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In the Einstein--Hilbert action, if the Ricci scalar R is replaced by a function of R and the trace of the stress--energy tensor T, the modified action in
f(R,T)− gravity is expressed asS=116π∫f(R,T)√−gd4x+∫Lm√−gd4x,
(1) where det
(gμν)=g . The source term of the matter Lagrangian densityLm defines a stress tensor asTμν=−2√−gδ(√−gLm)δgμν.
(2) Following Harko et al.'s approach [88], Eq. (2) reduces to
Tμν=gμνLm−2∂Lm∂gμν.
(3) Variation in the action with respect to
gμν yields the field equations(Rμν−∇μ∇ν)fR(R,T)+gμν◻fR(R,T)−12f(R,T)gμν=8πTμν−fT(R,T)(Tμν+Θμν),
(4) provided
fR(R,T)=∂f(R,T)/∂R andfT(R,T)=∂f(R,T)/∂T .∇μ denotes the covariant derivative, whereas the box operator◻ is defined as◻≡1√−g∂∂xμ(√−ggμν∂∂xν)
with
Θμν=gαβδTαβδgμν.
The conservation equation [128] yields
∇μTμν=fT(R,T)8π−fT(R,T)[(Tμν+Θμν)∇μlnfT(R,T)+∇μΘμν−12gμν∇μT].
(5) Therefore,
∇μTμν≠0 implies that the conservation equation no longer holds inf(R,T) theory. Using Eq. (3), the tensorΘμν is found to beΘμν=−2Tμν+gμνLm−2gαβ∂2Lm∂gμν∂gαβ.
(6) To complete the field equations, we assume an anisotropic fluid source
Tμν=(ρ+pr)uμuν−ptgμν+(pr−pt)gvμvν,
(7) where
uν is the four-velocity, satisfyinguμuμ=−1 anduν∇μuμ=0 ,ρ is the matter density, andpr andpt are the radial and transverse pressures, respectively. If we define the isotropic pressure as−P=Lm=(pr+2pt)/3 [88], then Eq. (6) reduces toΘμν=−2Tμν−Pgμν.
(8) Further, the functional
f(R,T) is chosen to bef(R,T)=R+2χT [88], whereχ is the coupling constant. Now the field equations (5) take the formGμν=8πTμν+χTgμν+2χ(Tμν+Pgμν).
(9) For
χ=0 , one can recover the general relativistic field equations. The linear expression inf(R,T) solves several cosmological and astrophysical related problems. By substitutingf(R,T)=R+2χT and Eq. (8) in Eq. (5), we obtain∇μTμν=−χ2(4π+χ)[gμν∇μT+2∇μ(Pgμν)].
(10) Thus, the conservation equation in Einstein's gravity can be recovered for
χ=0 . -
To determine the field equations we assume an interior space-time of the form
ds2−=eνdt2−eλdr2−r2(dθ2+sin2θdϕ2).
(11) For the space-time expressed in Eq. (11), the field equation (9) becomes
8πρeff=e−λ(λ′r−1r2)+1r2,
(12) 8πpreff=e−λ(ν′r+1r2)−1r2,
(13) 8πpteff=e−λ4(2ν″+ν′2+2(ν′−λ′)r−ν′λ′),
(14) where
ρeff=ρ+χ24π(9ρ−pr−2pt),preff=pr−χ24π(3ρ−7pr−2pt),pteff=pt−χ24π(3ρ−pr−8pt).
Now the decoupled field equations (12)-(14) are as follows:
ρ=e−λ48r2(χ+2π)(χ+4π)[rλ′{16(χ+3π)−rχν′}+16(χ+3π)(eλ−1)+rχ{2rν″+ν′(rν′+4)}],
(15) pr=e−λ48r2(χ+2π)(χ+4π)[r{χλ′(rν′+8)−2rχν″+ν′(20χ−rχν′+48π)}−16(χ+3π)(eλ−1)],
(16) pt=e−λ48r2(χ+2π)(χ+4π)[r{−λ′{r(5χ+12π)ν′+4(χ+6π)}+2r(5χ+12π)ν″+r(5χ+12π)ν′2+8(χ+3π)ν′}+8χ(eλ−1)].
