Realistic stellar anisotropic model satisfying Karmarker condition in f(R, T) gravity
Abstract
The present study explores the \(f(\mathcal {R},\mathcal {T})\) modified gravity on the basis of observational data for three different compact stars with matter profile as anisotropic fluid without electric charge. In this respect, we adopt the well-known Karmarker condition and assume a specific and interesting model for \(\mathrm {g}_{rr}\) metric potential component which is compatible with this condition. This choice further leads to a viable form of metric component \(\mathrm {g}_{tt}\) by utilizing the Karmarkar condition. We also present the interior geometry in the reference of Schwarzschild interior and Kohler–Chao cosmological like solutions for \(f(\mathcal {R},\mathcal {T})\) theory. Moreover, we calculate the spacetime constants by using the masses and radii from the observational data of three different compact stars namely 4U 1538-52, LMC X-4 and PSR J1614-2230. In order to explore the viability and stability of the obtained solutions, some physical parameters and properties are presented graphically for all three different compact object models. It is noticed that the parameters c and \(\lambda \) have some important and considerable role for these solutions. It is concluded that our obtained solutions are physically acceptable, bearing a well-behave nature in \(f(\mathcal {R},\mathcal {T})\) modified gravity.
1 Introduction
In last few decades, various extended theories of gravity have been proposed as the most promising candidates of mysterious dark energy for exploring the accelerated expansion aspects of our cosmos. There are several independent observations in the context of astrophysics that provide evidences about the accelerated expanding nature of space. These astrophysical experiments include the outcomes acquired from the supernova type Ia (SNIa) [1, 2, 3, 4, 5, 6, 7, 8], large scale construction surveys [9], X-ray brightness from galaxy erect [10], cosmic microwave background radiation (CMBR) [11, 12, 13], weak lensing and the baryon acoustic oscillation (BAO) surveys [14]. This phenomenon of accelerating cosmic expansion is regarded as an outstanding critical riddles of contemporary physics. It is argued that the expansion of cosmos is accelerated due to the presence of some uncertain dominant source of energy labeled as dark energy (DE). In order to incorporate this unusual motive of dynamical cosmos, some modifications in the Lagrangian density of Einstein’s general relativity have been presented in the literature. One of the way to deal with the problem is to modify the matter profile of the density by adding some DE terms as scalar field or cosmological constant, k-essence, canonical kinetic scalar term, quintessence, and different versions of chaplying gas etc. [15, 16, 17, 18, 19, 20]. While in the alternative approach, researchers extended the gravitational part by adding some extra degrees of freedom there, which provided the group of extended theories of gravity. In this respect, some leading examples include \(f(T,T_{G})\) theory, \(f(\mathrm {G})\) gravity, \(f(\mathcal {R})\) framework, braneworld scenarios, Kalb–Ramond background, Gauss–Bonnet gravity and Brans–Dicke gravity theories etc. [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. Based on these modifications, many other remarkable generalized modifications are also available, which help to explore various cosmic aspects successfully. These all modifications have passed different astrophysical and necessary solar system constraints and are regarded as viable candidates.
After the formulation of general theory of relativity (GR), the \(f(\mathcal {R})\) theory is regarded as one of the most interesting and viable extension of GR. Later on, its different modified versions have been presented by researchers that are also proved as successful in various respects. Particularly, its recent extension namely \(f(\mathcal {R},\mathcal {T})\) gravity, proposed by Harko et al. [37] almost 6 years ago, has attracted many researchers. This modification is based on a governing function \(f(\mathcal {R},\mathcal {T})\) depending on the Ricci scalar \(\mathcal {R}\) and the trace \(\mathcal {T}\) of anisotropic like energy-momentum tensor. They also derived the corresponding field equation by utilizing the metric potentials formalism and discussed the significance of this modification. Further, the authors have presented different models for \(f(\mathcal {R},\mathcal {T})\) function in separable form, i.e., \(f(\mathcal {R},\mathcal {T})=f_{1}(\mathcal {R})+f_{2}(\mathcal {T})\). It is considered as a very interesting modification because the resulting field equations have not much difficult form or un-handled order. There is a sufficient literature available where numerous cosmological aspects of this theory have been explored like expansion of universe due to accelerated matter, Birkhoff’s theorem, scalar field reconstructions, stability using cosmological perturbation, large scale structure, thermodynamical laws and its relevant features, constrains regarding solar system, stellar equilibrium configurations of compact stars, neutron stars, gravitational collapse phenomenon [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48].
