arX

iv:2

006.

0161

9v1

[gr

-qc]

30

May

202

0

Eur. Phys. J. C manuscript No.(will be inserted by the editor)

Anisotropic strange star with Tolman-Kuchowicz metricunder f(R, T ) gravity

Suparna Biswasa,1, Dibyendu Sheeb,2, B.K. Guhac,1, Saibal Rayd,2,3

1Department of Physics, Indian Institute of Engineering Science and Technology, Shibpur, B. Garden, Howrah 711103, WestBengal, India2Department of Physics, Government College of Engineering and Ceramic Technology, Kolkata 700010, West Bengal, India3Department of Natural Sciences, Maulana Abul Kalam Azad University of Technology, Haringhata 741249, West Bengal,India

Received: date / Accepted: date

Abstract In the current article, we study anisotropicspherically symmetric strange star under the background

of f(R, T ) gravity using the metric potentials of Tolman-

Kuchowicz type [1,2] as λ(r) = ln(1 + ar2 + br4) and

ν(r) = Br2+2 lnC which are free from singularity, sat-

isfy stability criteria and also well behaved. We calcu-late the value of constants a, b, B and C using matching

conditions and the observed values of the masses and

radii of known samples. To describe the strange quark

matter (SQM) distribution, here we have used the phe-nomenological MIT bag model equation of state (EOS)

where the density profile (ρ) is related to the radial

pressure (pr) as pr(r) =13 (ρ − 4Bg). Here quark pres-

sure is responsible for generation of bag constant Bg.

Motivation behind this study lies in finding out a non-singular physically acceptable solution having various

properties of strange stars. The model shows consis-

tency with various energy conditions, TOV equation,

Herrera’s cracking condition and also with Harrison-Zel′dovich-Novikov’s static stability criteria. Numerical

values of EOS parameter and the adiabatic index also

enhance the acceptability of our model.

1 Introduction

Einstein’s General Relativity (GR) consists of concep-

tual ingenuity and mathematical elegance in its ev-

ery steps. To describe a gravitating system and non-inertial frames from a classical view point at large scale

regimes, GR is a highly useful tool [3]. But inspite of its

beauty, singularity makes it stagnant [4] in few cases.

ae-mail: sb.rs2016@physics.iiests.ac.inbe-mail: dibyendu shee@yahoo.comce-mail: bkguhaphys@gmail.comde-mail: saibal@associates.iucaa.in

Also the theory is unable to behave rationally for thedescription of observational accelerating phase of the

Universe. It cannot describe properly the cosmic dy-

namics of the Universe without considering the exotic

form of matter-energy which is supposed to be known

as dark matter and dark energy [5,6,7,8,9,10,11,12,13,14]. Einstein’s GR does not consider the quantum

nature of matter and cannot be quantized in a conven-

tional way of renormalization. In this line of thinking it

was shown [15] that inclusion of higher order curvatureterms in Einstein-Hilbert action will make the system

renormalizable in one loop. On the other hand, to in-

clude quantum corrections one has to consider higher

order curvature invariants in the effective gravitational

action at low energy regime [16,17,18].

To overcome the above mentioned problems of GR,

several modified or alternative theories of gravitation

have been proposed from time to time by researchers.

The modification has been done by changing the ge-ometrical part of the Einstein field equations (EFE),

i.e., the action is modified by considering a general-

ized functional form of gravitational Lagrangian den-

sity. Some of the well known relevant alternative the-

ories of gravitation are f(R) gravity [19,20,21,22,23],f(T) gravity [24,25], f(R, T ) gravity [26,27,28,29,30,

31,32,33,34], f(T, T ) gravity [35,36,37,38,39,40], f(G)

gravity [41,42,43] and f(R,G) gravity [44] where R, T,

T and G are scalar curvature, torsion scalar, trace ofthe energy momentum tensor and Gauss-Bonnet scalar

respectively.

In the present work we are interested in f(R, T )

theory. This theory is developed by considering non-

minimal coupling between the Ricci Scalar R and traceof the energy momentum tensor T . Harko et al. [45] in-

troduced it first to tackle the problems in an efficient

way. In this theory matter is considered on an equal

2

footing with geometry. As a result one can explore many

interesting and novel features of the Universe, such as

the role of dark matter [46]. One can note that depen-

dency of T may come from the consideration of quan-

tum effects or from the presence of an imperfect fluid.

Neutron stars have been the subject of immense

study for the last few decades due to its small size (ra-

dius ∼ 11-15 km) and being the mass ∼ 1.4 − 2.0 M⊙a tremendous density (ρ ∼ 1017 kg/m3), which distortthe space-time geometry [47]. They take birth from the

gravitationally collapsing massive star (M > 8M⊙), af-

ter type II supernovae explosion [48]. Neutron stars en-

riched with neutrons, spin rapidly and often making

several hundred rotations per second. Sometimes neu-tron stars form radio pulsars, emitting radio waves.

The tremendous pressure and density is probably

responsible for phase transition of neutrons inside the

neutron stars to hyperon (Λ,Σ,Ξ,∆,Ω) and quark mat-ter (u, d, s). Cameron [49] predicted that conversion of

nucleon into hyperon is energetically more favourable.

Interior (specially core) of the neutron star contains

quark matter since they become free of interaction due

to high energy density and extreme asymptotic momen-tum transfer. The idea that quark matter may exist in

the core of the neutron star, has been suggested by sev-

eral scientists [50,51,52,53,54,55,56].

The energy level of the hyperon at the Fermi-surface

becomes higher than its rest mass, due to the tremen-dous density and as a result these particles could be de-

confined into strange quarks. Though quark stars may

contain up (u), down (d) and strange (s) quarks but

mostly they contain strange (s) quarks [51,57,58,59,60] since strange quarks are the most stable ones. Un-

der some conditions, up (u) and down (d) quarks can

also be transformed into strange (s) quarks. So once

up (u) and down (d) quarks of the quark star convert

into strange matter, the entire quark matter get con-verted into strange matter. In this way neutron stars

may be totally converted into strange quark star [61].