(17) To solve the field equations, we have to assume some of the physical quantities that satisfy strict physical constraints.
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The Kasner coordinate transformation [129] shows that the exterior Schwarzschild vacuum is class-two space--time. Using the following transformations,
X=Rsint√R2+16m2,Y=Rcost√R2+16m2,Z=∫√1+256m4(R2+16m2)3dR,
where
R=√8m(r−2m) andr2=x2+y2+z2 , the well-known Schwarzschild vacuum reduces tods2=−dx2−dy2−dz2+dX2+dY2−dZ2.
(18) This means that the Schwarzschild exterior space--time can be embedded into a six-dimensional pseudo-Euclidean manifold. This method was extended for the general four-dimensional space--time of the form (11) by Gupta & Goel [ 130]. The chosen coordinate transformations were
z1=keν/2cosh(tk),z2=keν/2sinh(tk),z3=f(r),z4=rsinθcosϕ,z5=rsinθsinϕ,z6=rcosθ,
which transform (11) into
ds2=(dz1)2−(dz2)2∓(dz3)2−(dz4)2−(dz5)2−(dz6)2,
(19) with
[f′(r)]2=∓[−(eλ−1)+k2eνν′2/4] .Equation (19) also implies that the interior line element (11) can be embedded in six-dimensional pseudo-Euclidean space. However, if
(dz3)2=[f′(r)]2=0 , it can be embedded in 5D Euclidean space, i.e.,[f′(r)]2=∓[−(eλ−1)+k2eνν′24]=0,
(20) which implies
eλ=1+k24ν′2eν
(21) i.e., Eq. (19) reduces to
ds2=(dz1)2−(dz2)2−(dz4)2−(dz5)2−(dz6)2,
(22) a class-one spacetime.
The same condition (21) was originally derived by Karmarkar [120] in the form of the components of the Riemann tensor as
RrtrtRθϕθϕ=RrθrθRϕtϕt+RrθθtRrϕϕt.
(23) Pandey & Sharma [ 121] pointed out that the Karmarkar condition is only a necessary condition to become class one; they discovered the sufficient condition as
Rθϕθϕ≠0 . Hence, the necessary and sufficient condition to be a class one is to satisfy both the Karmarkar and Pandey-Sharma conditions. In terms of the metric components, Eq. (23) can be written as2ν″ν′+ν′=λ′eλeλ−1,
(24) which, on integration, becomes the following:
eν=(A+B∫√eλ−1dr)2.
(25) where A and B are the constants of integration. In general relativity, there is no class-one exterior, as the existing Schwarzschild's exterior itself is a class-two space-time. It has been shown that the two class-one isotropic-neutral solutions in general relativity, i.e., the Schwarzschild uniform density model and Kohler-Chao infinite boundary model are the only two possible solutions [131]. However, Mustafa et al. [132] have also shown that these two isotropic-neutral solutions still exist in
f(R,T)− theory as well, although the constant density model in GR can have decreasing density inf(R,T)− gravity and the infinite boundary Kohler-Chao solution can have a finite boundary, where the pressure vanishes owing to thef(R,T)− term.Once one of the metric functions is determined, say, for
eλ , then the EoS is fixed. Using Eqs. (12), (13), and (25), we obtain8πpreff=−1r2+[c1r+1r2∫(1−8πr2ρeff)dr].[2B√eλ−1r(A+B∫√eλ−1dr)+1r2].
(26) However, because of the highly coupled and nonlinear field equations (15)-(17), finding an exact expression for the EoS in terms of
pr andρ is very difficult; instead, one can represent it graphically . -
Solving the field equations in
f(R,T)− gravity exactly is a challenging task because of the highly coupled nonlinear differential equations. To simplify the problem, we have adopted the embedding class-one approach, which is an application of four-dimensional space-time. Here, we propose a new metric function:eλ=1+ar2ebr2+cr4.
