A gravitational decoupling MGD model in modified $f(R,T)$ gravity theory

Einstein’s famous theory of gravity comprises of reasonable creativity and scientific style in its each progression. To represent non-inertial systems and floating structures from a standard standpoint at large scale phases, Einstein’s gravity theory is a very helpful implement [1]. However, despite its excellence, the geometrical singularities make it stationary in certain situations [2]. Moreover, the theory cannot continue normally for the representation of the period of observational acceleration of the Universe. Cannot adequately depict the enormous elements of the Universe without taking into consideration the exotic type of energy-matter that should be known as the dark side (dark-energy and dark-matter) [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Einstein’s theory of gravity does not account into the matter quantum kind and cannot be quantified in a renormalization traditional method. In this reasoning, it has been indicated [15] that taking into account of higher-order curvature and flow terms in Einstein-Hilbert’s action will make it renormalizable framework into a loop. Then again, to incorporate quantum adjustments, the higher-order curvature and flow invariants in the effective gravitational action at the low energy phase must be considered [16, 17, 18]. In spite of the fact that the theory stood the trial of time and has various applications in physical cosmology, it can’t clarify the most serious issue in physical cosmology, i.e., the kind of cosmological constant. The acceleration of the universe at the present age can’t be clarified without summoning new types of matter as well as energy [19]. There are two possible cases to address this exceptional issue: the first case, numerous candidates for dark energy have been suggested in the literature, such as f-essence, k-essence, quintessence, spintessence, tachyons field, ghost field, Chaplygin gas [20]. In the second case, we can persuade ourselves that we are living in a Universe with a cosmological constant or scalar field that accelerates the Universe on a larger scale. There are numerous studies on the nature of dark energy and cosmic expansion based on different methods. These researches can be organized as follows: to modify all cosmic energy by including new elements of dark energy and to modify the action of Einstein-Hilbert to obtain various types of modified theories that include $f(R)$ gravity [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 41, 40, 39, 42, 43, 44], $f(\mathbb{T})$ gravity [45, 46, 47, 48, 49], $f(R,T)$ gravity [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68], $f(\mathbb{T})$ gravity [69, 70], $f(\mathbb{T},T)$ gravity [71, 72, 73, 74, 75, 76, 77, 78], $f(\mathcal{G})$ gravity [79, 80, 81], $f(R,\mathcal{G})$ gravity [82], where $R$, $T$, $\mathbb{T}$ and $\mathcal{G}$ are Ricci’s scalar, trace of stress-energy tensor, torsion scalar and Gauss-Bonnet scalar respectively. The $f(R)$ theory is an appropriate theory that modifies Einstein’s gravity with the substitution of action $R$, by a general expression of scalar curvature $R$ [83, 84, 85, 86, 87, 88]. Starobinsky [89] assumed a cosmological model of inflation employing $f(R)$ gravity. Starobinsky [90], Hu and Sawicki [91] inserted cosmological models employing nonlinear expressions $f(R)$.

In the same spirit, Capozziello and his accomplices [92] exhibited a review of $f(R)$ gravity and talked about the clarification of dark energy and inflation periods, depicting the quickened periods of the Universe, with regards to $f(R)$ gravity. They likewise depicted some significant constituents of inflation, cosmography and quintessence in the context of $f(R)$ gravity theory. Cruz-Dombriz and his partners [93] figured the limits on model variables of modified theories that include quintessence, $f(R)$ and $f(\mathbb{T})$ and utilized cosmography as an instrument to separate these theories. Furthermore, Aviles et al. [94] examined the development of the appropriate type of gravity theory $f(\mathbb{T})$ in a cosmographic manner. The first associated the capacity $f(\mathbb{T})$ and it’s differential coefficients with the cosmographic variables and fixed the cosmographic restrictions which are free of the model and estimated that the obtained outcomes were reliable with another cosmological model. D’Agostino and Luongo [95] considered teleparallel dark energy simultaneously with a non-minimal coupling among the scalar field and torsion and contrasted this situation with that of the quintessence. They exhibited that this energy has an enormous set of arrangements than controlled by minimal quintessence. For this purpose, the same authors investigated the rate of perturbation development and contrasted the outcomes with another cosmological model.