The neutron stars may very well contain quark matter

in their cores, which ought to be in a color supercon-ducting state [62,63,64,65]. But still now, no one has

been able to predict (theoretically or experimentally)

the exact density at which expected phase transition to

quark matter occurs.

Quantum Chromodynamics (QCD) deals with thequark confinement mechanism. At asymptotic density,

the ground state of QCD with a vanishing strange quark

mass is the color-flavor-locked (CFL) phase. According

to Chodos et al. [66], a strongly interacting particle canbe defined as a finite region of space, confined with

fields and bag constant is such a candidate. As a result,

bag constant affects the energy momentum tensor of

the star and even the spacetime geometry. According to

the MIT bag model, the universal pressure Bg known

as bag constant, is responsible for quark confinement

and it is defined as the difference between energy den-

sity of the perturbative and non-perturbative quantumchromodynamics vacuum. According to Farhi [67] and

Alcock [60], for a stable strange quark matter (SQM),

the bag constant should be within the following range

(55-75) MeV/fm3 under GR. However CERN-SPS andRHIC [68] provided a wide range of this constant.

Abbas et al. [69] using MIT bag model, investigated

anisotropic charged strange stars in f(T ) gravity em-

ploying the diagonal tetrad field of static space-time.

Biswas et al. [70], have established a new model forhighly compact anisotropic strange star using the met-

ric potentials, given by Krori-Barua [71] under f(R, T )

gravity. Besides these, several studies are available in

literature survey [72,73,74,75,76,77,78] based on strangestar.

Inhomogeneous matter distribution and its evolu-

tion changes the fabric of the spacetime of compact ob-

jects and leads to the anisotropic features. Anisotropy

plays an important role to study the stellar properties.Anisotropic pressure measures the difference between

radial component of pressure and tangential compo-

nent of pressure. Ruderman [79] first argued that at

very high density (> 1015 gm/cc) anisotropy in pres-sure arises and we have to treat nuclear interactions

relativistically. Different physical properties of a stel-

lar object like energy density, total mass, gravitational

redshift and frequency of oscillation of the fundamen-

tal mode are affected by the anisotropy. The reasonsbehind the anisotropy are provided by different scien-

tists in their research work from time to time. Usov [80]

suggested that strong electric field is the producer of

anisotropy. According to Weber [81] the anisotropy inpressure inside a strange star is due to immense mag-

netic field of the neutron star. Sokolov [82] argued that

different types of phase transitions can be taken as re-

sponsible for this anisotropy. Also pion condensation

can be taken as one of the reason behind the anisotropy,predicted by Sawyer [83]. According to Kippenhahn and

Weigert [84], the anisotropy may arise due to presence

of a type 3A superfluid or existence of solid stellar core.

The literature survey regarding anisotropic fluid sphereare provided in the following references [85,86,87,88,

89,90,91,92,93,94,159,96,97,98,99,100,101,102,103].

In this background, we shall summarize our research

work as follows. Basic mathematics behind f(R, T ) grav-

ity has been discussed in Sect. 2. Solutions to EFEs forstrange star and the proper choice of EOS for SQM dis-

tribution have been explained in Sect. 3. Sect. 4 con-

tains the boundary conditions from whics we have cal-

3

culated all the model parameters. In Sect. 5, different

physical features like density and pressure (5.1), energy

conditions (5.2), have studied along with different sta-

bility criteria like Herrera’s cracking conditions (5.3.1),

TOV equation (5.3.2), adiabatic index (5.3.3), equa-tions of state parameter (5.3.4) and Harrison-Zel′dovich-

Novikov static stability criteria (5.3.5) etc. The effec-

tive mass, compactness, i.e., the mass-radius ratio from

which one can calculate the gravitational as well as thesurface redshift, has been discussed in Sect. 5.4. Finally,

in Sect. 6, we have provided an overall conclusion of

our detailed study from different aspect of strange star

model.

2 Mathematics behind f(R, T ) gravity

Following Harko [45], we can represent action in f(R, T )

theory as

S =

∫

d4x£m

√−g +

1

16π

∫

d4xf(R, T )√−g. (1)

Here the arbitrary function f(R, T ) contains the

Ricci scalar R as well as T , the trace of the energy mo-

mentum tensor, £m is the matter Lagrangian density

which represents the possibility of non-minimal cou-

pling between matter and geometry, g is the determi-nant of gµν metric (c = G = 1).

Varying the action (1) w.r.t. gµν metric, we can de-

rive modified EFEs for f(R, T ) gravity as

fR(R, T )Rµν − 12gµνf(R, T ) + fR(R, T )(gµν−∇µ∇ν)

= 8πTµν − TµνfT (R, T )−ΘµνfT (R, T ), (2)

where fT (R, T ) = ∂f(R,T )∂T

, fR(R, T ) = ∂f(R,T )∂R

, ≡∂µ(

√−ggµν∂ν)√−g

, Rµν represents the Ricci tensor,∇µ is the

covariant derivative w.r.t. the symmetry connected to

gµν , Θµν = gαβδTαβ/δgµν and Tµν = gµν£m − 2∂£m

∂gµν

represents the stress-energy tensor.

From Eq. (2), the covariant divergence becomes [104]

∇µTµν = fT (R,T )8π−fT (R,T ) [(Θµν + Tµν)∇µ ln fT (R, T )

− 12gµν∇µT +∇µΘµν ]. (3)

Eq. (3) states that in f(R, T ) theory of gravity,

energy-momentum tensor is not conserved where as itremains conserved in general relativity.

For a perfect anisotropic fluid, the energy-momentum

tensor takes the following form

Tµν = (ρ+ pt)uµuν − ptgµν + (pr − pt)vµvν , (4)

with uµ∇νuµ = 0 and uµuµ = 1. Here ρ(r), pr(r), pt(r),

uµ, vµ stand for the energy density, radial pressure and

tangential pressure, four-velocity and radial four-vector

respectively for a static fluid source. Besides these, we

have another condition Θµν = −2Tµν − pgµν .