(27) The physical motivation for using this ansatz is that it not only is a new metric function but also satisfies the required criteria of
grr , i.e.,eλ(0)=1 , and being an increasing function of r. Satisfying these properties increases the probability of obtaining well-behaved solutions.Using Eq. (27) in Eq. (25), we get
eν=[A+B√2c√aebr2+cr4F(2cr2+b2√2c)]2,
(28) where
F(x) is Dawson's integral defined byF(x)=e−x2∫x0eτ2dτ=√π2e−x2erfi(x).
Here
erfi(x) is the usual imaginary error function. The variations in the metric functions are presented in Fig. 1.Figure 1. (color online) Variation in metric functions with radial coordinate for Vela X-1 (
M=1.77±0.08M⊙,R=9.56±0.08km ) withb=0.0005/km2 andc=0.000015/km4 .Plugging the metric functions into the field equations (15)-(17), one can write
ρ=√aebr2+cr46(χ+2π)(χ+4π)f3(r)(ar2ebr2+cr4+1)2[(χ+3π)×f2(r)F(2cr2+b2√2c)+√cf1(r)],
(29) pr=√aebr2+cr46(χ+2π)(χ+4π)f3(r)(ar2ebr2+cr4+1)2[√cf4(r)+2√2aBf5(r)ebr2+cr4F(2cr2+b2√2c)],
(30) Δ=r2√aebr2+cr4(aebr2+cr4−b−2cr2)2(χ+4π)f3(r)(ar2ebr2+cr4+1)2[F(2cr2+b2√2c)×√2aBebr2+cr4+2√c(A√aebr2+cr4−B)],
(31) pt=pr+Δ.
(32) where
f1(r)=4A(χ+3π)(ar2ebr2+cr4+2br2+4cr4+3)×√aebr2+cr4+Bχ(2ar2ebr2+cr4+br2+2cr4+3),f2(r)=2√2aBebr2+cr4(ar2ebr2+cr4+2br2+4cr4+3),f3(r)=√2B√aebr2+cr4F(2cr2+b2√2c)+2A√c,f4(r)=B[χ(10ar2ebr2+cr4−br2−2cr4+9)+24π(ar2ebr2+cr4+1)]−4A√aebr2+cr4[3π×(1+ar2ebr2+cr4)−r2χ(b−aebr2+cr4+2cr2)],f5(r)=r2χ(b−aebr2+cr4+2cr2)−3π(ar2ebr2+cr4+1).
The variations in density, pressures, EoS, anisotropy, and EoS parameter are displayed in Figs. 2, 3, 4, 5, and 6, respectively. The nonpolytropic nature of the EoS can be clearly seen in Fig. 4, i.e., the pressure vanishes when the surface density is greater than zero.
Figure 2. (color online) Variation in density with radial coordinate for Vela X-1 (
M=1.77±0.08M⊙,R=9.56±0.08km ) withb=0.0005/km2 andc=0.000015/km4 .Figure 3. (color online) Variation in pressures with radial coordinate for Vela X-1 (
M=1.77±0.08M⊙,R=9.56±0.08km ) withb=0.0005/km2 andc=0.000015/km4 .Figure 4. (color online) Equation of state for Vela X-1 (
M=1.77±0.08M⊙,R=9.56±0.08km ) withb=0.0005/km2 andc=0.000015/km4 . -
Any new solutions must be analyzed through various physical tests. After satisfying all the physical constraints, one can proceed further with modeling physical systems.
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All the physical compact stars are believed to be in equilibrium states. Such equilibrium states can be tested using the equation of hydrostatic equilibrium or the modified TOV equation, which is expressed as
−ν′2(ρ+pr)−dprdr+2Δr+χ3(8π+2χ)ddr(3ρ−pr−2pt)=0.
(33) Here, the first term is gravity (
Fg ), second term is the pressure gradient (Fh ), third term is anisotropic force (Fa ), and last term is the additional force (Fm ) inf(R,T)− gravity. The fulfillment of the modified TOV equation is presented in Fig. 7. It shows that the forces owing to gravity, pressure gradient, andFm are the highest inχ=−1 ; however, the anisotropic force is the lowest. This enables it to hold more mass than others for small values ofχ . Asχ increases,Fg,Fm , andFh decrease althoughFa increases slightly; therefore, the maximum mass that can be held by the system also reduces. -
We are aware of
f(R,T)− gravity as an extension of general relativity, which provides a constraint on the maximum speed limit. All the particles with nonzero rest mass travel at subluminal speeds, i.e., at less than the speed of light (causality condition). The velocity of sound in a medium must also satisfy the causality condition, and it determines the stiffness of the related EoS. Therefore, one can determine the speed of sound in a stellar medium to relate its stiffness. The most stiff EoS is Zeldovich's fluid (pz=ρz ), in which sound travels exactly at the speed of light. The speed of sound can be determined asv2r=dprdρ,v2t=dptdρ.