In addition, the treatment of related open research in the cosmological domain is also a promising methodology [24] that is closely related to the $f(R,T)$ theory. This theory is created by considering non-minimal coupling among the trace of the stress-energy tensor $T$ and Ricci Scalar $R$. Harko and his colleagues [50] acquainted it first to handle the problems in a proficient manner, introducing a new modification of Einstein’s theory of general relativity and the $f(R,T)$ theory of gravity. In this $f(R,T)$ theory of gravity, the matter is well-respected on an equivalent equilibrium with configuration. Thus one can investigate numerous fascinating and novel highlights of the Universe, for example, the dark matter part [96]. One can take note of that reliance of $T$ may originate from the thought of quantum impacts or from the nearness of a flawed fluid. The concept of coupling of the matter and geometry fields in a non-minimal manner returns to Rastall [97], who tested the stress-energy conservation law in the bended spacetime just because, a hypothesis upheld by the particle creation during the Universe development [96, 98, 99, 100, 101]. Along these lines, it was appeared by Smalley [102] that a model of Rastall’s gravity, where the uniqueness of the stress-energy tensor is relative to the Ricci scalar gradient, can be logical from a variational rule.

Besides, some static spherically symmetric arrangements of the Rastall field equations have been acquired. For instance, by uncommon adjustment of the parameters of the Rastall hypothesis, a vacuum arrangement having a similar configuration as the Schwarzschild-de Sitter arrangement in the theory of general relativity acquired with cosmological constant acting as a source [103]. In this regard, Bronnikov et al [104] examined the problem of finding static spaces with spherical symmetry in Rastall theory in the presence of a free scalar field or with self-interaction. It was discovered that it is possible to obtain specific arrangements in which part of these arrangements are equivalent to the arrangements that are acquired with respect to the k-essence hypothesis. Furthermore, asymptotically level through-gap arrangements are examined in [105] where it is demonstrated that the Rastall hypothesis parameters influence the wormholes space-time parameters.

Nevertheless, despite its great versatility recently as claimed by H. Velten and T. R. P. Caram$\hat{\text{e}}$s [106] the Cosmological inviability of $f(R,T)$ gravity theory. In their work they tested the general functional

$$f(R,T)=R+\lambda T+\gamma_{n}T\,^{n},$$

(1)

where $R$ is the Ricci scalar, $T\equiv g^{\mu\nu}T_{\mu\nu}$ the trace of the energy-momentum tensor, $\lambda$, $\gamma$ and $n$ constants with $n$ restricted to be $n\geq 0$. It was shown by the authors that the general model (1) for different values of the parameter $n$ does not match the observational data. Hence, it is not possible to explain the existences of dark components ı.e, dark energy and dark matter. Of course, this antecedent reduce the utility of $f(R,T)$ gravity theory as an alternative to GR to explain the accelerated era of the Universe. However, the general functional given by (1) and others models with logarithmic form containing the contribution of the trace of the energy-momentum tensor have been used to study the feasibility in the line of compact structures describing stellar interiors in the arena of $f(R,T)$ theory. In this regards, the formulation of a suitable solution of a spherically symmetrical self-gravitating structure has still been difficult due to the presence of non-linearity in the field equations. Many studies have been made in the literature to address this problem. Mak and Harko [107] acquired a precise anisotropic disposition of the field equations and establish a positively finished performance of pressure and density that contribute to the nucleus of astrophysical objects. Gleiser and Dev [108] studied anisotropic self-gravitating frame algorithm with a mass-to-radius ratio $M/R=4/9$ and acquired steady model for small estimates of the adiabatic index. Kalam and his colleagues [109] discovered compact models with regards to the anisotropic system utilizing Krori and Barua metric. Maurya et al. [110] examined compact stars anisotropic solutions in terms of charge distribution. Moreover, in [111, 112] were obtained compact objects representing real celestial bodies such as neutron and quark stars. These solutions were study within the background of $f(R,T)$ gravity theory by employing the so called embedding class I approach or Karmarkar condition [113].

