The criteria of the anisotropic quark star models in Rastall gravity Takol Tangphati1Ayan Banerjee2 Sudan Hansraj2 and Anirudh Pradhan3 1Theoretical and Computational Physics Group

2025-05-03 0 0 964.83KB 9 页 10玖币

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The criteria of the anisotropic quark star models in Rastall gravity

Takol Tangphati,1, ∗Ayan Banerjee,2, †Sudan Hansraj,2, ‡and Anirudh Pradhan3, §

1Theoretical and Computational Physics Group,

Theoretical and Computational Science Center (TaCS), Faculty of Science,

King Mongkut’s University of Technology Thonburi, 126 Prachauthid Rd., Bangkok 10140, Thailand

2Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science,

University of KwaZulu–Natal, Private Bag X54001, Durban 4000, South Africa

3Centre for Cosmology, Astrophysics and Space Science,

GLA University, Mathura-281 406, Uttar Pradesh, India

(Dated: October 5, 2022)

Quark stars are terrestrial laboratories to study fundamental physics at ultrahigh densities and

temperatures. In this work, we investigate the internal structure and the physical properties of quark

stars (QSs) in the Rastall gravity. Rastall gravity is considered a non-conservative theory of gravity,

which is an effective gravity theory at high energy density, e.g., relevant to the early universe and

dense, compact objects. We derive the hydrostatic equilibrium structure for QSs with the inclusion of

anisotropic pressure. More speciﬁcally, we ﬁnd the QS mass-radius relations for the MIT bag model.

We focus on the model depending on the Rastall free parameter ηand examine the deviations from

the General Relativity (GR) counterparts.

I. INTRODUCTION

In order to construct realistic models capable of rep-

resenting physically viable conﬁgurations of matter re-

quires solving a complicated system of gravitational

ﬁeld equations. In this regard, the standard theory of

gravity proposed by Einstein has generated over 120 ex-

act solutions to the ﬁeld equations for the well studied

isotropic ﬂuid case. What has proved elusive is ﬁnd-

ing exact solutions with realistic equations of state since

the isotropy equation is complicated. In the literature,

there is no exact solution known for the simplest linear

barotropic case. When models display an equation of

state, it is often discovered after an assumption of one

of the gravitational potentials is made [1–3] and is usu-

ally very complicated. Relaxation of the condition of

isotropy carries the elevated prospects of ﬁnding exact

solutions with the caveat of introducing imperfect ﬂu-

ids.

In the 1970s, some six decades after the ﬁrst per-

fect ﬂuid solutions emerged, the concept of pressure

anisotropy was introduced through the observations of

Ruderman [4], and Canuto [5]. Bowers and Liang [6]

reported the ﬁrst such solutions and thereafter numer-

ous exact solutions emerged on account of the now triv-

ial ﬁeld equations [7–12]. Solution ﬁnding is trivial be-

cause any metric can solve the anisotropic ﬁeld equa-

∗takoltang@gmail.com

†ayanbanerjeemath@gmail.com

‡hansrajs@ukzn.ac.za

§pradhan.anirudh@gmail.com

tions. Such solutions must still be examined against

the conditions for physical acceptability. In most cases,

there are violations of these conditions, or an equation of

state is not realizable. The possibility of anisotropy car-

ries the major mathematical advantage of studying exact

solutions with equations of state, given that two degrees

of freedom are inherent in the ﬁeld equations. Various

conjectures have been made regarding the question of

what could give rise to anisotropy. For example, pion

condensation in neutron stars could account for unequal

pressure stresses [13]. It is also known that Boson stars

display pressure anisotropy [14]. Therefore there are

solid motivations to investigate anisotropic stellar dis-

tributions.

It is widely believed that general relativity may need

augmentation at the scale of the universe and that it is

also not well tested in extreme gravity regimes such as

near black holes. Additionally, the explanation of the

accelerated cosmic expansion evades general relativity

unless an appeal to exotic matter, such as dark energy

and dark matter, is made. Rastall [15,16] argued that

the vanishing divergence of the Einstein tensor does

not necessarily imply the vanishing divergence of the

energy-momentum tensor. There is room for the covari-

ant divergence of the energy-momentum tensor to vary

as the gradient of the Ricci scalar in his proposal. In

such a scenario, energy conservation must be sacriﬁced.

However, when studying closed compact objects such

as stars, it is not unreasonable for net energy production

or loss as such are not genuinely isolated systems. Be-

sides, non-conservative systems have been an active and

fertile area of research for a long time. This property is

arXiv:2210.01372v1 [gr-qc] 4 Oct 2022

a feature of quadratic Weyl conformally invariant grav-

ity [17] and its generalisation by Dirac [18,19], f(R,T)

theory [20] and trace-free gravity [21–23]. The theories

mentioned above have emerged due to efforts to correct

shortcomings in general relativity.