(34) Fig. 8 presents a plot of the speed of sound with respect to the radial coordinate. It can be seen that the speed of sound is maximum for
χ=−1 and decreases with increase inχ . This imply that the solution leads to a stiffer EoS at small values ofχ .Figure 8. (color online) Variation in speed of sound with radial coordinate for Vela X-1 (
M=1.77±0.08M⊙,R=9.56±0.08km ) withb=0.0005/km2 andc=0.000015/km4 .The speed of sound can also be related to the stability of the configuration. As per Abreu et al. [133], the stability factor can be defined as
v2t−v2r . So long asvr>vt or−1⩽v2t−v2r⩽0 , the system is generally considered stable, otherwise it is considered unstable. The variation in the stability factor is displayed in Fig. 9, which clearly indicates that the solution is stable. -
Another parameter that determines the stability and stiffness of an EoS is the adiabatic index, which is defined as the ratio of specific heat at constant pressure to that at constant volume. For any fluid distribution, the adiabatic index can be determined as [134]
γ=pr+ρprv2r.
(35) In polytropic cases, the pressure and density are linked by the EoS
pr=Kργ , and the adiabatic indexγ is constant throughout the stellar interior. It is well-known that white dwarf stars are supported by degenerate electron pressure, and the adiabatic indexes for nonrelativistic and ultrarelativistic electrons are 5/3 and 4/3, respectively. As per Bondi's perceptions, a polytropic stellar fluid distribution is stable ifγ>4/3 in the Newtonian limit. Ifγ⩽1 , contraction is possible, and it is catastrophic ifγ<1 . This was extended by Chan et al. [135] to anisotropic fluids; however, it is no longer valid for anisotropic fluids, for which the stable limit ofγ depends on the nature of anisotropy and its initial configuration. If anisotropyΔ>0 , the stable limit will still beγ>4/3 ; however, ifΔ<0 , stability is still possible even ifγ<4/3 . Variation in the adiabatic index is presented in Fig. 10. For various values ofχ , the central adiabatic index is accumulated around 2 and increases outward. Unlike the polytropic case, the adiabatic index is an increasing function of r and its central values decide whether the stellar configuration is stable or not. -
This criterion, originally established by Chandrasekhar, is used to analyze the stability of stellar configurations under radial perturbations [136]. Harrison et al. [137] and Zeldovich & Novikov [ 138] simplified this method later. The static stability criterion imposed the condition that if
∂M/∂ρc is greater than zero, the system is stable; otherwise, it is unstable. To verify it, we have calculated the mass as a function ofρc , expressed asM(ρc)=R2(1−11+aR2ebR2+cR4).
(36) Here, a is a very complicated function of
ρc and, therefore, we avoid a definition. The variation of mass with respect to the central density is presented in Fig. 11, from which one can conclude that the stability is enhanced with increase inχ . This is because the range central density is higher for saturating the mass whenχ=1 than whenχ=−1 . This implies that the stable range of density during radial oscillation is greater for higher values ofχ . Thus, it can be concluded that this solution is stable under radial perturbations. -
After confirming all the stability tests, the nature of matter content, i.e., either normal (baryonic, hadronic, etc.) or exotic (dark matter, dark energy, etc.), can be identified using energy conditions. The satisfaction or violation of certain energy conditions will imply the nature of matter. These energy conditions are expressed as follows:
Null:ρ+pr⩾0,ρ+pt⩾0,Weak:ρ+pr⩾0,ρ+pt⩾0,ρ⩾0,Strong:ρ+pr⩾0,ρ+pt⩾0,ρ+pr+2pt⩾0,Dominant:ρ⩾|pr|,ρ⩾|pt|.