The existence of accurate inside solutions of self-gravitating frameworks within the sight of anisotropy has been achieved in different manners. In this respect, the Minimal Gravitational Decoupling (MGD) procedure showed up as importantly settled in finding the physically feasible answers for spherically symmetric astrophysical geometry. Ovalle [114] used this methodology to find precise answers to compact astrophysical objects as part of Braneworld. However, the complete details of the MGD approach in the Brane-World problem is given in Ref. [115]. Moreover, this MGD approach is a very powerful technique for exploring new spherically symmetrical solutions of Einstein’s field equations. The principal property of this method is that a straightforward solution can be prolonged to more complicated areas. Ovalle and his collaborators [116, 117, 118, 119, 120, 121, 122, 123, 124] have done pioneering works in the context of MGD as well as in the extended MGD scenario. The starting point of this procedure is to couple to the seed energy-momentum tensor $T^{(\text{m})}_{\mu\nu}$ representing a perfect fluid (this represents the simplest case, however it can be anisotropic also) an additional source $\theta_{\mu\nu}$. This is done through a dimensionless parameter, namely $\alpha$. So, the extended or total energy-momentum tensor now reads $T^{(m)}_{\mu\nu}$ $\longrightarrow$ $T^{(tot)}_{\mu\nu}$ = $T^{(m)}_{\mu\nu}$ + $\alpha\theta_{\mu\nu}$. The $\alpha$ parameter plays an important role into the feasibility of describing anisotropic compact structures. It sign and magnitude depends on the mechanism selected to close the $\theta$-system in order to determine the $\theta_{\mu\nu}$ components and the decoupler function. In this respect, one has several ways to face the closure of the set of equations corresponding to the additional source $\theta_{\mu\nu}$. For example in the seminal work [125] the mimic constraint procedure was proposed and employed to deform Tolman IV solution, what is more the same technique was used in [126], [127] and [128] to extend the Durgapal-Fuloria, Heintzmann IIa and Charged Heintzmann IIa solution respectively, to an anisotropic domain. Basically, this approach consists of matching the isotropic pressure $p$ with the $\theta^{r}_{r}$ component or the seed energy-density $\rho$ with the temporal component $\theta^{t}_{t}$, arriving to an algebraic equation in the former or a differential equation in the second case. It should be noted that all this explanation is within the framework of GR. In the case of modified gravity theories the resulting equations leading to the determination of the deformation function $f(r)$ can be different. So, this way yields to find the concrete form of the minimal deformation function $f(r)$. Nevertheless, as was shown in [129, 130] also it is possible to impose an adequate form for $f(r)$ respecting all the physical and mathematical requirements in order to close the $\theta$-system of equations. So, in the previous works by using the mimic constraint procedure naturally a lower and upper bounds appear when $p=\theta^{r}_{r}$, namely $0<\alpha<1$. This restriction on $\alpha$ are to ensure a positive defined anisotropy factor $\Delta$ everywhere. In the case when $\rho=\theta^{t}_{t}$ to guarantees $\Delta>0$ the parameter $\alpha$ is allowed to take negative values. On the other hand, the methodology following in [129, 130] is not too much restrictive in the choice of $\alpha$ (sign and magnitude), because it depends on the behaviour of the model.

In these circumstances, Casadio and his partners [131] implemented external responses to the self-gravitating frame with spherical symmetry using the gravitational decoupling method and established a geometry of naked singularity at the Schwarzschild radius. Ovalle et al [125] expanded the isotropic arrangement by incorporating an anisotropy employing the MGD technique for a static astrophysical object system. According to this technique, Sharif and Sadiq [132] studied a charged anisotropic spherical solution and further analyzed the stability criteria based on sound velocity, availability conditions.

