numerical solution and it can be ignored.
− ξ − 1 −
K 00 = − , K 22 = K 33 = r e−η/2 ,
2 eη/2 sin θ
2 5 Physical viability of the model
+ M
Q2
K 00 = − 3 − r We have adopted the numerical approach to obtain the solu-
r2 r 3
−1/2 tion of the wave equation which leads to the expression for
2M Q2 the Brans–Dicke scalar field Φ. To do this, we chose the
× 1− + 2 − r2 ,
r r 3 initial condition Φ(0) = Φ0 = 1.001134 and Φ (0) = 0.
+ 1 + 2M Q2 2 1/2 Before embarking on a discussion of the physical viabil-
K 22 = K = r 1− + 2 − r .
sin2 θ 33 r r 3
ity of our model we like to comment on the magnitude of
123Eur. Phys. J. C (2021) 81:729 Page 9 of 19 729
the BD coupling constant, ω B D . A review of the literature stellar modeling as the TOV will be unable to balance at the
reveals that lower bound on the Brans–Dicke parameter is center. In order to nullify the anisotropy at r = 0, we must
|ω B D | > 40, 000. This bound arises from the Doppler track- have pr (0) = pt (0), which leads to
ing data of the Cassini spacecraft. However, the shortcom-
ing of such experiments is that they are restricted to “weak- (A + C)(6 a Φ0 + m 2φ Φ02 − 32 B π − 9 ψ0 )
field” limits and cannot reveal spatial or temporal deviations b= . (64)
6 C Φ0
of the gravitational constant on significantly larger scales.
First year data from WMAP restricted the BD constant to
ω B D > 1000 with further refinements coming through an Now the central value of E 2 is given by
array of observational data pinning ω B D > 120. A more
recent attempt in refining the bounds on ω B D uses data from −6 b C Φ0 + 6 a (A + C) Φ0 + (A + C) E 10
E 2 (0) = .
CMB (WMAP 5 yr, ACBAR 2007, CBI polarization, the (A + C)
BOOMERANG 2003 ) and the LSS data (galaxy clustering (65)
power spectrum from SDSS DR4 LRG data). This puts a
restriction of ω B D < −120.0 or ω B D > 97.8 which differs where E 10 = (m 2φ Φ02 − 32 B π − 9 ψ0 ). As we can see
in at least two orders of magnitude from |ω B D | > 40000 from above Eq. (65) that E 2 (0) is nonvanishing. However,
[65,66]. The range of ω B D (5; 20) chosen in this work is in incorporating Eq. (64) into Eq. (65), we obtain E 2 (0) = 0,
agreement with the solar system constraints of the Brans– as required. We will now discuss the regularity of these phys-
Dicke (BD) gravity presented by Perivolaropoulos [67], ical parameters throughout the stellar interior for particular
where ω B D −3/2 for all m φ 2 × 10−6 ≡ 2 × 102 m AU , compact object PSR J1903+327.
where m AU = 10−27 GeV. The mass of the scalar particle
m φ is taken to be 0.002 ≡ 2 × 105 m AU and is within the
predicted range for neutron stars according to Popchev et 5.2 Regularity
al. [68] i.e. 102 m AU m φ 109 m AU or in dimensionless
unit 10−6 m φ 10. Therefore, the range of Brans–Dicke (i) Metric functions at the centre, r = 0: we observe from
parameter ω B D used in this work is physically motivated for Fig. 1 (left panel) that the metric functions at the centre r = 0
the chosen scalar particle mass m φ = 0.002. assume finite values and are smooth and continuous through-
out the interior of the stellar configuration. We conclude that
5.1 Central values of the physical parameters the metric functions are free from singularity and positive at
the centre.
In the interior of the compact star, the central values of all the (ii) The density as a function as the radial coordinate is dis-
physical parameters must be finite and non-singular. Firstly, played in Fig. 1 (right panel). We observe that the density
we find the central values for both metric functions at r = 0
√ assumes a finite value at the centre and decreases monoton-
as: eη(0) = 1 and ξ(r ) = 2 ln[A + a B/b]. Since r = 0 is a ically towards the stellar surface. Furthermore, we observe
zeros for the function Φ (r ), then by the factor theorem we that a higher value for Φ0 leads to a higher density at each
can write: Φ (r ) = r ψ(r ), where ψ(r ) is any function of r . point of the stellar interior.