Following Harko et al. [45], f(R, T ) takes the form

f(R, T ) = R+ 2χT. (5)

Here χ is a constant arises due to the coupling be-

tween matter and geometry in modified gravity.

Substituting Eq. (5) in Eq. (2), we get

Gµν = 8πTµν + χTgµν + 2χ(Tµν + pgµν). (6)

where Gµν is known as the Einstein tensor. Now, sub-

stituting χ = 0 in the Eq. (6), we get different results

which represent the results of GR.

Combination of Eqs. (3) and (5) gives the following

result

(8π + 2χ)∇µTµν = −2χ

[

∇µ(pgµν) +1

2gµν∇µT

]

. (7)

Incorporation of χ = 0 in Eq. (7) shows the invari-

ance of energy-momentum tensor in general relativity.

3 Solution of Einstein’s field equations

To study a strange star, we consider the static, spheri-

cally symmetric stellar configuration, for which the lineelement of the interior space-time, can be described as

the following metric:

ds2 = eν(r)dt2 − eλ(r)dr2 − r2(dθ2 + sin2θdφ2). (8)

Here the metric potentials λ(r) and ν(r) are chosen

as λ(r) = ln(1 + ar2 + br4) and ν(r) = Br2 + 2 lnC,

known as Tolman-Kuchowicz metric potentials [1,2].The constants a, b, B and C can be determined us-

ing matching conditions. The non-zero components of

energy-momentum tensor can be written as T 00 = ρ(r),

T 11 = −pr(r) and T 2

2 = T 33 = −pt(r).

For an anisotropic spherically symmetric unchargedstellar system, combining Eqs. (4), (6) and (8), we get

the EFE as follows

e−λ

r2(−1 + eλ + rλ′) = 8πρ+ 2χ

[

2ρ− pr+2pt

3

]

= 8πρeff ,(9)

e−λ

r2(1− eλ + rν′) = 8πpr − 2χ

[

ρ− 4pr+2pt

3

]

= 8πpeffr ,(10)

e−λ

4r

[

2(ν′ − λ′) + (2ν′′ + ν′2 − λ′ν′)r]

=

8πpt − 2χ[

ρ− pr

3 − 5pt

3

]

= 8πpefft . (11)

where ′ expresses the differentiation of the respective

parameters w.r.t. r.

The strange quark matter distribution inside the

strange star has been generated by the simplest phe-

nomenological MIT bag model EOS [66,67,60]. The

4

quark pressure of SQM, considering the quarks are non-

interacting, massless and including all the corrections

of energy and pressure, can be defined as

pr(r) =∑

f=u,d,s

pf −Bg, (12)

where pf is the individual pressure of all three flavors

of quarks and Bg is the bag Constant.

The relation between the individual quark pressure

pf and energy density of individual quark flavor is given

by pf = 13ρ

f . Therefore deconfined quarks inside thebag have the total energy ρ as follows

ρ =∑

f=u,d,s

ρf +Bg. (13)

Using Eqs. (15) and (16) we have the MIT bag EOSas

pr(r) = α [ρ(r) − 4Bg] , (14)

where α is a constant having numerical value 0.28 for

the massive strange quarks with mass 250 MeV and 13

for massless strange quarks.

MIT bag model equation of state (EOS) for strange

quark matter (SQM) can be written as

pr(r) =1

3[ρ(r) − 4Bg] . (15)

Inserting λ(r) = ln(1+ar2+ br4) and ν(r) = Br2+2 lnC in Eqs. (9)-(15), we get

ρeffr = 1X

[

12b2Bgr8Y + 24abBgr

6Y − 4B2bχr6+

12a2Bgr4Y − 4B2aχr4 + 24bBgr

4Y + 36Bπbr4 + 23Bbχr4

+24aBgr2Y − 4B2χr2 + 36Bπar2 + 19Baχr2 + 72πbr2

+54bχr2 + 12BgY + 36Bπ + 15Bχ+ 36πa+ 27aχ]

, (16)

peffr = 1X

[

−12b2Bgr8Y − 24abBgr

6Y + 4B2bχr6−12a2Bgr

4Y + 4B2aχr4 − 24br4BgY + 12Bπbr4 − 3Bbχr4

−24ar2BgY +Br2(4Bχ+ 12aπ + aχ) + 24πbr2

−14bχr2 − 12BgY + 12π(B + a) + 5Bχ− 7aχ]

, (17)

pefft = B2br6+B2ar4+B2r2+Bar2−2br2+2B−a8π(br4+ar2+1)2 . (18)

where, X = 16π(12π+5χ)(br4+ar2+1)2, Y = (16π2+

16πχ+ 3χ2)

The anisotropic stress can be obtained as

∆ = [pefft − peffr ] = 3X

[

4b2Bgr8Y + 8abBgr

6Y + 8B2πbr6

+2B2bχr6 + 4a2Bgr4Y + 8B2πar4 + 2B2aχr4 + 8bBgr

4Y

−4Bπbr4 +Bbχr4 + 8aBgr2Y + 8B2πr2 + 2B2χr2

+4Bπar2 + 3Baχr2 − 24πbr2 − 2bχr2 + 4BgY + 12Bπ

+5Bχ− 12πa− aχ] , (19)

where, X = 16π(12π+5χ)(br4+ar2+1)2, Y = (16π2+

16πχ+ 3χ2).

4 Matching conditions

4.1 Interior space-time

The effective density at the center from Eq. (16) is

ρeff0 = ρeff (r = 0).

=192π2Bg+192πBgχ+36Bgχ

2+36Bπ+15Bχ+36πa+27aχ16π(12π+5χ) . (20)

Here we assume the coupling constant χ = 1. Anisotropic

condition, i.e., at the center (r = 0) radial pressure bal-

ances the tangential pressure as peffr (r = 0) = pefft (r =

0). Using these conditions, we get

Bg = 12(a−B)π−5B+a

64π2+64π+12 . (21)

4.2 Exterior space-time

As there is no mass in exterior region, matter geome-

try coupling constant χ becomes zero, leading all the

components of the energy momentum tensor Tµν =(ρ + pt)uµuν − ptgµν + (pr − pt)vµvν to be zero. So

at the exterior Schwarzschild solution is as follows

ds2 =(

1− 2Mr

)

dt2 − dr2

(1− 2Mr )

− r2(dθ2 + sin2 θdφ2),(22)

where M represents total mass of the stellar system.