Fig. 12 indicates that all the energy conditions are satisfied by the solution and, therefore, the matter content is normal.
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The boundary conditions ensure that the internal
M− and external space--time geometryM+ match in a smooth manner. The matching condition procedure permits the determination of the total set of constant parameters that depict the stellar model and the macrophysical observables i.e., the total mass M and the radius R of the anisotropic relativistic fluid sphere. To so do, we use the well-known Israel-Darmois junction conditions [126, 127]. As usual, we assume that the exterior space--time geometry in the form of Schwarzschild's vacuum is expressed asds2+=(1−2mr)dt2−(1−2mr)−1dr2−r2(dθ2+sin2θdϕ2).
(37) However, we must keep in mind that to avoid singularity, one must satisfy the condition
r>2m . Therefore, at the surface of the stellar objectΣ≡r=R , the Israel-Darmois matching conditions lead to, first, the interiorM− and exteriorM+ manifolds expressed by Eqs. (27)-(28) and (37), respectively, induced onΣ as an intrinsic curvature described by the metric tensorgμ,ν . The continuity of the metric tensor components across the boundaryΣ establishes the first fundamental form, which implies thatds2|Σ=0 . Second, the conditions also induceM− andM+ onΣ as an extrinsic geometry represented by the extrinsic curvature tensorKμ,ν , whereμ andν run over(x1,x2,x3)=(r,θ,ϕ) . The continuity of the componentKr,r of the extrinsic curvature tensor across the boundary of the stellar objectΣ infers the second fundamental form, which perusespr(r)|Σ=0 , and the continuity of theKθ,θ andKϕ,ϕ components prompt the determination of the complete mass interior to the spherem(R)=M . In this respect, the following remarks are appropriate. In the first situation, to acquire the complete mass governed by the anisotropic relativistic fluid sphere, at least three equivalent methods can be used: the first method is to integrate the energy densityρ over the range[0,R] , the second method is to impose the continuity of the radial components of the metric tensorgμ,ν acrossΣ , and the third method is based on the continuity of the angular components of the extrinsic curvature tensorKμ,ν . In the second situation, a vanishing radial pressure at the boundary of the stellar structure determines the radius R, i.e., the size of the stellar object; further, it avoids an indeterminate expansion of the stellar model surrounding the matter field within the region0⩽r⩽R . Therefore, at the surfaceΣ≡r=R , we obtain the first fundamental formds2−|r=R=ds2+|r=R , which explicitly readse−λ(R)=1−2MR=eν(R).
(38) Using Eq. (38), we obtain
a=2Me−R2(b+cR2)R2(R−2M),
(39) A=√1−2MR−B√2c√aebR2+cR4F(2cR2+b2√2c).
(40) Generally, when modeling compact stars, the pressure at the surface should vanish, i.e., the second fundamental form
pr(R)=0 . This condition allows us to determine one more constant as follows:B=4√c√1−2MR√aR2ebR2+cR4[R2χ(2cR2−aebR2+cR4+b)−3π(aR2ebR2+cR4+1)][√cR{χ(bR2+2cR4−9−10aR2ebR2+cR4)−24π(aR2ebR2+cR4+1)}+2√2aRebR2+cR4F(2cR2+b2√2c){3π(aR2ebR2+cR4+1)−R2χ(b−aebR2+cR4+2cR2)}−2aR√2e−(bR2+cR4)F(2cR2+b2√2c){3π(aR2e−(bR2+cR4)+1)−R2χ(b−ae−(bR2+cR4)+2cR2)}]−1.
(41) The parameters b and c are treated as free, whereas M and R are obtained from observed evidence.
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There are several ways of determining the stiffness of an EoS, e.g., by determining the adiabatic index, sound speed, etc. However, the sensitivity to stiffness is found to be very sharp in
M−R andM−I graphs. In fact, theM−I graph is the most effective and sensitive to the stiffness of an EoS. Fig. 14 displays the variation in mass with respect to the radius. In the preceding sections, we have already noted that the EoS is the stiffest forχ=−1 , and asχ increases, the stiffness reduces. Therefore, the mass that can be held by the corresponding EoS will also reduce asχ increases. The same observations can be made from theM−R curve in Fig. 14. To compare it with theM−I curve, one must establish how to determine the moment of inertia (I). Adopting the Bejger & Haensel [ 139] formula, one can determine the value of I corresponding to a static solution. It is expressed asI=25(1+(M/R)⋅kmM⊙)MR2.