In the present article the gravitational decoupling by means of MGD is combined with the $f(R,T)$ gravity methodology to investigate the the possibility of obtaining in analitycal way compact structures describing real static and spherically symmetric self-gravitating configurations such as neutron or quark stars. As we will see, the general field equations in the picture of $f(R,T)$ gravity theory are more complicated than the GR ones. So, the gravitational decoupling by MGD approach depends on the election of the $f(R,T)$ functional. In this respect we have selected $f(R,T)=R+2\chi T$, being $R$ the Ricci scalar, $T$ the trace of the energy-momentum tensor and $\chi$ a dimensionless constant. This simple choice allows to decouple the $f(R,T)$ sector from the $\theta$-sector, what is more the $f(R,T)$ contribution is carried out into the $\theta$-sector contributing in the final form of the decoupler function $f(r)$. As pointed out above, in [106] was proved that the model given by (1) is not viable from the cosmological point of view (this model matches our election when $\gamma_{n}=0$). Nevertheless, the failure to explain the problems at the cosmological level does not imply that $f(R,T)$ gravity theory should be discarded from the astrophysical plane. In the literature there are some novel works devoted to the study of compact structures within the arena of $f(R,T)$ gravity theory employing a linear functional in $R$ and $T$, showing the feasibility of the model. For example in [133] $f(R,T)$ filed equations were solved numerically finding that the maximum mass allowed for neutron and strange stars are $1.538M_{\odot}$ and $2.017M_{\odot}$ respectively. Furthermore, in[134, 135, 136, 137, 138] were considered the inclusion of different ingredients such as electric charge and anisotropies to solve the problem in analitycal form. Furthermore, the study of collapsed configurations using different models for the $f(R,T)$ functional and investigations on exotic objects such as gravastar can be found at [139, 140]. Hence, all these good antecedents motivate and support the present investigation. Moreover, gravitational decoupling by MGD has shown to be a powerful and versatile method to work out in different context. Therefore, despite the simple choice considered by us about the $f(R,T)$ functional, this work presents a first step in the understanding of how to decouple gravitational sources which represent analytical models for the description of real celestial bodies. What is more, in the limit $\{\alpha\rightarrow 0,\chi\rightarrow 0\}$ general relativity results are recovered, hence the obtained macro-physical observables such as the the mass $M$ and radius $R$ as well as the central data of the thermodynamic variables driven the micro-physical process, namely $\{\rho(0),p_{r}(0)\}$ can be compared with those provided by Einstein theory to check the possibility of exceeding the limits established for these quantities within the framework of classical theory of gravity.

So, we have organized the present paper as follows: fundamental mathematical detailing of $f(R,T)$ gravity is introduced in Section II. In Section III we formulate fundamental stellar equations in $f(R,T)$ gravity theory and present the solution of the Einstein field equations for multiple sources. In Section IV we discuss the gravitational decoupling by the MGD approach in the $f(R,T)$ gravity structure, and present the matching conditions of our stellar model in Section V. In Section VI we incorporate the decoupler MGD procedure, so as to get some of the limitations imposed on the field equations, namely specific the mimic constraints. Section VII discusses the ansatz for the metric function $g_{rr}$ suggested by Korikina and Orlyanskii according to our solutions. Section VIII We examine the dynamical equilibrium of the stellar system. Then in Sections IX, X and XI we analysis the physical acceptability and stability of the stellar system, by investigating energy conditions, causality condition, surface redshift, adiabatic index and fitting of observed values in $M-R$ curves is given in Section XII. Finally, the concluding remarks close this paper in the last section.

Let us present $f(R,T)$ gravity. We introduce the modified gravity theory and its extension to $f(R,T)$ theory. The complete action in this theory is given by

	$\displaystyle S$	$\displaystyle=$	$\displaystyle\frac{1}{16\pi}\int f(R,T)\sqrt{-g}~{}d^{4}x+\int\mathcal{L}_{m}\sqrt{-g}~{}d^{4}x$		(2)
			$\displaystyle+\alpha\int\mathcal{L}_{\theta}\sqrt{-g}~{}d^{4}x.$

Here the generic function $f(R,T)$ contains the trace of the stress-energy tensor $T$ as well as $R$, the Ricci scalar, $\mathcal{L}_{m}$ is the matter Lagrangian density which represents the possibility of non-minimal coupling between geometry and matter, $\mathcal{L}_{\theta}$ is the Lagrangian density of a new sector and $g$ stands for the determinant of the metric $g_{\mu\nu}$. In this respect, the stress-energy tensor is described by the following expressions

	$\displaystyle T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\,\frac{\delta(\sqrt{-g}\,\mathcal{L}_{m})}{\delta g^{\mu\nu}},$		(3)
	$\displaystyle\theta_{\mu\nu}=-\frac{2}{\sqrt{-g}}\,\frac{\delta(\sqrt{-g}\,\mathcal{L}_{\theta})}{\delta g^{\mu\nu}}.$		(4)