We then obtain values for density and pressures at the centre
r =0 (iii) Pressure at the centre r = 0: Figure 2 (left panel) shows
us that both the radial and tangential pressure are regular
6 b C Φ0 + 6 a (A + C) Φ0 − 3 (A + C) ψ0
ρ(0) = + B, at the centre of the star and decrease smoothly towards the
32 π (A + C) boundary. The vanishing of the radial pressure at some finite
(61) value, r = r defines the boundary of the fluid sphere. It
2 b C Φ0 + 2 a (A + C) Φ0 − (A + C)ψ0 is widely acknowledged that while the radial pressure van-
pr (0) = − B,
32 π (A + C) ishes at the boundary the tangential pressure need not. A
(62) phenomenological explanation is provided in the work by
pt (0) Boonserm et al. [78]. They point out that if the tangential
14 b C Φ0 − 10 a (A + C)Φ0 − (A + C) pt10 pressure does not vanish at the boundary of the star then
= + B. matching to the Schwarzschild exterior would imply that
32 π (A + C)
(63) the electric field is discontinuous at the stellar surface. This
would point to the existence of a nonzero surface charge den-
where, Φ(0) = Φ0 , (0) = 0 , and pt10 = (2 m 20 Φ02 − sity. To avoid this peculiarity the interior spacetime can be
17ψ0 ). Now we observe that pr (0) = pt (0) which leads matched to a Reissner-Nordström-de Sitter (RNdS) exterior
to (0) = 0. This situation may create certain problems in implying that the stellar object is electrically charged. The
123729 Page 10 of 19 Eur. Phys. J. C (2021) 81:729
nonzero transverse pressure at the boundary is then balanced fluid sphere:
by the transverse stress of the electric field in the exterior.
E2 E2
NEC : ρ + ≥ 0, WEC : ρ + pi + ≥ 0,
(iv) The anisotropy parameter is presented in Fig. 2 (right 8π 8π
panel) and it is clear that > 0 at each interior point of the E2
stellar configuration. The anisotropy parameter vanishes at SEC : ρ + pi + ≥ 0. (66)
4π
the center of the star and increases to a maximum for some i
finite radius, r < r where denotes the boundary of the It is clear from Fig. 4. (left panel) that all three energy condi-
star. A positive value for ( pt > pr ) signifies a repulsive tions are satisfied at each interior point of the configuration.
force due to anisotropy. It is clear that the increase in the
anisotropy parameter especially towards the surface layers
lead to greater stability in these regions. It is interesting to 5.6 TOV equation
observe that a larger value for Φ0 leads to increased values
for the anisotropy parameter thus stabilising the fluid even It is well known that in the absence of any dissipative effects
further. such as heat flow the equilibrium of a gravitationally bounded
charged fluid configuration is characterised by the resultant
of the gravitational force, Fg , the hydrostatic Fh force, the
5.3 Equation of state
force due to anisotropy, Fa and the electrostatic interaction,
Fe vanishing at each interior point of the star. The modified
The role of the equation of state (EoS) has been demon-
TOV equation in Brans–Dicke gravity is given by
strated in many models of compact objects within the frame-
work of classical general relativity and modified theories of ξ 2 1(Φ)
− pr − ( pr + ρ) + ( pt − pr ) + T1
gravitation. A barotropic EoS of the form p = p(ρ) points 2 r
strongly to the type of matter making up the star. Recently, the ξ 2 qq
+ T11Φ − T00Φ + T11Φ − T22Φ + = 0,
colour-flavoured locked-in EoS was utilised to model com- 2 r 4 πr 4
pact objects. This particular EoS is a generalisation of the (67)
MIT bag model and attempts to connect the microphysics to
where,
macrophysics of the fluid configuration. Figure 3 (left panel)
shows the variation of the ratio p/ρ with r/R. We note that FhB D = − pr + T11Φ ;
the pressure is less than the density at each interior point ξ
of the configuration. This ratio is also positive everywhere FgB D = − pr + ρ − T11Φ + T00Φ ;
2
inside the star. 2 qq
FaB D = pt − pr + T11Φ − T22Φ ; FeB D =
r 4 πr 4
5.4 Electric field
We note the contribution of the scalar field to the hydrostatic,
gravitational and anisotropic forces respectively. In a recent
The trend in the electric field and the charge density are shown
study, Herrera [79] pointed out an interesting observation
in Fig. 3 (right panel). We observe that the electric field van-
regarding the nonappearance of the tangential pressure in the
ishes at the centre of the star and increases monotonically
gravitational force term. In the case of pt > pr , anisotropic
towards the surface. It is well-known that intense electric
spheres are more compact than their isotropic counterparts.