From matching conditions, the metric coefficients gtt,

grr and ∂gtt∂r

are continuous between the interior and

exterior regions, at the surface, i.e., r = ℜ (where ℜ is

the radius). Hence, by comparing Eqs. (8) and (22), weget

gtt = 1− 2M

ℜ = eBℜ2+2 lnC ,

grr = 1− 2M

ℜ = e(1+ar2+br4),

∂gtt∂r

=2M

ℜ2= 2BℜeBℜ2+2 lnC .

(23)

Again, at the surface (r = ℜ) effective radial pres-

sure vanishes, i.e., peffr (r = ℜ) = 0. Comparing this

condition with Eq. (21) and using Eq. (23) we can repre-

sents the unknown constants a, b, B,C and Bg in termsof M and ℜ.

Variations of the metric potentials eν and eλ with

the radial coordinate (vide Fig. 1) imply that eν(r)|r=0

has a non-zero positive value and eλ(r)|r=0 = 1. These

are the necessary conditions for the solution to be freefrom physical and geometrical singularity. The metric

potentials increase non-linearly from the center of the

star and achieve maximum value at the surface of the

star. Variation of Bg with the coupling constant χ hasalso shown in Fig. 2 (right panel). From the display, one

can note that χ effectively reduces the Bg value. Here,

for simplification, we have assumed χ = 1 for which

5

Fig. 1 Variation of e−λ, eν w.r.t. the radial coordinate r forthe strange star candidate PSRJ 1614 − 2230.

Fig. 2 Variation of Bg w.r.t χ for the strange star candidatePSRJ 1614 − 2230.

Bg ∼ 47 MeV/fm3. But, while approaching χ → 0,

Bg increases and reaches it’s maximum at χ = 0 as ∼68 MeV/fm3.

5 Physical characteristics of the proposed

model

In this section we shall check the physical validity of our

model and also whether our proposed fluid distribution

is in stable equilibrium or not.

5.1 Density and pressure

We know that the density and pressure are the most im-

portant physical quantity in the study of a stellar sys-tem. Since we consider anisotropic fluid distribution in

the modified gravity in the specific form f(R, T ), so we

have to consider the effective value of those quantities.

Eqs. (16)-(18) provide us the effective density, effective

radial pressure and effective tangential pressure. Since,

as an example for strange star candidate, we have con-

sidered PSRJ 1614−2230, it will be most effective if we

look at the numerical values of those effective quantitiesfor this star. From Table 2 it is noticeable that effec-

tive central density is 7.412×1014 gm/cm3 where as its

value at the surface of the star is 4.85× 1014 gm/cm3.

Though the decreasing nature of the effective density isvery prominent but very high matter density through-

out the stellar system make it as a possible candidate of

ultra dense strange quark star [105,106]. The graphical

variation of ρeff (Fig 3, left panel) demonstrates that

it is maximum at the center and decreases graduallywith the radius of the star.

The variation of the effective radial pressure peffr

as well as the transverse pressure pefft (Fig 3, middle

panel) also maintain the same pattern, i.e., they aremaximum at the center and decreases nonlinearly with

the radius of the star. Their finite value at the center

of the star makes it a singularity free model of strange

star. The graphical variation of the pressures clearly

indicate that at the surface, effective radial pressurevanishes prominently whereas effective tangential pres-

sure retain a finite value which make the shape of the

star spheroidal in nature [107,108,103]. The origin of

this spheroidicity makes it a topic of future research.The anisotropic stress is represented in Eq. (19).

The beauty of this expression is that the anisotropy

will be directed inward when ∆ < 0, i.e. pefft < peffr

whereas ∆ > 0, i.e. pefft > peffr implies that direc-

tion of the anisotropy will be outward [109]. From thegraphical variation of the anisotropic stress (Fig 3, right

panel), it is clear that at the origin anisotropy van-

ishes, i.e. at the center the effective perpendicular pres-

sures are equal. Also we can observe that anisotropyincreases nonlinearly with the increase of the radius of

the star. The positive value of the anisotropic stress

helps to construct more compact object as described

by Gokhroo and Mehra [110]. The graphical represen-

tation of anisotropic stress shows that it achieves max-imum value at the surface of the star, which according

to the Deb et al. [161] is an inherent property of any

ultra dense compact object.

5.2 Energy Conditions

The distribution of matter in the space-time, as mea-

sured by an observer, is termed as energy conditions.

The positivity of these conditions implies the flow ofmatter should be along null or time-like world line.

Our proposed anisotropic fluid distribution will be con-

sistent with the energy conditions, i.e., the null en-

6

Fig. 3 Variation of ρeff (upper panel), peffr , p

efft (middle

panel) and anisotropic stress (lower panel) w.r.t. the radialcoordinate r for the strange star candidate PSRJ 1614-2230.

ergy condition (NEC), weak energy condition (WEC),

strong energy condition (SEC) and dominant energy

condition (DEC) only if it satisfies the following in-

Fig. 4 Variation of the different energy conditions w.r.t. theradial coordinate r for the strange star candidate PSRJ 1614-2230.

equalities simultaneously:

NEC : ρeff ≥ 0,

WEC : ρeff + peffr ≥ 0, ρeff + pefft ≥ 0,

SEC : ρeff + peffr + 2pefft ≥ 0,

DEC : ρeff − |peffr | ≥ 0, ρeff − |pefft | ≥ 0.

(24)

But several matter configurations violate the strong

energy condition, e.g. for a scalar field with a posi-tive potential and any cosmological inflationary pro-

cess [112] can violate this condition. In such a situation

we have to use an alternative theory of gravity since vi-

olation of SEC will violate the classical regime of GTR.From the graphical variation (Fig. 4) it is clear that

our model satisfies all the energy conditions for coupling

constant χ = 1 which will represent a stable anisotropic

strange star configuration of our consideration.