(42) The change in I with respect to mass is presented in Fig. 15. Again, we can verify that the EoS is most stiff for small values of
χ . The transition at the peak of theM−I curve is sharper than that of theM−R curve, which indicates its sensitivity to the stiffness of the EoS.Further, the generated
M−R curve is also fit with observed results for a few well-known compact stars. As examples, we have matched the results obtained for PSR J1614-2230, Vela X-1, Cen X-3, and SAX J1808.4-3658. From theM−R curve fits for these compact stars, it is possible to predict the probable range of I for the abovementioned objects from theM−I curve. -
In this study, we have successfully embedded the class-one technique in the area of
f(R,T)− gravity. This procedure not only simplifies the exploration of new exact solutions within thef(R,T)− theory but also facilitates the investigation of the theory of compact stars in the same realm. The solution was analyzed through various physically stringent conditions such as the causality condition, energy conditions, satisfaction of TOV equation, stability criteria through Bondi's condition, Abreu et al. condition, and static stability criterion. As we increase the coupling constantχ from−1 to 1, the density, pressures,ω , speed of sound, and interior redshift increase. This increase in energy density may lead to the generation of exotic particles such as quarks, hyperons [140, 141], and kaon condensation [142], which soften the EoS. Therefore, in theM−R andM−I curves, one can see thatMmax andImax increase with a decrease in the coupling constant, which is a direct consequence of the stiffening of the EoS at lowχ . This outcome can also be cross-checked with the internal velocity of sound. From Fig. 8, we can see thatvr/t(χ=−1) is always greater thanvr/t(χ=1) , i.e., the EoS is stiffer for the former condition as compared to the latter.Although the stiffness of the EoS is enhanced by the low coupling constant, the stability is compromised. In Fig. 10, the central value
Γc(χ=1)=1.977 is greater than that atΓc(χ=−1)=1.848 . Forχ=−1 , the valueΓc(χ=−1)=1.848 is comparatively closer to the Bondi limit, i.e.,Γ=1.333 . Therefore,χ=−1 is more sensitive towards radial oscillations and, hence, the stability. On the other hand, the range of central density is smaller forχ=−1 than forχ=1 . This means that during radial perturbations, the range of stable density perturbation is more forχ=1 than forχ=−1 , thereby enhancing its stability. Further, the solution fulfills all the energy conditions. The anisotropy decreases with decrease in the coupling parameter. This leads to the conclusion that the anisotropy in pressure reduces with increase in the stiffness of the EoS. The surface redshifts predicted form the solution in far within Ivanov's limit, i.e.,zs=3.842 [143].In Table 1, we have presented some of the physical parameters of four compact stars. We have also presented how the radius, central and surface densities, central pressure, and moment of inertia vary with the
f(R,T)− coupling constant. As the stiffness increases with decrease inχ , the central and surface densities, central pressure, and radius increase while the stability is compromised. For−1⩽χ⩽1 , we have predicted the radii of the compact stars. All these results are accurate and in agreement with the observed values of the masses and radii. Thus, one can undoubtedly conclude that the solution might have astrophysical significance.Objects χ M( M⊙ )Predicted radius/km ρ0 (MeV/fm3 )ρs (MeV/fm3 )p0 (MeV/fm3 )I×1044 (gcm2 )PSR J1614-2230 −1.0 1.97 ± 0.0410.13 621.11 376.32 72.57 18.51 −0.5 9.89 581.58 353.52 63.20 17.52 0 9.66 549.65 330.71 56.10 16.77 0.5 9.47 519.24 313.98 50.28 16.04 1.0 9.28 491.87 295.74 45.44 15.59 VELA X-1 −1.0 1.77 ± 0.089.76 551.66 362.96 56.48 15.29 −0.5 9.55 518.29 339.95 49.12 14.62 0 9.34 488.38 319.24 43.36 13.94 0.5 9.15 461.91 303.13 38.76 13.56 1.0 8.99 437.75 285.87 34.85 13.01 SAX J1808.4-3658 −1.0 0.9 ± 0.37.91 431.10 339.05 20.76 4.74 −0.5 7.69 403.98 317.68 18.14 4.51 0 7.54 380.96 298.78 16.12 4.34 0.5 7.42 359.60 282.34 14.50 4.17 1.0 7.26 340.69 266.73 13.19 4.00 CEN X-3 −1.0 1.49 ± 0.089.29 502.10 352.97 41.72 11.24 −0.5 9.03 471.03 330.02 36.47 10.85 0 8.85 443.42 311.65 31.83 10.42 0.5 8.66 418.10 294.44 28.60 10.11 1.0 8.53 397.39 278.37 25.98 9.72 Table 1. Prediction of radius for few well-known compact stars and their corresponding central densities and pressures for various values of χ.