The study of compact stars formation and phenomenon of gravitational collapse is regarded as one of the most fascinating subjects in modern cosmology and astronomy. In 1916, Carl Schwarzschild presented the exact solution of interior of symmetric star spherically by using the uniform density based matter profile [49]. In 1939, Oppenheimer and Snyder [50] contributed to explore the gravitational collapse with homogeneity based dust sphere. In literature [51, 52, 53, 54, 55], hundreds of static analytic models representing the relativistic stars have been constructed by introducing bulk viscous effects, anisotropic pressures, charge, multilayered fluids and equations of state etc. The discussion of ultra densities for matter profile of order \(10^{15}\;{\text {g cm}}^{-3}\) provides a new idea that radial pressure profile and tangential pressure profile are unequal. In this context, Ruderman [56] was the pioneer who predicted that in the high density, relativistic interaction of nuclear matter yields anisotropy as an inherent feature. Lobo [57] used barotropic EoS for exploring some compact object models which is defined as \(p=\epsilon \rho ;~ -1<\epsilon <-1/3\), for the dark stars existence. He also presented a modified version of Mazur–Mottola gravastar model by utilizing the junction conditions to match Schwarzschild vacuum solution with static line element. Egeland [58] explained Neutron stars by taking mass-radius relationship into account. In another study [59], Mak and Harko proposed a spherically symmetric model exhibiting the characteristics of strange stars. Further, Rahaman et al. [60] discussed the possibility of compact stars formation by using Krori–Barua model in the presence of Chaplygin gas matter distribution. In modified gravitational frameworks like f(R) gravity and scalar-tensor theories, much work has already been done for modeling the massive as well as neutron stars [61, 62, 63, 64, 65]. Hossein et al. [66] have developed some compact stars models with anisotropic matter and variable cosmological constant where they assumed a linear equation of state. In different collaborations [67, 68], Herrera discussed some interesting anisotropic solutions for static as well as non static sources. In the context of GR and modified gravitational frameworks of f(T) and f(G) theories, much work has already been done by the researchers [69, 70, 71, 72, 73, 74, 75, 76]. Zubair and Abbas [77] have explored some interior compact star models in f(R) extension of GR by including Krori and Barua solutions and discussed different physical features as well as the stability of the obtained models.
Naidu and Govender [78] investigated two new stellar models with same radii and mass but distinct pressure distributions. In another study, Zubair et al. [51, 52, 53, 54, 55] investigated the possible existence of compact stars in f(R,T) framework by taking the analytic models of Krori and Barua line element into account and presented a detailed analysis of the models by discussing some physical features. In 2013, Herrera and Barreto [79] acquired a mathematical procedural technique to produce a relativistic-polytropes with radial and tangential pressure profiles by using curvature coordinates. Recently, a considerable approach for deriving the analytic solutions of Einstein field equations, representing the compact objects, has been proposed by the researchers namely the Karmarkar condition [80, 81, 82, 83, 84, 85, 86, 87]. This condition was firstly proposed by Karmarkar [88] and is regarded as a compulsory condition for a spherically symmetric spacetime to be of embedding class-I. It is basically a mathematical tool which helps us in obtaining the exact solutions of field equations. In literature [89, 90, 91, 92, 93], this condition has been used by numerous researchers for discussing the compact stars models. In the present paper, we shall adopt the Karmarkar condition to develop the analytic solutions representing compact objects in \(f(\mathcal {R},\mathcal {T})\) gravity.