5.3 Stability Criteria

5.3.1 Herrera’s concept for stability analysis

The stability of our model can be checked by the help ofcausality condition of Herrera. According to this condi-

tion, square of the radial and tangential speed of sound

for anisotropic fluid distribution should follow the re-

sults 0 < v2rs < 1 and 0 < v2ts < 1. Another concept

for checking the stability of a stellar system is knownas cracking concept. It appears when the equilibrium

configuration of the fluid distribution has been per-

turbed. According to cracking concept the region for

which square of the transverse speed of sound is smallerthan the square of the radial speed of sound, is a poten-

tially stable region [113,114,115,116,117]. For the sta-

ble matter distribution Herrera [114] and Andreasson [117]

7

Fig. 5 Variation of v2rs and v2ts (upper panel), |v2rs−v2ts| (lowerpanel) w.r.t. the radial coordinate r for the strange star can-didate PSRJ 1614-2230.

demands that |v2rs − v2ts| ≤ 1. The cracking has to ap-

pear either from the anisotropy of fluid distribution or

due to the emission of incoherent radiation.

In Fig. 5, variations of v2rs and v2ts with respect tothe radius of the star has been shown and it is clear that

they remain within their specified range (0, 1) through-

out the stellar system, which confirm the causality con-

dition. Also it is observed clearly that the term |v2rs−v2ts|confirm the cracking concept.

5.3.2 The generalized Tolman-Oppenheimer-Volkoffequation

The modified form of the energy momentum tensor in

the framework of f(R, T ) gravity is presented in Eq. (7).

Hence the modified form of the Tolman-Oppenheimer-

Volkoff equation, which describes the stability of the

proposed model, is given by

− (ρ+pr)ν′

2 − dpr

dr+ 2

r(pt − pr)

+ χ(8π+2χ)

[

ddr(pr+2pt

3 )− dρdr

]

= 0. (25)

Fig. 6 Variation of different forces w.r.t. the radial coordi-nate r for the strange star candidate PSRJ 1614-2230.

Here the first term represents the gravitational force

(Fg), second term denotes the hydrodynamic force (Fh)

and third term indicates the anisotropic force (Fa). Thefourth term arises due to coupling between the matter

and the geometry, which can be termed as the force due

to modified gravity (Fmg). So we can conclude that sum

of all the forces are zero, i.e., Fg + Fh + Fa + Fmg = 0,which implies that our system is completely stable.

Fg = − (Bbr4+Bar2+2br2+B+a)Br

(br4+ar2+1)2ξ , (26)

Fh = r(2Bb2r6+3Babr4+6b2r4+Ba2r2+2Bbr2+6abr2+Ba+2a2−2b)2(br4+ar2+1)3ξ ,(27)

Fa = 2r

[

8b2χBgξr8+12bξ( 16

3 aχBg+B2)r6+Zr4+Y r2+8Bgχ2+Xχ+P

2(12π+5χ)ξ(br4+ar2+1)2

−Bbr4+Bar2+2br2+B+a4(br4+ar2+1)2ξ +Bg

]

, (28)

Fmg = − [B2b2r8+bB(Ba−4b)r6−3b(Ba+6b)r4+Pr2−B2+Q]rχ(12π+5χ)ξ(br4+ar2+1)3 , (29)

where, ξ = (χ+4π), P = (−B2a+(−a2+4b)B−18ba),

Z = [8Bg(a2+2b)χ2+

(

32Bg(a2 + 2b)π + 3B2a+ 4bB

)

χ+

12B2aπ], Y = [16aBgχ2+(64πaBg+3B2+7Ba+2b)χ+

12π(B2+Ba−2b)], P = 24(B− a2 )π, Q = Ba−6a2+6b

and X = (32πBg + 10B + a).

Fig. 6 shows the variation of different forces with

respect to the radius of the star. From the pictorial rep-resentation it is clear that inward pull of Fg is counter

balanced by the combined effect of Fh, Fa and Fmg

which acts along outward direction. For χ = 0 the usual

form of TOV equation as in GR will be retrieved be-

cause in this situation the force due to modified gravityvanishes, i.e., effect of coupling vanishes.

5.3.3 Adiabatic Index

The term adiabatic index Γ , ratio of two specific heat,

represents the stiffness of the EOS for a given den-

8

Fig. 7 Variation of the adiabatic indices Γr and Γt w.r.t. theradial coordinate r for the strange star candidate PSRJ 1614-2230.

sity profile [118]. It can be used to study stability ofa relativistic and non relativistic fluid sphere. Accord-

ing to Chandrasekhar [119] the dynamical stability of

the stellar model can be checked against an infinitesi-

mal radial adiabatic perturbation. Later on several re-

searchers [120,121,122,123,124,125,126,127] used theidea to different astrophysical system.

According to this stability condition Γ must be greaterthan 4

3 . For an anisotropic fluid sphere Γ can be rep-

resented as Γr and Γt, the radial adiabatic index and

tangential adiabatic index respectively [128]. Chan et

al. [115] and Heinzmann [129] predicted that adiabatic

index should exceed 43 inside an anisotropic, relativistic

and dynamically stable stellar system. Γr and Γt can

be expressed mathematically as follows

Γr =ρeff+peff

r

peffr

[

dpeffr

dρeff

]

, (30)

Γt =ρeff+p

efft

pefft

[

dpefft

dρeff

]

. (31)

Graphical representations of radial and tangential

adiabatic indices are shown in Fig. 7. One can note that

throughout the stellar distribution adiabatic indices are

greater than 43 and hence the model represents a stable

configuration.

5.3.4 Equation of State (EOS)

The simplest form of the barotropic equation of state

can be represented as pi = ωiρ, where ωi are the param-

eters along the radial and transverse direction. Since

we have considered the anisotropic strange star under

f(R, T ) gravity our radial and transverse EOS param-eter will be given by as follows:

ωeffr =

peffr

ρeff, ωeff

t =pefft

ρeff. (32)

Fig. 8 Variation of the EOS parameter ωeffr and ω

efft

w.r.t. the radial coordinate r for the strange star candidatePSRJ 1614-2230.