Following Harko et al. [50] and using $T=g^{\mu\nu}T_{\mu\nu}$ with respect to the situation of the matter Lagrangian density $\mathcal{L}_{m}$ depends solely on the metric tensor $g_{\mu\nu}$, the stress-energy tensors expressed in Eqs. (3) and (4) can be written as follows

	$\displaystyle T_{\mu\nu}$	$\displaystyle=$	$\displaystyle g_{\mu\nu}\mathcal{L}_{m}-\frac{2\,\partial(\mathcal{L}_{m})}{\partial g^{\mu\nu}},$		(5)
	$\displaystyle\theta_{\mu\nu}$	$\displaystyle=$	$\displaystyle g_{\mu\nu}\mathcal{L}_{\theta}-\frac{2\,\partial(\mathcal{L}_{\theta})}{\partial g^{\mu\nu}}.$		(6)

The full action (2) can be varied regarding the metric tensor $g^{\mu\nu}$ to acquire the general gravitational field equations for $f(R,T)$ gravity as

	$\displaystyle\left(R_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}\right)f_{R}(R,T)+\Box f_{R}(R,T)g_{\mu\nu}-\frac{1}{2}f(R,T)g_{\mu\nu}$
	$\displaystyle=8\pi\,(T_{\mu\nu}+\alpha\,\theta_{\mu\nu})-f_{T}(R,T)\,\left(T_{\mu\nu}+\Theta_{\mu\nu}\right),~{}~{}~{}~{}~{}$		(7)

where $f_{R}(R,T)={\partial f(R,T)}/{\partial R}$, $f_{T}(R,T)={\partial f(R,T)}/{\partial T}$ and $\nabla_{\mu}$ is the covariant derivative related to the Levi-Civita connection of metric tensor $g_{\mu\nu}$ and the D’Alambert operator $\Box$ which is defined as

$\displaystyle\Box\equiv\partial_{\mu}(\sqrt{-g}~{}g^{\mu\nu}\partial_{\nu})/\sqrt{-g},~{}~{}\textrm{and}~{}~{}\Theta_{\mu\nu}=g^{\mu\nu}\delta T_{\mu\nu}/\delta g^{\mu\nu}.$

In obtaining the covariant derivative expression of the stress-energy tensor as well as the algebraic function extract, the covariant derivative of formula (II) is performed as follows

	$\displaystyle\nabla^{\mu}T_{\mu\nu}$	$\displaystyle=$	$\displaystyle\frac{f_{T}(R,T)}{8\pi-f_{T}(R,T)}\bigg{[}(T_{\mu\nu}+\Theta_{\mu\nu})\nabla^{\mu}\ln f_{T}(R,T)$		(8)
		$\displaystyle+$	$\displaystyle\nabla^{\mu}\Theta_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\nabla^{\mu}T-\frac{8\pi\,\alpha}{f_{T}\left(R,T\right)}\nabla^{\mu}\theta_{\mu\nu}\bigg{]}.$

It is obvious to see that the stress-energy tensor $T_{\mu\nu}$ in $f(R,T)$ gravity theory is not preserved, as in other modified gravity theories [141, 142]. Now using Eq. (5), we define the tensor formula $\Theta_{\mu\nu}$ as follows

$$\Theta_{\mu\nu}=-2T_{\mu\nu}+g_{\mu\nu}\mathcal{L}_{m}-2g^{\alpha\beta}\,\frac{\partial^{2}\mathcal{L}_{m}}{\partial g^{\mu\nu}\,\partial g^{\alpha\beta}}.~{}~{}~{}$$

(9)

To describe inside spacetime of the spherically symmetric and static astrophysical structure in Schwarzschild-like coordinates ($t,r,\theta,\phi$), we use line element as follows

$$ds^{2}=-e^{\nu(r)}\,dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),$$

(10)

where $e^{\nu(r)}$ and $e^{\lambda(r)}$ depict the gravitational potentials of astrophysical configuration. In the further investigation, we adopt the geometrical units $G=c=1$. We suppose that the inside of spherical object is filled of an ideal fluid source, in the current study we are taking into consideration the stress-energy tensor in the following form

$$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}-pg_{\mu\nu},$$

(11)

where the covariant component ${u_{\nu}}$ denote the 4-velocity, fulfilling $u_{\mu}u^{\mu}=-1$ and $u_{\nu}\nabla^{\mu}u_{\mu}=0$. Here, $\rho$ and $p$ represent the pressure and matter density for isotropic matter. In the current investigation, according to the definition proposed by Harko et al. [50], we assume $\mathcal{L}_{m}=-p$ and applying the tensor (9) we find