fields can lead to instabilities within the stellar core. The
In Fig. 4 (right panel), we can see that the combined forces
presence of charge as high as 1020 coulombs can generate
of electric, hydrostatic and anisotropic counter-balanced the
quasi-static equilibrium states. These high charge densities
gravity so that the configuration is under equilibrium. It can
are linked to very intense electric fields which in turn induce
also be seen that when changing the scalar field contribution
pair production within the star thus leading to an unstable
(by changing the initial condition of the Brans–Dicke scalar
core.
field Φ0 ) all the forces also change proportionally. This con-
tribution from the scalar field is a mechanism which enables
5.5 Energy conditions the system to support a larger mass within each concentric
shell i.e. the equation of state will be stiffened.
In order for physical admissibility of our models the solution
should satisfy the following energy conditions, viz., (i) null
energy condition (NEC), (ii) weak energy condition (WEC) 5.7 Causality
and (iii) strong energy condition (SEC). In order to satisfy
the above energy conditions, the following inequalities must In order to prevent superluminal speeds within the stellar
be hold simultaneously at each interior point of the charged fluid we require that the speed of sound be less than that
123Eur. Phys. J. C (2021) 81:729 Page 11 of 19 729
Fig. 1 The metric potentials and matter density for PSR J1903+327 plotted against r/R by taking Φ0 = 1.001134, ω = 5, m φ = 0.002, B =
0.0381, β = −0.0092819, b = 0.0035/km2 , α = 0.9951, B = 99.54 MeV / f m 3
ˆ = ( pt − pr )/ pt are plotted against r/R by taking the same values as in Fig. 1 for PSR J1903+327
Fig. 2 The pressure and
of light everywhere inside the star. The speed of sound for 5.9 Stability through adiabatic index
the charged fluid sphere should be monotonically decreasing
√
from centre to the boundary of the star (v = dp/dρ < 1). It Within the Newtonian formalism of gravitation, it is also well
is clear from Fig. 5 (left panel) that the speed of sound is less known that there has no upper mass limit if the EoS has an
than 1 throughout the interior of the matter distribution. This adiabatic index Γ > 4/3 where
implies that our fluid model fulfills causality requirements
throughout its interior. p + ρ dp
Γ = (68)
p dρ
the definition of which arises from an assumption within the
Harrison-Wheeler formalism [81]. A perturbative study of
dissipative collapse by Chan et al. [82] in which gravita-
tional collapse proceeds from an initially static configura-
5.8 Stability factor
tion Eq. (68) follows from the EoS of the unperturbed, static
matter distribution. Equation (68) is modified in the presence
We observe from Fig. 5 (right panel) that our model satisfies
anisotropic fluids (radial and transverse stresses are unequal)
the Herrera cracking condition −1 < vt2 − vr2 < 0. Abreu
and we can write
et al. [80] showed that stable and unstable patches can arise
within the stellar fluid and their existence depends on the 4 4 pr − pt
Γ > − (69)
relative sound speeds in the radial and tangential directions. 3 3 r pr max
In particular, potential unstable regions occur when the tan-
gential component of the speed of sound exceeds the radial It is well-known that a bounded charged configuration can
component. be treated as an anisotropic system. In the special case of
123729 Page 12 of 19 Eur. Phys. J. C (2021) 81:729
Fig. 3 The p/ρ and electric field intensity (E 2 ) with charge density (σ ) are plotted against r/R by taking the same values as in Fig. 1 for PSR
J1903+327