For the graphical test and investigation of the physical reasonable grounds of the accomplished solutions, we have selected the physical profiles of four well-known compact stellar systems, namely, PSR J1614-2230, Vela X-1, Cen X-3, and SAX J1808.4-3658. In this regard, we have assumed the radius--radius element of the metric function (
eλ ) to be in the new formeλ=1+ar2ebr2+cr4 and provided the time--time element of the metric function (eν ) as expressed explicitly in Eq. (28). Further, we exhibit the anisotropic impacts on the physical systems imposed by theχ− coupling constant of thef(R,T) gravity theory. Thus, we have established the most significant salient features that describe the stellar system, fulfilling all the general necessities to guarantee a respectful framework.The comportment of physical amounts of time--time and radius--radius components, namely,
eν andeλ , respectively, with respect to the radial coordinate r for Vela X-1 (M=1.77±0.08M⊙,R=9.56±0.08km ) is represented in Fig. 1. This figure illustrates that both the metric functions are limited at the origin and monotonically expanding towards the point of confinement at the surface. Moreover, it may very well be seen from Eqs. (27) and (28) thateλ(r=0)=1 andeν(r=0)≠0 , which shows that this stellar system is realistic and agreeable. Hence, Figs. 2 and 3 clearly display that all the thermodynamic observables, i.e., energy densityρ , radial pressurepr , and transverse pressurept , are well-defined within the stellar structure. In this context, it is worth mentioning that all the quantities mentioned have their maximum values at the core of the stellar structure and monotonically decreasing comportment with increasing radius towards the surface. At this stage, it merits mentioning that the present stellar model shows a positive anisotropy factorΔ ; from Fig. 3, it can be seen that whenpt>pr ,Δ>0 . Thus, the stellar structure is found to be a repulsive force that neutralizes the gravitational slant. This reality permits the building of a progressively compact stellar configuration. From Figs. 1, 2, and 3, we can confirm that the stellar system is absolutely devoid of physical or geometrical singularities for all chosen values of theχ− coupling constant of thef(R,T)− gravity theory. Fig. 5 exhibits the variation in the anisotropy parameterΔ with the radial coordinate for Vela X-1. The anisotropy vanishes at the center, and, consequently, it is defined as a positive, increasing function towards the surface of the stellar structure. In addition, Fig. 6 indicates that the values of the EoS parametersωr=pr/ρ andωt=pt/ρ are under 1, demonstrating that Zeldovich's condition is fulfilled everywhere within the stellar structure in the context of thef(R,T)− gravity theory.Further, the equilibrium study of the model for the stellar system is established using the generalized TOV equation that originates from the modified type of energy conservation equation for the energy-momentum tensor in the area of the
f(R,T)− gravity theory, as expressed in Eq. (10). In this regard, it is easy to see from Fig. 7 that the modified TOV equation permits exploration under various forces that perform on the stellar structure. In this event, the stellar structure is under the effect of four different forces, namely, gravity (Fg ), pressure gradient (Fh ), anisotropic force (Fa ), and the additional force (Fm ) inf(R,T)− gravity. The forces caused by gravity and pressure gradient and the additional force are the highest forχ=−1 ; however, the anisotropic force is the lowest. This enables it to hold more mass than the others for small values ofχ . Asχ increases,Fg ,Fm , andFh decrease althoughFa increases slightly. Thus, the maximum mass that can be held by the system also reduces. On the other hand, we investigated the stability of realistic and compact stellar structure solutions using the causality condition and stability factor, i.e., the stability criterion through Bondi's condition and the Harrison-Zeldovich-Novikov static stability criterion corresponding to theχ− coupling constant off(R,T)− gravity. Fig. 8 displays the performances of the radial (vr ) and transverse (vt ) speeds of sound with respect to the radial coordinate r, according