Being motivated from the above literature, in the present manuscript, we will discuss the formation of compact stars in \(f(\mathcal {R},\mathcal {T})\) modified gravity by considering three different models of compact stars. For this purpose, we will consider static spherically geometry filled with anisotropic matter contents and also take the well-known Karmarkar condition. In the up-coming section, we will define the mathematical structure of \(f(\mathcal {R},\mathcal {T})\) gravity and formulate its field equations. In the same section, we will present the Karmarkar condition briefly and calculate isotropic like Class-I solutions with two different cases of \(f(\mathcal {R},\mathcal {T})\) gravity. In Sect. 3, we will discuss the set up for a new family of embedding class-I solutions along with the physical boundary conditions. In Sect. 4, we will present the analysis of obtained solutions using three different models for compact stars like 4U 1538-52, LMC X-4 and PSR J1614-2230 by investigating different physical properties analytically and graphically. In the same section, we will discuss the stability of obtained model using different viable measures. In last, we will conclude the whole discussion by focusing on the main achievements.
2 \(f(\mathcal {R},\mathcal {T})\) gravity field equations and anisotropic fluid distribution
3 Isotropic like class-I solutions for \(f(\mathcal {R},\mathcal {T})\) theory
3.1 Schwarzschild interior solution for \(f(\mathcal {R},\mathcal {T})\) theory
3.2 Kohler–Chao cosmological like solution for \(f(\mathcal {R},\mathcal {T})\) theory
4 A new setup for a family of embedding class-I models
In this section, we shall study new stellar models which are based on anisotropic fluid and are compatible with the Karmarker condition. It is a considerable point that the pressure anisotropy, i.e., \(\triangle \) has an important role in the discussion of gravitational collapse process. Recently, Naidu and Govender [78] studied the collapse dynamics which is related to its radial pressure and density function of radial coordinate under the stellar fluid. Further, they have assumed a linear form of equation of state within a static configuration which can be presented by the expression \(p_{r}=\epsilon \rho -\varepsilon \), with \(\epsilon \) and \(\varepsilon \) as constants. They argued that the succeeding collapse is responsive to the correlation of energy density function and radial pressure. Further, they discussed the impact of parameter \(\epsilon \) on the temperature description of collapsing configuration.
4.1 Physical boundary conditions
Calculated values of \(a,\, A\) and B for different three small values of \(\lambda \), under three different well-known compact stars 4U 1538-52, LMC X-4 and PSR J1614-2230
Object | \(a\;({\text {km}}^{-2})\) | \(b\;({\text {km}}^{-2})\) | c | A | \( B\;({\text {km}}^{-1})\) | \(R\;({\text {km}})\) | \(M\;(M_{\odot })\) |
---|---|---|---|---|---|---|---|
4U 1538-52 (\(\lambda =0.35\)) | 0.1216 | 0.00025 | 0.1800 | 29.9455 | 0.04246 | 7.866 | 0.87 |
LMC X-4 (\(\lambda =0.35\)) | 0.1826 | 0.00020 | 0.1500 | 46.0060 | 0.04282 | 8.300 | 1.04 |
PSR J1614-2230 (\(\lambda =0.35\)) | 0.6123 | 0.00010 | 0.1000 | 177.8551 | 0.04553 | 9.690 | 1.97 |