For simplicity we have considered the fluid distribu-tion to be spatially homogeneous, though different lit-

erature survey [130,131,132,133] shows the possibility

of space and time dependency of ω. From the graphical

representation of EOS (Fig. 8) it is clear that 0 < ωi <13 , i.e., the matter content of our proposed model isnon-exotic in nature [102] and decreases monotonically

towards the surface of the star.

5.3.5 Harrison-Zel′dovich-Novikov static stability

criteria

According to Harrison et al. [134] and Zel′dovich and

Novikov [135] the adiabatic index of a slowly deformed

matter is comparable with that of a pulsating star. Thisresult states that for a stable configuration the nature

of the mass will be increasing with respect to the central

density (i.e., dMdρc

> 0) and for an unstable configurationdMdρc

< 0 which can be provided as follows

M(ρc) =(Ψ−80πρc+15B+36Bg)R

3

2[ΨR2−80πR2ρc+15BR2+36R2Bg−36π−27] , (33)

dMρc

dρc= 72(12π+5)πR3(4π+3)

[ΨR2−80πR2ρc+15BR2+36R2Bg−36π−27]2 , (34)

where Ψ = −36πR2b + 192π2Bg − 192π2ρc − 27R2b +

36Bπ + 192πBg.

Fig. 9 conveys that though dMdρc

decreases with theincrease of ρc, it remains always positive throughout

the stellar structure. So our model is stable according

to Harrison-Zel′dovich-Novikov condition.

5.4 Compactification factor and surface redshift

We can calculate the effective mass for our static, spher-

ically symmetric and anisotropic fluid distribution through

9

Fig. 9 Variation of dMdρc

w.r.t. ρc for the strange star candidatePSRJ 1614-2230.

the relation

M eff =

∫ ℜ

0

4πr2ρeffdr

=

∫ ℜ

0

4πr2[

ρ+χ

4π

(

2ρ− pr + 2pt3

)]

dr

= m+

∫ ℜ

0

r2χ

(

2ρ− pr + 2pt3

)

dr. (35)

From the Fig. 10 (left panel) we can see that M eff

increases monotonically with the radius of the star and

as r → 0, M eff → 0. We can observe in Eq. (35) that

effect of χ is very much prominent. The term with χrepresents mass due to modified gravity, without this

term we will get the traditional mass as in the classical

physics. As χ → 0 the mass due to modified gravity will

also be zero.

The factor Mℜ defines the compactification of the

stellar system. Buchdahl [136] in one of his pioneering

work derived an upper limit for the allowed mass to

radius ratio, i.e., Mℜ < 4

9 , for a static spherically sym-

metric perfect fluid star. Later on Mak et al. [137] gen-eralized it for charged sphere. Jotania and Tikekar [138]

classifies the stellar objects into different categories on

the basis of the values of Mℜ as follows - (i) normal star:

M/ℜ ∼ 10−5, (ii) white dwarf: M/ℜ ∼ 10−3, (iii) neu-

tron star: 10−1 < M/ℜ < 1/4, (iv) ultra dense compactstar: 1/4 < M/ℜ < 1/2 and (v) black hole:M/ℜ = 1/2.

The expression of the compactification factor is given

by

u(r) = Meff (r)ℜ = 1

ℜ [m+∫ ℜ0 r2χ

(

2ρ− pr+2pt

3

)

dr]. (36)

The graphical representation of the above factor for

our model is shown in Fig. 10 (lower panel). We can notevery clearly that it is a monotonic increasing function

of the radius of the star and its maximum value implies

that our model corresponds to an ultra dense compact

object. Here also the effect of χ is prominent. It plays

a significant role in determining the compactification

factor of the strange stars in f(R, T ) gravity.

Fig. 10 Variation of mass (upper panel), compactness (lowerpanel) w.r.t. the radial coordinate r for the strange star can-didate PSRJ 1614-2230.

The surface redshift (Zs) and gravitational redshift

(Z) are represented respectively as

Zs =1√

1− 2 u− 1, (37)

Z = e−ν(r)

2 − 1 =1√

eBr2+2 ln(C)− 1. (38)

Barraco and Hamity [139] proved that for an isotropicstar and in absence of cosmological constant Zs < 2.

According to Bohmer and Harko [140] surface redshift

for an anisotropic star can reach maximum higher value

Zs ≤ 5, in presence of cosmological constant. Though

according to Ivanov [89] maximum acceptable value ofsurface redshift will be 5.211.

We have plotted the gravitational redshift Z for our

model in Fig. 11, where it is clear that it decreases with

the radius of the star. It’s value on the surface of the

10

star, i.e., the surface redshift is 0.271 which strongly

confirms the acceptance of our model as a strange star.

Fig. 11 Variation of gravitational redshift w.r.t. the radialcoordinate r for the strange star candidate PSRJ 1614-2230.

6 Discussions and concluding remarks

Motivation of this work is to study the relativistic strange

star under the framework of f(R, T ) gravity where spher-

ically symmetric spacetime is of Tolman-Kuchowicz type.With the help of SQM distribution, demonstrated as in

the famous MIT bag EOS [Eq. (15)], we explore strange

quark star. As anisotropic matter distribution is con-

sidered here, incorporation of the EOS does not makethe system over determined. Solving EFEs [Eqs. (9)-

(11)] in modified gravity, using Tolman-Kuchowicz met-

ric as well as MIT bag EOS, we have obtained differ-

ent physical parameters. Numerical values of different

physical quantities have been calculated and shown intabular form and their graphical variations have been

shown w.r.t. the radial co-ordinate r. Though graph-

ical representation has been made for the candidate

PSR J 1614 2230, we have shown the respective valuesfor different strange star candidates, like V ela X − 1,

PSR J 1903 + 327, Cen X − 3 and SMC X − 1 also.