$\displaystyle\Theta_{\mu\nu}=-2T_{\mu\nu}-p\,g_{\mu\nu}.$

(12)

The algebraic function $f(R,T)$ can be chosen in many ways corresponding to feasible models. In the present work, we have considered the algebraic function as

$$f(R,T)=R+2\chi T$$

(13)

in order to establish the effective energy-momentum tensor ${T}^{eff}_{\mu\nu}$, where $\chi$ indicates a coupling constant. This choice of $f(R,T)$ can be use to resolve the cosmological constant problem, where the cosmological constant is represented by an effective value proportional to Hubble function ($H=\dot{a}/a$) i.e. $\Lambda_{eff}\propto H^{2}$ [50, 51]. By using this linear function and with the help of the general function of motion (II), Einstein’s tensor is explicitly reduced as

	$\displaystyle G_{\mu\nu}=8\pi\,(T_{\mu\nu}+\alpha\,\theta_{\mu\nu})+\chi Tg_{\mu\nu}+2\chi(T_{\mu\nu}+p\,g_{\mu\nu})$
	$\displaystyle=8\pi\tilde{T}_{\mu\nu}+8\pi\,\alpha\,\theta_{\mu\nu}={T}^{eff}_{\mu\nu}.~{}~{}~{}$		(14)

It should be noted that when the algebraic function reduces to curvature scalar $R$, the field equations (II) turning directly to Einstein field equations. Examining such specific linear supposition has generally acknowledged addressing cosmological as well as stellar solutions. By exchanging the value of the functional $f(R,T)$ in Eq. (8), we get

	$\displaystyle\nabla^{\mu}T_{\mu\nu}=-\frac{1}{2\,\left(4\pi+\chi\right)}\chi\bigg{[}g_{\mu\nu}\nabla^{\mu}T+2\,\nabla^{\mu}(pg_{\mu\nu})$
	$\displaystyle+\alpha~{}\frac{8\pi}{\chi}\nabla^{\mu}\theta_{\mu\nu}\bigg{]}.$		(15)

However, this Eq. (II) can be written in the following achievable form $\nabla^{\mu}{T}^{eff}_{\mu\nu}$=$0$, which depicts the conservation of the stress-energy tensor ${T}^{eff}_{\mu\nu}$ for the effective distribution of the matter. Note that if we disable the coupling $\alpha$ and the free parameter $\chi$ takes the value zero, the standard results due to the general relativity can be achieved.

As we are focused on examining a spherical fluid, we consider the static, spherically symmetric astrophysical structure, for which the most general line element of the inside spacetime, can be depicted as the following metric

$${ds}^{2}=-{{e}^{\nu(r)}}{{dt}^{2}}+{{e}^{\eta(r)}}{{dr}^{2}}+{r}^{2}({{d\theta}^{2}}+{{\sin}^{2}}\theta~{}{{d\phi}^{2}}).$$

(16)

where $\nu$ and $\eta$ are metric potentials and depend only on the radial coordinate r, for which the staticity of spacetime is guaranteed. For a spherically symmetric stellar system, combining Eqs. (11), (12), (II) and (16), we arrive at the following differential equations

	$\displaystyle 8\pi(\bar{\rho}+\alpha~{}\theta^{t}_{t})$	$\displaystyle=$	$\displaystyle{{\rm e}^{-\eta}}\left({\frac{\eta^{{\prime}}}{r}}-\frac{1}{r^{2}}\right)+\frac{1}{r^{2}},~{}~{}~{}~{}~{}$		(17)
	$\displaystyle 8\pi(\bar{p}-\alpha~{}\theta^{r}_{r})$	$\displaystyle=$	$\displaystyle{{\rm e}^{-\eta}}\left({\frac{\nu^{{\prime}}}{r}}+\frac{1}{r^{2}}\right)-\frac{1}{r^{2}},~{}~{}~{}~{}~{}$		(18)
	$\displaystyle 8\pi(\bar{p}-\alpha~{}\theta^{\varphi}_{\varphi})$	$\displaystyle=$	$\displaystyle\frac{{\rm e}^{-\eta}}{4}\left(2\nu^{{\prime\prime}}+\nu^{\prime 2}+2{\frac{\nu^{{\prime}}-\eta^{{\prime}}}{r}}-\nu^{\prime}\eta^{\prime}\right)$		(19)