4U 1538-52 (\(\lambda =0.45\)) | 0.1216 | 0.00025 | 0.1800 | 32.0451 | 0.04553 | 7.866 | 0.87 |
LMC X-4 (\(\lambda =0.45\)) | 0.1826 | 0.00020 | 0.1500 | 49.0519 | 0.04572 | 8.300 | 1.04 |
PSR J1614-2230 (\(\lambda =0.45\)) | 0.6123 | 0.00010 | 0.1000 | 188.7054 | 0.04832 | 9.690 | 1.97 |
4U 1538-52 (\(\lambda =0.55\)) | 0.1216 | 0.00025 | 0.1800 | 33.8717 | 0.04820 | 7.866 | 0.87 |
LMC X-4 (\(\lambda =0.55\)) | 0.1826 | 0.00020 | 0.1500 | 51.8354 | 0.04835 | 8.300 | 1.04 |
PSR J1614-2230 (\(\lambda =0.55\)) | 0.6123 | 0.00010 | 0.1000 | 198.5136 | 0.05084 | 9.690 | 1.97 |
Calculated values of \(a,\, A\) and B for different three large values of \(\lambda \), under three different well-known compact stars 4U 1538-52, LMC X-4 and PSR J1614-2230
Object | \(a \; ({\text {km}}^{-2})\) | \(b\;({\text {km}}^{-2})\) | c | A | \( B\;({\text {km}}^{-1})\) | \(R\;({\text {km}})\) | \(M\;(M_{\odot })\) |
---|---|---|---|---|---|---|---|
4U 1538-52 (\(\lambda =2.00\)) | 0.12168 | 0.00025 | 0.1800 | 49.6093 | 0.07119 | 7.866 | 0.87 |
LMC X-4 (\(\lambda =2.00\)) | 0.18269 | 0.00020 | 0.1500 | 75.6308 | 0.07093 | 8.300 | 1.04 |
PSR J1614-2230 (\(\lambda =2.00\)) | 0.61229 | 0.00010 | 0.1000 | 278.3272 | 0.07136 | 9.690 | 1.97 |
4U 1538-52 (\(\lambda =4.00\)) | 0.12168 | 0.00025 | 0.1800 | 58.4452 | 0.08410 | 7.866 | 0.87 |
LMC X-4 (\(\lambda =4.00\)) | 0.18269 | 0.00020 | 0.1500 | 88.8503 | 0.08347 | 8.300 | 1.04 |
PSR J1614-2230 (\(\lambda =4.00\)) | 0.61229 | 0.00010 | 0.1000 | 319.7631 | 0.08202 | 9.690 | 1.97 |
4U 1538-52 (\(\lambda =6.00\)) | 0.12168 | 0.00025 | 0.1800 | 62.7608 | 0.09041 | 7.866 | 0.87 |
LMC X-4 (\(\lambda =6.00\)) | 0.18269 | 0.00020 | 0.1500 | 95.2703 | 0.08956 | 8.300 | 1.04 |
PSR J1614-2230 (\(\lambda =6.00\)) | 0.61229 | 0.00010 | 0.1000 | 339.1949 | 0.08701 | 9.690 | 1.97 |
Calculated values of different physical properties at center and boundary for different three values of \(\lambda \)
Models | \(e^{\nu (r=0)}\) | \(e^{\mu (r=0)}\) | \(\rho _{c}\,({\text {g/cc}})\) | \(p_{r_{c}}\,({\text {dyne/cm}}^{2})\) | \(p_{t_{c}}\,({\text {dyne/cm}}^{2})\) | \(\rho _{R}\,({\text {g/cc}})\) | \(p_{r_{c}}/\rho _{0}=p_{t_{c}}/\rho _{0}\) |
---|---|---|---|---|---|---|---|
4U 1538-52 (\(\lambda =0.35\)) | 1.0 | 0.635094 | \(10.1893\times 10^{15}\) | \(5.1871\times 10^{35}\) | \(5.1871\times 10^{35}\) | \(3.39813\times 10^{14}\) | 0.0512 |
LMC X-4 (\(\lambda =0.35\)) | 1.0 | 0.588418 | \(10.6924\times 10^{15}\) | \(6.3032\times 10^{35}\) | \(6.3032\times 10^{35}\) | \(3.39756\times 10^{14}\) | 0.0589 |
PSR J1614-2230 (\(\lambda =0.35\)) | 1.0 | 0.354732 | \(16.2936\times 10^{15}\) | \(15.9502\times 10^{35}\) | \(15.9502\times 10^{35}\) | \(3.53990\times 10^{14}\) | 0.0978 |
4U 1538-52 (\(\lambda =0.45\)) | 1.0 | 0.625062 | \(09.8315\times 10^{15}\) | \(5.0893\times 10^{35}\) | \(5.0893\times 10^{35}\) | \(3.26101\times 10^{14}\) | 0.0511 |
LMC X-4 (\(\lambda =0.45\)) | 1.0 | 0.578418 | \(10.3176\times 10^{15}\) | \(6.0143\times 10^{35}\) | \(6.0143\times 10^{35}\) | \(3.25889\times 10^{14}\) | 0.0582 |
PSR J1614-2230 (\(\lambda =0.45\)) | 1.0 | 0.341994 | \(15.7891\times 10^{15}\) | \(15.3191\times 10^{35}\) | \(15.3191\times 10^{35}\) | \(3.39658\times 10^{14}\) | 0.0970 |
4U 1538-52 (\(\lambda =0.55\)) | 1.0 | 0.616863 | \(09.4913\times 10^{15}\) | \(4.8655\times 10^{35}\) | \(4.8655\times 10^{35}\) | \(3.13246\times 10^{14}\) | 0.0510 |