It has been shown that Rastall gravity does admit the

accelerated expansion of the universe. The basic argu-

ment is that the nonconserved elements of the energy-

momentum tensor play the roles of dark energy to sup-

port cosmic inﬂation [24–29]. Additionally, it has been

shown that stellar models are well behaved in theory

[30,31]. At the geometric level, Rastall gravity is equiv-

alent to Einstein gravity [32], unimodular theory as well

as f(R,T)theory; however, the point that the physical

consequences are dramatically different has been ampli-

ﬁed several times [33]. The one notable negative feature

of Rastall gravity is the absence of Lagrangian - various

efforts at constructing a suitable Lagrangian have failed

thus far, and some wonder if this is even possible. Nev-

ertheless, the many positive features of the theory can-

not be ignored. Various aspects of this theory including

theoretical and observational once have been reported

in recent literature [34–37].

To create a conﬁguration for QS in the current inves-

tigation that complies with the available observational

data on the (M−R)relations, we use the quark matter

as our primary input. The paper is formatted as follows:

For a static and spherically symmetric line element, we

describe the issue and derive the associated ﬁeld equa-

tions using Rastall gravity in Section II. Section III is

devoted to prescribing a quark matter EoS with proper

boundary conditions. We numerically solve these equa-

tions and provide our critical ﬁndings in Section IV. We

also look at the stability of compact conﬁgurations of-

fered by the MIT bag model EoS in this part. In the con-

clusion section V, we summarise the ﬁndings.

II. FIELD EQUATIONS IN RASTALL GRAVITY MODEL

In this section, we introduce fundamental aspects of

Rastall gravity theory. As explained above, the possibil-

ity that the divergence of the energy-momentum tensor

is proportional to the gradient of the Ricci scalar with-

out violating the vanishing divergence of the Einstein

tensor also exists. That is Rastall postulated the condi-

tion ∇bTab ∝∇Rb.ˆ

A Effectively the gravitational ﬁeld

equations may be written in the form [15,16,38]

Gµν −γgµνR=8πGTµν, (1)

where γ≡η−1/2 is the Rastall parameter and ηis

the Rastall free parameter. Gµν and Tµν are the Einstein

tensor and the energy-momentum tensor, whereas gµν

is the metric tensor, Ris the scalar curvature, and Gis

Newton’s gravitational constant, respectively.

The Rastall ﬁeld equations from Eq. (1) reduce to the

standard general relativity (GR) equations of motion

when η=1. It is also noteworthy that the conserva-

tion law of the energy-momentum tensor is no longer

valid, and the divergence of the energy-momentum ten-

sor given by

∇µTµν =1

2 η−1

2η−1!∇νT6=0 (2)

is sometimes called the non-conservation of the energy

equation. We avoid the value η=1/2 since the diver-

gence becomes singular in this case. In the following

context, we choose units with G=c=1 throughout the

paper.

In the current investigation, we consider the

anisotropic form of the energy-momentum tensor

given by

Tµν = (ρ+p⊥)uµuν+p⊥gµν + (pr−p⊥)χµχν, (3)

where ρis the energy density, pris the radial pressure,

p⊥is the tangential pressure, uµis the 4-velocity of the

ﬂuid, and χµis the space-like unit vector in the radial

direction. One can adjust the ﬁeld equations of Rastall

gravity from Eq. (1) into the GR ﬁeld equation as follows

[38],

Gµν =8π˜

Tµν, (4)

where ˜

Tµν is the effective energy-momentum tensor,

Tµν =Tµν −1

2 η−1

2η−1!gµν T. (5)

This leads to a novel version of the conservation law of

the energy-momentum tensor for Rastall gravity given

∇µ˜

Tµν =0. (6)

The effective energy density ˜

ρ, radial pressure ˜

pr, and

tangential pressure ˜

p⊥can be written in terms of the tra-

ditional energy density ρ, radial pressure pr, and tan-

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ThecriteriaoftheanisotropicquarkstarmodelsinRastallgravityTakolTangphati,1,AyanBanerjee,2,SudanHansraj,2,andAnirudhPradhan3,§1TheoreticalandComputationalPhysicsGroup,TheoreticalandComputationalScienceCenter(TaCS),FacultyofScience,KingMongkut'sUniversityofTechnologyThonburi,126PrachauthidRd.,Bangk...

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The criteria of the anisotropic quark star models in Rastall gravity Takol Tangphati1Ayan Banerjee2 Sudan Hansraj2 and Anirudh Pradhan3 1Theoretical and Computational Physics Group

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