where

	$\displaystyle\bar{\rho}$	$\displaystyle=$	$\displaystyle\rho+\frac{\chi}{8\,\pi}(3\rho-p),$		(20)
	$\displaystyle\bar{p}$	$\displaystyle=$	$\displaystyle p-\frac{\chi}{8\,\pi}(\rho-3p)$		(21)

Then from the system (17)-(19), we can write the corresponding conservation law $\nabla^{\mu}T^{(eff)}_{\mu\nu}=0$ as,

	$\displaystyle-\alpha\left[\frac{\nu^{\prime}}{2}\left(\theta^{t}_{t}-\theta^{r}_{r}\right)-\frac{d\theta^{r}_{r}}{dr}+\frac{2}{r}\,(\theta^{\varphi}_{\varphi}-\theta^{r}_{r})\right]-\frac{d\bar{p}}{dr}$
	$\displaystyle-\frac{\nu^{\prime}}{2}(\bar{\rho}+\bar{p})=0.$		(22)

As a whole the equations system (17)-(19), we have three non-linear differential equations for seven unknown functions: $\{\eta,\nu\}$, $\{\rho^{(\text{eff})},p^{(\text{eff})}\}$ and $\{\theta^{t}_{t},\theta^{r}_{r},\theta^{\varphi}_{\varphi}\}$, where these grouped functions in braces represent the metric potentials, the thermodynamic observables and the extra source components, respectively. So as to get these unknown we choose a systematic methodology. Moreover, the equations system (17)-(19), which includes total energy density, total radial and total tangential pressures, can be classified as follows

	$\displaystyle\rho^{\rm{(eff)}}$	$\displaystyle=$	$\displaystyle\bar{\rho}+\alpha~{}\theta^{t}_{t}=\rho+\frac{\chi}{8\,\pi}(3\rho-p)+\alpha~{}\theta^{t}_{t},$		(23)
	$\displaystyle p^{(\text{eff})}_{r}$	$\displaystyle=$	$\displaystyle\bar{p}-\alpha~{}\theta^{r}_{r}=p-\frac{\chi}{8\,\pi}(\rho-3p)-\alpha~{}\theta^{r}_{r},$		(24)
	$\displaystyle p^{(\text{eff})}_{t}$	$\displaystyle=$	$\displaystyle\bar{p}-\alpha~{}\theta^{\varphi}_{\varphi}=p-\frac{\chi}{8\,\pi}(\rho-3p)-\alpha~{}\theta^{\varphi}_{\varphi}.$		(25)

Once more, we remark that when $\theta^{r}_{r}\neq\theta^{\varphi}_{\varphi}$, anisotropic behavior appears in the system due to the present of the $\theta$-sector. So, we defined the effective anisotropy factor $\Delta^{(\text{eff})}$, in order to measure the anisotropy behavior as follows

$$\Delta^{(\text{eff})}=p^{(\text{eff})}_{t}-p^{(\text{eff})}_{r}=\alpha\left(\theta^{r}_{r}-\theta^{\varphi}_{\varphi}\right).$$

(26)

In this respect, the equations system (17)-(19) could undoubtedly be dealt as an anisotropic fluid, which should make it possible to consider five unknown functions, viz., the two metric potentials and the three total functions expressed in the equations system (17)-(19). However, we will perform an alternate methodology, as clarified below.