LMC X-4 (\(\lambda =0.55\)) | 1.0 | 0.569217 | \(09.9671\times 10^{15}\) | \(5.7575\times 10^{35}\) | \(5.7575\times 10^{35}\) | \(3.13041\times 10^{14}\) | 0.0577 |
PSR J1614-2230 (\(\lambda =0.55\)) | 1.0 | 0.330763 | \(15.3123\times 10^{15}\) | \(14.7345\times 10^{35}\) | \(14.7345\times 10^{35}\) | \(3.26264\times 10^{14}\) | 0.0962 |
5 Analysis of the physical properties of the \(f(\mathcal {R},\mathcal {T})\) stellar model
In this section , we explore our results in more detail by focusing on some physical aspects and necessary properties of the obtained \(f(\mathcal {R},\mathcal {T})\) stellar configuration using three different stellar objects observation data values. For this purpose, we shall present the discussions about some physical measures analytically and graphically by taking different values of parameter \(\lambda \). Here we shall consider three different models for stellar objects like 4U 1538-52, LMC X-4 and PSR J1614-2230.
5.1 Evolution of metric functions, energy density and pressure components
Calculated results for three different compact stars in \(f(\mathcal {R},\mathcal {T})\) theory under Karmarkar condition
Expressions | \(\lambda =2.00,\;\lambda =4.00\), and \(\lambda =6.00\) | ||
---|---|---|---|
4U 1538-52 model | LMC X-4 model | PSR J1614-2230 model | |
\(\rho \) | \(\rho >0\) | \(\rho >0\) | \(\rho >0\) |
\(p_{r}\) | \(p_{r}>0\) | \(p_{r}>0\) | \(p_{r}>0\) |
\(p_{t}\) | \(p_{t}<0\) (near to boundary) | \(p_{t}<0\) (near to boundary) | \(p_{t}<0\) (near to boundary) |
\(\triangle (r)\) | \(\triangle (r)<0\) (throughout) | \(\triangle (r)<0\) (throughout) | \(\triangle (r)<0\) (throughout) |
Gradients | Gradients\(<0\) | Gradients\(<0\) | Gradients\(<0\) |
\(\mathcal {F}_{\mathrm {a}}, \mathcal {F}_{\mathrm {h}}\) and \(\mathcal {F}_{\mathrm {g}}\) | Forces are balance | Forces are balance | Forces are balance |
Energy conditions | Energy conditions \(>0\) | Energy conditions \(>0\) | Energy conditions \(>0\) |
m(r), u(r), z(r) | \(m(r)>0, u(r)>0, z(r)>0\) | \(m(r)>0, u(r)>0, z(r)>0\) | \(m(r)>0, u(r)>0, z(r)>0\) |
\(w_r\) | \(0\le w_r<1\) (satisfied) | \(0\le w_r<1\) (satisfied) | \(0\le w_r<1\) (satisfied) |
\(w_t\) | \(-0.01\le w_t<1\) (not satisfied) | \(-0.02\le w_t<1\) (not satisfied) | \(-0.03\le w_t<1\) (not satisfied) |
\(v^{2}_{r}\) | \(0\le v^{2}_{r}<1\) (satisfied) | \(0\le v^{2}_{r}<1\) (satisfied) | \(0\le v^{2}_{r}<1\) (satisfied) |
\(v^{2}_{t}\) | \(0\le v^{2}_{t}<1\) (satisfied) | \(0\le v^{2}_{t}<1\) (satisfied) | \(0\le v^{2}_{t}<1\) (satisfied) |
\(v^{2}_{t}-v^{2}_{r}\) | \(-1\le v^{2}_{t}-v^{2}_{r}>0\) (not satisfied) | \(-1\le v^{2}_{t}-v^{2}_{r}>0\) (not satisfied) | \(-1\le v^{2}_{t}-v^{2}_{r}>0\) (not satisfied) |
\(\Gamma _{r}\) | \(\Gamma _{r}<4/3\) (not satisfied) | \(\Gamma _{r}<4/3\) (not satisfied) | \(\Gamma _{r}<4/3\) (not satisfied) |
The behavior of anisotropy function for different three values of \(\lambda \), under three different well-known compact stars 4U 1538-52, LMC X-4 and PSR J1614-2230
Object | \(\lambda =0.35\) | \(\lambda =0.45\) | \(\lambda =0.55\) |
---|---|---|---|
4U 1538-52 | \(\triangle >0\), for \(0.01\le r\le 7.866\) | \(\triangle >0\), for \(0.01\le r\le 7.866\) | \(\triangle >0\), for \(0.01\le r\le 7.866\) |