Fig. 4 The energy conditions and TOV-equation are plotted against r/R and r respectively by taking the same values as in Fig. 1 for PSR J1903+327
Fig. 5 The speed of sound and stability factor are plotted against r/R by taking the same values as in Fig. 1 for PSR J1903+327
isotropic pressure ( pr = pt ) the classical Newtonian result 5.10 Stability through Mass-central density (M − ρc ) curve
holds from Eq. (69). Observations of Eq. (69) indicate that
instability is increased when pr < pt and decreases when Now, we focus on the M − ρc function dubbed as static
pr > pt . Figure 6 (left panel) confirms that our model is sta- stability criterion which is a noteworthy thermodynamically
ble against radial perturbations at each interior point within quantity in order to give more insight into the stability of the
the stellar fluid. compact celestial structure. This static stability criterion has
been developed and made more accessible by Harrison and
co-workers [88] and Zeldovich and Novikov [89] after sug-
123Eur. Phys. J. C (2021) 81:729 Page 13 of 19 729
gestions by Chandrashekhar [90,91] in order to portray the The effective mass of the charged fluid sphere can be deter-
stability of gaseous celestial configuration according to radial mined as:
pulsations. In this respect, the formula associated between the
gravitational mass M and the central density ρc is given as
R E2 R
m e f f = 4π ρ+ r 2 dr = 1 − e−η(R) (75)
follows, 0 8π 2
∂ M(ρc ) where e−η is given by the equation (37) and compactness
> 0, (70)
∂ρc u(r ) is defined as:
which must be satisfied in order to describe the solutions of m e f f (R) 1
static and stable celestial configurations or otherwise unsta- u(R) = = 1 − e−η(R) . (76)
R 2
ble if
5.12 Redshift
∂ M(ρc )
< 0, (71)
∂ρc The maximum possible surface redshift for a bounded con-
figuration with isotropic pressure is Z s = 4.77. In the work
under radial perturbation. We present the variation of the of Bowers and Liang they showed that this upper bound can
gravitational mass M with respect to central density ρc in be exceeded when the radial and transverse pressures are dif-
Fig. 6 (right panel). It shows that in the present study the ferent [76]. In particular, when > 0 ( pt > pr ) the surface
gravitational mass M is an increasing function with regard redshift is greater than its isotropic counterpart. Studies show
to central density for the initial condition values of the BD that for strange quark stars the surface redshift is higher in low
scalar field Φ0 i.e., Φ0 = 1.001134, by tuning ω B D to 5. mass stars with the difference being as high as 30% for a 0.5
This confirms the static stability criterion of the stellar system solar mass star and 15% for a 1.4 solar mass star. It appears
against radial perturbations. We can see that the solution takes that higher redshift predictions in low mass stars appear to
its stability with this specific choice of the initial condition be an anomaly. In a recent study Chandra et al. [77] have
of Φ0 and we also find that the celestial bodies become more used gravitational redshift measurements to determine the
massive according to increasing central density. mass-radius ratio of white dwarfs. Using data of over three
thousand catalogued white dwarfs they were able to deter-
5.11 Effective mass and compactness parameter for the mine the mass-radius relation over a wide range of stellar
charged compact star masses. Their improved technique entailed the cancelling of
random Doppler shifts by averaging out the apparent radial
As a starting point we recall that the maximal absolute limit velocities of white dwarfs with similar radii enabling them
of mass-to-radius (M/R) ratio for a static spherically sym- to measure the associated gravitational redshift.