Geometry of the space-time can be specified by the

metric potentials ν(r) and λ(r). From variation of ν(r)

and λ(r) in Fig. 1, it’s clear that eν(r)(r = 0) 6= 0 and

positive Also eλ(r)(r = 0) = 1, which are the necessary

conditions to achieve the non-singular solutions fromphysical as well as geometrical aspects. Both the func-

tions eν(r)) and eλ(r) increases non-linearly toward the

surface from the center.

In f(R, T ) gravity, a constant χ arises due to the

coupling between matter and geometry. Assuming χ =

1 in anisotropic condition, we can calculate the bag con-

stant Bg which remains in the range (47-48) MeV/fm3,

though the predicted range for stable SQM matter dis-

tribution is (55-75) MeV/fm3 [67,60] in GR. However,

very recently Aziz et. al [141] have discussed about thepossible wide range (41.58-319.31) MeV/fm3 for Bg on

the basis of observational data for different strange star

candidates. In our model the coupling parameter χ ef-

fectively reduces to Bg in case of f(R, T ) gravity. In-corporating χ = 0 in anisotropic condition, we get Bg

in the range (68-69) MeV/fm3. Biswas et al. [70] have

previously shown that χ is blameworthy behind this

effective reduction of Bg. The coupling constant plays

a very crucial role to study the strange star. Depend-ing on its value we can study different aspect of Bag

constant.

Bauswein and his co-workers [142,143] predicted a

strange star-strange star merger process which increasesthe probability of SQM hypothesis if one can detect

Gravitational Wave from this merging phenomenon. It

is to note that some simple features of the GW signals

may reveals whether Strange Star (SS) or Neutron Star

(NS) merger had produced the emission. In case of SSmergering, the maximal frequency during the inspiral

and the frequency of ringdown of the postmerging rem-

nant are higher than those of NS merging. A particular

choice of the EOS will make the frequencies similar forSS and NS merging procedure. In such a case one can

take the help of Gravitational Wave luminosity spec-

trum to reveal the features. As a result to differenti-

ate between NS and SS merging one can use the ratio

of energy emitted during the inspiral phase to the en-ergy radiated away in the postmerger stage. Hopefully,

the upcoming advanced GW detectors LIGO [144,145]

and VIRGO [146] may provide more valuable data re-

garding the signals from high density compact objectbinaries [147].

Other forms of self-bound matter like pion conden-

sation may lead to the stellar objects similar to SS [148,

149,150]. So we can generalize our study to these states

of matter also provided we have the knowledge of thespecific EOS. In addition to GW measurements the

cooling history of NS, SS and self-bound pion condensed

stars are different [81,151,152,153]. Since merger rem-

nants are the promising source to develop better under-standing regarding the origin of GW spectra, one can

claim in future a better understanding of SS and BH

through SS-BH merger.

Now the current study can be summarized as fol-

lows:

(i) Density and pressure: The effective density,

effective radial and tangential pressure are maximum

at the center of the star and decrease monotonically

11

Table 1 Mass and radius of different strange star candidates [154]

Case Stars Mass Radius Mℜ

Bg(χ = 1) Bg(χ = 0) Zs

(M⊙) (ℜ in km) (MeV/fm3) (MeV/fm3)

I PSR J 1614 2230 1.97±0.04 10.3 0.191 47.55 68.01 0.271II V ela X − 1 1.77±0.08 9.99 0.177 47.64 68.23 0.2432III PSR J 1903 + 327 1.667±0.021 9.82 0.17 47.67 68.4 0.23IV Cen X − 3 1.49±0.08 9.51 0.157 47.59 68.41 0.206V SMC X − 1 1.29±0.05 9.13 0.1413 47.334 68.17 0.18

Table 2 Determination of model parameters a, b, C, B, central density, surface density, radial pressure for different strangestar candidates

Case a b B C ρeff (r = 0) ρeff (r = ℜ) peffr (r = 0)

(km−2) (km−4) (km−2) (gm/cm3) (gm/cm3) (dyne/cm2)

I 0.00459 0.0000118 0.00292 0.6731 7.412×1014 4.85×1014 6.0331×1034

II 0.0044 0.000011 0.00275 0.7005 7.1131×1014 4.871×1014 5.286×1034

III 0.0043 0.0000106 0.00267 0.715 6.967×1014 4.879×1014 4.924×1034

IV 0.0042 0.0000099 0.002523 0.7393 6.71×1014 4.88×1014 4.311×1034

V 0.00397 0.0000091 0.002363 0.7676 6.41×1014 4.863×1014 3.653×1034

towards the surface of the star, shown in the Fig. 3as well as in Table 2. However, it is to be noted that

the effective density at the surface of the star reduces to

34.56 % of its value at the center. The anisotropic stress

is minimum at the center of the star and increases non-

linearly with the radius of the star and acquire the max-imum value at the surface of the system. The tremen-

dous pressure at the center make the system ultra com-

pact object. The spheroidicity in our system arises due

to this maximum anisotropic force at the surface of thestar. The pictorial variation of EOS Fig. 8 states clearly

that 0 < ωi <13 , i.e., our system consists of non-exotic

matter.

(ii) TOV equation: In f(R, T ) gravity, our pro-

posed model satisfies the force condition, i.e., the gener-

alized TOV equation [Eq. 25]. From Fig. 6, it is distinct

that the effect of gravitational force (Fg) stabilizes theunified effect of hydrostatic force (Fh), anisotropic force

(Fa) and the newly modified gravity force (Fmg), orig-

inated due to the matter-geometry coupling in f(R, T )

gravity.

(iii) Energy Conditions: Our system is consis-

tence with all the energy conditions, e.g. NEC, WEC,SEC and DEC for both the radial and tangential cases

of pressure. Variations of all these conditions have been

shown graphically (Fig. 4) which re-confirms the phys-

ical stability and acceptability of proposed model.