LMC X-4 | \(\triangle >0\), for \(0.01\le r\le 8.300\) | \(\triangle >0\), for \(0.01\le r\le 8.300\) | \(\triangle <0\), for \(0.01\le r\le 1.060\) |
PSR J1614-2230 | \(\triangle >0\), for \(0.01\le r\le 9.690\) | \(\triangle <0\), for \(0.01\le r\le 3.650\) | \(\triangle <0\), for \(0.01\le r\le 5.350\) |
5.2 Equilibrium condition
5.3 Energy conditions
5.4 Mass function, compactness parameter and gravitational red-shift function
Here we shall explore the impact of parameter \(\lambda \) on the obtained solutions by discussing the behavior of corresponding mass function, compactness parameter and gravitational red-shift function versus radial coordinate graphically. It can be observed from the Figs. 9, 10, 11 and 12 that mass function and compactness parameter exhibit a regular increasing behavior with the increasing values of small or large \(\lambda \) choices while the gravitational red-shift function indicates the decreasing behavior in both cases of parameter \(\lambda \). Here it can also be seen that the \(\lambda \) has an impact on the obtained solutions, i.e., if we consider large \(\lambda \) values, then function are showing gradually increasing or decreasing behavior as compared to its small values. These behavior show that all these physical parameters satisfy the required condition for compact stars in \(f(\mathcal {R},\mathcal {T})\) theory.
5.5 Equation of state
The development of Herrera cracking criterion, i.e., \(-1\le v^{2}_{t}-v^{2}_{r}\le 0\) for different three values of \(\lambda \), under three different well-known compact stars 4U 1538-52, LMC X-4 and PSR J1614-2230
Object | \(\lambda =0.35\) | \(\lambda =0.45\) | \(\lambda =0.55\) |
---|---|---|---|
4U 1538-52 | Satisfy, for \(0.01\le r\le 7.866\) | Satisfy, for \(0.01\le r\le 7.866\) | Satisfy, for \(0.01\le r\le 7.866\) |
LMC X-4 | Satisfy, for \(0.01\le r\le 8.300\) | Satisfy, for \(0.01\le r\le 8.300\) | Not satisfy, for \(0.01\le r\le 0.750\) |
PSR J1614-2230 | Satisfy, for \(0.01\le r\le 9.690\) | Not satisfy, for \(0.01\le r\le 2.500\) | Not satisfy, for \(0.01\le r\le 3.650\) |
Here we discuss the graphical development of two different ratios, i.e., \(w_r\) and \(w_t\) versus radial coordinate. It is observed from the right penal of Figs. 11 and 12, both ratios \(\frac{p_r}{\rho }\) and \(\frac{p_t}{\rho }\) exhibit monotonically decreasing behavior less than 1 against different small and large values of parameter \(\lambda \) which is regraded as one of the necessary condition for compact stars. It can also be noticed that these ratios decrease more rapidly as \(\lambda \) increases.
5.6 Causality stability analysis
5.7 Adiabatic index stability analysis
In Figs. 15 and 16, we describe the graphical behavior of \(\Gamma _{r}\) parameter versus radial coordinate along with different small and large values of parameter \(\lambda \). It can be observed that the expression \(\Gamma _{r}\) shows monotonically increasing behavior with values greater than \(\frac{4}{3}\) for all 4U 1538-52, LMC X-4 and PSR J1614-2230 models as shown by the green, blue and red curves, respectively. Hence, it can be concluded that the adiabatic index is compatible with the stability condition in the reference of \(f(\mathcal {R},\mathcal {T})\) theory.