metric isotropic fluid model is given by 2M/R ≤ 8/9 [73]. The gravitational surface red-shift (Z s ) is given as:
In the case of charged fluid spheres [74] showed that there
exists a lower bound for the mass-radius ratio Z s = (1 − 2 u)−1/2 − 1, (77)
Q 2 (18R 2 + Q 2 ) M
≤ , (72) From Eq. (77), we note that the surface redshift depends upon
2R (12R + Q )
2 2 2 R
the compactness u, which implies that the surface redshift for
for the constraint Q < M. any star cannot be arbitrarily large because compactness u
However this upper bound of the mass-radius ratio for satisfies the Buchdhal maximal allowable mass-radius ratio.
charged compact star was generalized by [75] who proved However, the value for the surface redshift for the different
that compact objects have been calculated when Φ0 = 1.001134
with ω B D = (5, 50) are as follows: (i) 0.466 and 0.467 for
2
M 2R + 3Q 2 2 2 PSR J1903+327, (ii) 0.418 and 0.419 for Cen X-3, (iii) 0.384
≤ + R + 3Q .
2 (73) and 0.384 for EXO 1785-248, (iv) 0.350 and 0.350 for LMC
R 9R 2 9R
X-4. The graphical behavior for gravitational redshift for PSR
The Eqs. (72) and (73) imply that J1903+327 is shown by Fig. 7 (left panel). Furthermore, the
behavior of the scalar field Φ(r ) with respect to r is shown
Q 2 (18R 2 + Q 2 ) in Fig. 7 (right panel). From this graph, it is clear that the
2R 2 (12R 2 + Q 2 ) functional form of the scalar field Φ(r ) is well-fitted with
2
M 2R + 3Q 2 2 2 numerical data which also approves the validity of our new
≤ ≤ + R + 3Q 2 (74) solutions.
R 9R 2 9R
123729 Page 14 of 19 Eur. Phys. J. C (2021) 81:729
Fig. 6 The adiabatic index and mass-central density curves are plotted against r/R and ρc by taking the same values as in Fig. 1 for PSR J1903+327
Fig. 7 The gravitational redshift profile and scalar field Φ(r ) are plotted against r/R by taking the same values as in Fig. 1 for PSR J1903+327
5.13 The effect of BD scalar field Φ, and BD-parameter Our survey on M−R and M−I curves is highly significant
ω B D on the M−R and M−I curves for the stellar systems which clearly show the state (more or
less) of compact celestial bodies via the maximum bound of
In this section, we examine the M−R and M−I diagrams the total mass as well as the efficacy and the sensitivity to the
resulted from our stellar model in the background of BD stiffness of an EoS. In this regard, from Fig. 8, we show the
gravity with a massive field via the embedding approach. variation of the total mass M in [M ] versus the total radial
In this respect, we provide an instructive explanation of the coordinate R in [km] and the maximum moment of inertia
influences included by the choices made on different param- I in [×1045 g cm2 ], for all chosen values of the parameters
eters viz., the initial condition of the BD scalar field Φ0 , Φ0 , ω B D and B. In the present BD gravity stellar model via
BD-parameter ω B D and the total external bag pressure B the embedding approach, we observe from M−R curves fea-
(or bag constant), in order to give a more achievable scenario tured in Fig. 8 (left panel) the contributions due to ω B D where
and efficient astrophysical stellar system. On the other hand, we have set Φ0 to 1.001134 and B to 99.54 MeV / f m 3 . We
for determining the stiffness of an EoS, we can analyze the note that as the parameter ω B D decreases from 20 to 5, the
moment of inertia I associated with a static celestial solu- most extreme value of mass M increases with an increase
tion which could give a precise instrument via adopting the in the total radial coordinate R, which culminates in more
Bejger and Haensel concept [87], given by, massive compact celestial bodies. We can conclude that as
ω B D increases the corresponding radius R decreases and
the most extreme value of mass M also decreases, which
gives us also a celestial system less compact and less mas-
2 (M/R) · km sive. This shows that the parameters Φ0 , B and ω B D will
I = 1+ M R2. (78)
5 M affect the maximum mass limit as well as compactness of
the objects. On the other hand, the variation of the maximum
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