(iv) Stability: To examine the stability of the sys-

tem, we have checked Herrera’s cracking condition andthe causality conditions as |v2rs − v2ts| ≤ 1, 0 < v2rs < 1

and 0 < v2ts < 1. Fig. 5 also manifest that all the in-

equality criteria are well satisfied This enhances the

physical acceptability of the system with the backgroundof sound velocity of the system.

From Fig. 7, it’s clear that both the adiabatic indices

(i.e., Γr and Γt) remain within the critical value 43 [128]

through out the system which establish that infinites-

imal adiabatic perturbation can’t distort the stabilityof the model. Harrison-Zel′dovich-Novikov’s static sta-

bility criteria is well satisfied, Fig. 9 shows that dmdρc

is

always positive throughout the system.

(v) Buchdahl condition: Variation of the effec-

tive mass has been featured in Fig. 10 (upper panel)which emphasizes the regularity of M(r)eff as for r →0,M(r)eff → 0. According to Buchdahl condition [136],

for static spherically symmetric perfect fluid distribu-

tion the mass radius ratio, i.e., Mℜ ≤ 4

9 (Table 1) whichis also satisfied for every strange star candidate.

(vi) Compactification factor and redshift: We

have studied the compactification factor u and shown

graphically in Fig. 10 (lower panel), where the revealed

features highly recommend for a strange star candi-date. The gravitational redshift (Fig.11) reduces con-

tinuously from center to surface. The surface redshift

is high enough (∼ 0.18 − 0.27) for all the stars which

strongly indicates the possibility of stable star configu-ration.

At this point we would like to conduct a compara-

tive study between our work in f(R, T ) gravity and that

of the investigations by other authors [156,157,158], es-

pecially by Asthashenok et al. [157] in f(R) gravitywhich are as follows:

(i) In f(R) gravity, f(R) contains 1st and higher order

terms of scalar curvature R, connected through a con-

12

stant parameter α. Setting α = 0 gives the simpler form

as f(R) = R. But in our work of f(R, T ) gravity, where

Ricci scalar R and the trace of the energy-momentum

tensor T are connected through a coupling constant pa-

rameter χ, arises due to the matter-geometry coupling.

(ii) In the above study, α is the key factor and dif-

ferent results with the variation of α have been investi-

gated in their study whereas in our case the key factor

is χ and setting χ = 0 gives the results for GR as wellas non-zero χ represents the alternative gravity.

(iii) The study [157] is based on general relativistic

stars like neutron stars and quark stars. It’s not explic-

itly clear whether the model is energetically stable or

not. Our study is specifically based on strange quarkstars where we have calculated different unknown pa-

rameters like the matter density, radial and tangential

pressure, value of EOS parameter, redshift etc. Besides,

consistency of our model with various stability condi-tions at a time, is also worthy to mention.

(iv) Though Asthashenok et al. [157] in their study

have also mentioned about the MIT bag model EOS

and and hence bag constant Bg. But, here we have cal-

culated the range for Bg in f(R, T ) gravity. We havealso shown that value of Bg reduces in f(R, T ) due to

the effect of χ. Setting χ = 0 one can get the higher

value of Bg which exactly matches to the predicted

range [67,60].

In connection to the above comparative discussion

we would also like to make comments on the current

article regarding strange stars that the study shows re-

markably different results in comparison to the earlier

works [159,160,161,162,163]. Here, one can find a veryinteresting feature that the value of bag constant Bg

gets effected under modified gravity. However, the new

results can be mentioned as follows:

(i) In this study, we have shown the existence of sta-

ble strange quark stars with lower bag value in the range(47-48) Mev/fm3 whereas earlier works obtained [159,

160] or assumed [161,162,163] Bg in the higher range

for strange star models in GR [159,160] as well as in the

modified theories of gravity [161,162,163]. Our calcula-tions in this study, therefore, clearly illustrate the ef-

fect of f(R;T ) gravity. Here, the matter-geometry cou-

pling constant χ, effectively reduces the value of Bg.

Besides, minimization of the matter-geometry coupling

(i.e. χ = 0) in this study, gives back the higher val-ues of Bg in the range (55-75) Mev/fm3 as proposed by

Farhi [67] and Alcock [60] for stable strange star model.

(ii) Following the literature [159,160,161,162,163],

we have studied all the energy conditions, TOV equa-tion, Herrera’s cracking conditions, Buchdahl limit, adi-

abatic index and the EOS parameter. We have also

shown the variations of all these parameters w.r.t. r

for strange star candidate PSR J 1614 2230. Our pro-

posed model of anisotropic strange star with Tolman-

Kuchowicz metric under modified f(R, T ) gravity, shows

it’s consistency with all the stability criteria and indi-

cates the physical acceptability and stability. Proposedmodel also satisfy the Harrison-Zel’dovich-Novikov cri-

teria for stability. It’s mention worthy that no earlier

work [159,160,161,162,163] shows it’s consistency with

all the static stability criteria as our study does effec-tively.

We finally comment that our model is completely

free from both physical and geometrical singularities

and represents a strange quark star with highly dense

SQM distribution. It can be observed that under f(R, T )gravity, study of anisotropic strange star using Tolman-

Kuchowicz metric, provides such a model which satisfy

all the stability criteria and perfect for studying dif-

ferent features of anisotropic strange stars. Along withthis, another fascinating feature is the reduction in bag

value. The coupling constant χ caused due to the cou-

pling between matter and geometry, effectively reduces

Bg and also the value of v2rs, which is constant (13 ) in

GR. For both the cases, χ = 0 retrieves the result thatexactly matches to GR results. However, as our model

works well within the low range (47-48) MeV/fm3 for

χ = 1 and (68-69) MeV/fm3 for χ = 0, therefore the

present investigation may be generalized by consider-ing more realistic EOS obtained from QCD simula-

tions [155] to make it also suitable for the higher range

(41.58-319.31) MeV/fm3 [141].

Acknowledgement

SR is thankful to the Inter University Centre for As-

tronomy and Astrophysics (IUCAA) for providing Vis-

iting Associateship. SB is thankful to DST-INSPIRE

[IF 160526] for financial support and all type of facili-ties for continuing research work.

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