6 Conclusion
Energy density function and pressure profile It is evident from Tables 3 and 4 that all the necessary conditions for a physical model like the behavior of energy density function as well as both pressure components like radial and tangential profiles to be positive, have been achieved throughout in the stellar interior for all 4U 1538-52, LMC X-4 and PSR J1614-2230 compact stars models. Also, these functions provided their compatible maximum values at the center and then decreased monotonically as the radial coordinate increases towards the boundary surface. Further the ratio of central pressure and density satisfied the Zeldovich’s condition, i.e., \(p_{rc}/ \rho _{c}\le 1\). It is also concluded from Tables 3 and 4 that the radial derivatives of energy density, radial as well as tangential pressure components showed negative behavior for U 1538-52, LMC X-4 and PSR J1614-2230 compact stars models in \(f(\mathcal {R},\mathcal {T})\) theory. It is seen that the parameter \(\lambda \) has a significant effect on the obtained spherically symmetric solutions.
Anisotropic pressure It is noticed that the measure of anisotropic pressure \(\triangle (r)\) is positive in few cases when the small values of \(\lambda \) has been taken into account. In case \(\lambda >1\), the anisotropic pressure shows negative behavior for all three compact stars.
Equilibrium condition It is seen from Tables 3 and 4 that the forces \(\mathcal {F}_{\mathrm {a}},\,\mathcal {F}_{\mathrm {h}}\) and \(\mathcal {F}_{\mathrm {g}}\) satisfied the equilibrium condition for U 1538-52, LMC X-4 and PSR J1614-2230 models. It is also noticed from the same table that these forces balanced each other’s effect and hence leaving a stable configuration for all choices of small and large \(\lambda \) values. These forces have been illustrated graphically in the Figs. 7 and 8.
Energy conditions It is seen from the Tables 3 and 4 that all the functions \(\rho \), \(p_r\), \(p_t\), \(\rho -p_{r}\), \(\rho -p_{t}\), \(\rho -p_{r}-2p_{t}\), which are presented graphically and analytically satisfied the respective bounds namely NEC, WEC, SEC and DEC.
Mass function, compactness parameter and gravitational red-shift function In the present scenario, mass-radii function, compactness parameters and gravitational parameters have been investigated graphically. Tables 3 and 4 along with the corresponding graphs indicated that the mass-radii function m(r) and compactness parameter u(r) remained positive, regular and increasing and satisfy the Buchdahl limit, i.e., \(u(r)\le 8/9\). Further, the gravitational red-shift function z(r) showed positive decreasing behavior for all these compact star models and \(\lambda \) choices.
Equation of state In radial and tangential directions, we have investigated the equation of state (EoS) parameters \(w_r\) and \(w_t\) graphically. From Table 2, it is noticed that the value of these parameters remained positive inside the stellar objects and also less than 1 for all three U 1538-52, LMC X-4 and PSR J1614-2230 models in all cases.
Causality stability analysis It is concluded from Tables 4 and 6 that the radial and tangential speeds of sound for the considered compact stars, denoted by \(v^{2}_{r}\) and \(v^{2}_{t}\), satisfied the condition of decreasing behavior with velocities less than 1 for all choices of \(\lambda \). Furthermore, another necessary condition \(-1\le v^{2}_{t}-v^{2}_{r}\le 0\), the difference of tangential velocity and radial velocity is satisfied for the \(\lambda =0.35\) only.
Adiabatic index stability analysis In the reference of adiabatic index, it is seen from Tables and the corresponding graphs that the adiabatic index \(\Gamma _{r}\), satisfied the condition \(\Gamma _{r}<4/3\) and showed an increasing development versus radial coordinate for all choices of \(\lambda \).
Notes
Acknowledgements
The research work of G. Mustafa and Xia Tiecheng has been supported by the National Natural Science Foundation of China under Grant Nos. 11271008, 61072147, and 11975145 “M. Zubair thank the Higher Education Commission, Islamabad, Pakistan for its financial support under the NRPU project with grant number 5329/Federal/NRPU/R&D/HEC/2016”
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