to these criteria, for the compact stellar configuration; it is clear that they remain within their predetermined range[0,1] throughout the stellar framework, which affirms the causality condition and, furthermore, validates the acceptability of the subsequent anisotropic solution of our stellar system. Moreover, it can be seen that the speed of sound is maximum forχ=−1 and decreases with increase inχ , which implies that our solution leads to a stiffer EoS at small values of theχ− coupling constant. The solution can also be obtained for static and stable astrophysical structures, as the stability factor, which can be defined asv2t−v2r , lies between the bounds−1 to0 for various values of theχ− coupling parameter, displayed in Fig. 9. In addition, for a noncollapsing stellar fluid distribution, the adiabatic index should also be greater than4/3 forΔ>0 according to the stability criteria via Bondi’s perceptions, which can be observed clearly from Fig. 10; therefore, our stellar system is generally stable. In Fig. 11, we present the plot of the variation in mass with respect to the central density, which satisfies the Harrison-Zeldovich-Novikov static stability criterion. From this figure, one can conclude that the stability is enhanced with increase inχ . This holds on the grounds that the range of central density is greater for saturating the mass whenχ=1 than whenχ=−1 . This infers that the stable range of density during radial oscillation is more applicable for larger values ofχ . Thus, we can conclude that our solution is completely stable under radial perturbations. Subsequently, we investigated the profile of the thermodynamic quantities that prompt a well-behaved and positively defined energy--momentum tensor throughout the interior of the compact stellar structure, which is satisfied simultaneously by the inequalities named ECs that govern them. Hence, in Fig. 12, we present the plots of the left-hand sides of these inequalities, which verify that all the ECs are achieved at the astrophysical inside and, consequently, corroborate the physical accessibility of the compact stellar internal solution. The fulfillment of the redshift with respect to the radial coordinate r for Vela X-1 is illustrated in Fig. 13. This figure demonstrates surface redshift within typical values resulting from Ivanov’s effect, which strongly validates our compact stellar system.Figure 13. (color online) Variation in redshift with radial coordinate for Vela X-1 (
M=1.77±0.08M⊙,R=9.56±0.08km ) withb=0.0005/km2 andc=0.000015/km4 .Further, we have generated
M−R curves from our solutions in the area off(R,T) gravity theory and found a perfect fit for certain compact stellar spherical systems such as PSR J1614-2230, Vela X-1, Cen X-3, and SAX J1808.4-3658. Therefore, we have predicted the corresponding radii and their respective moment of inertia from theM−I curve by varying the coupling constantχ as a free variable. TheM−R andI−M curves are presented in Figs. 14 and 15. These curves indicate that our solution predicted the radii in good agreement with the observational data.Finally, we wish to remark that all anisotropic spherically symmetric solutions established in this study, which satisfy the well-behaved stellar interiors obtained in the area of
f(R,T) gravity theory using the embedding class-one procedure, fulfill and share all the physical and mathematical features necessary in the study of compact stellar spherical systems, leading to an understanding of the evolution of realistic compact stellar spherical systems. In this regard, thef(R,T) gravity theory is a promising principle for envisaging the existence of the compact stellar spherical systems characterized by anisotropic matter distributions, which meet the notable and tried general necessities and whose effects can be contrasted with the well-described GR.We are very grateful to the honorable referees and the editor for their relevant suggestions that have considerably improved our work in terms of the quality of research and presentation.
Exploring physical properties of compact stars in
f(R,T)
-gravity: An embedding approach
- Received Date: 2020-04-09
- Available Online: 2020-10-01
Abstract: Solving field equations exactly in