In this part, we use the minimal geometric deformation gravitational decoupling procedure as previously mentioned. So, for extending spherical and static isotropic fluid solutions to anisotropic areas, it is necessary that the gravitational decoupling by MGD approach becomes a straightforward and efficacious implement. Let us consider the perfect fluid solution $\{\xi,\mu,\bar{\rho},\bar{p}\}$ by deactivating the coupling $\alpha$, so that the canonical line element (16) becomes

$${ds}^{2}=-{{e}^{\xi(r)}}{{dt}^{2}}+\frac{{dr}^{2}}{\mu(r)}+{r}^{2}({{d\theta}^{2}}+{{\sin}^{2}}\theta~{}{{d\phi}^{2}}),$$

(27)

where $\mu(r)=1-2m/r$ is the standard term for the mass function. Now, in order to see the $\theta$-sector impacts generated by the gravitational source $\theta_{\mu\nu}$ in the equations system (17)-(19), we have to incorporate the coupling parameter $\alpha$ in the perfect fluid distribution. In order to incorporate the impacts of the new sector $\theta_{\mu\nu}$ in uncharged anisotropic model, we consider the geometric deformation functions ${\xi}$ and ${\mu}$ as

	$\displaystyle\xi\rightarrow\nu$	$\displaystyle=$	$\displaystyle\xi+\alpha h$		(28)
	$\displaystyle\mu\rightarrow e^{-\eta}$	$\displaystyle=$	$\displaystyle\mu+\alpha f,$		(29)

where $f$ and $h$ represent the deformations related to the components of the radial and temporal line elements, respectively. It should be noted that the previous deformations are purely radial functions, this characteristic ensures the spherical symmetry of the solution. The assumed MGD method relates to set $h=0$ or $f=0$, for this situation the deformation will be performed only on the radial component, remaining the temporal one unaltered, that is to say that $h=0$. In this respect where $h=0$, we have

$$\mu(r)\rightarrow e^{-\eta(r)}=\mu(r)+\alpha f(r),$$

(30)

which is known as the Minimal Geometric Deformation (MGD).

Using now this Eq. (30) is easy to see that the equations system (17)-(19) splits in two equations systems. The first of them is

$\displaystyle 8\,\pi\,\bar{\rho}$

$\displaystyle=$

$\displaystyle-\frac{\mu^{\prime}}{r}-\frac{\mu}{r{{}^{2}}}+\frac{1}{r^{2}}=8\,\pi\rho+\chi\,(3\rho-p),$

(31)

$\displaystyle 8\,\pi\,\bar{p}$

$\displaystyle=$

$\displaystyle\mu\left({\frac{{\nu}^{{\prime}}}{r}}+\frac{1}{r^{2}}\right)-\frac{1}{r^{2}}=8\,\pi\,p-\chi\,(\rho-3p),$

(32)

	$\displaystyle 8\,\pi\,\bar{p}$	$\displaystyle=$	$\displaystyle\frac{\mu}{4}\left({2\nu}^{{\prime\prime}}+{\nu}^{{\prime\,2}}+2\frac{{\nu}^{{\prime}}}{r}\right)+\frac{\mu^{\prime}}{4}\,\left({\nu}^{{\prime}}+\frac{2}{r}\right)$		(33)
			$\displaystyle\hskip 85.35826pt=8\,\pi\,p-\chi\,(\rho-3p),~{}~{}~{}~{}~{}$

which corresponds to $\alpha=0$, implying that the distribution of the fluid matter is perfect.

From now we will call the equations system (31)-(33) as modified $f(R,T)$ gravity system. Furthermore, with the considerations relating to Eqs.(20) and (21), the quantities $\rho$ and $p$ can be written in terms of metric potentials as follows

	$\displaystyle\rho=\frac{1}{(8\pi+4\chi)\,(8\pi+2\chi)\,r^{2}}\,\big{\{}(8\pi+3\chi)(1-\mu^{\prime}\,r-\mu)$
	$\displaystyle+\chi\,(\mu\,\nu^{\prime}\,r+\mu-1)\,\big{\}},~{}~{}~{}~{}~{}$		(34)
	$\displaystyle p=\frac{1}{(8\pi+4\chi)\,(8\pi+2\chi)\,r^{2}}\,\big{\{}(8\pi+3\chi)\,(\mu\,\nu^{\prime}\,r+\mu-1)$
	$\displaystyle+\chi\,(1-\mu^{\prime}\,r-\mu)\,\big{\}}.~{}~{}~{}~{}$		(35)

while the pressure isotropy equation $G^{1}_{1}=G^{2}_{2}$ reduces to

$$\begin{split}4\left(1-\mu\right)+2r\left(\mu^{\prime}-\mu\nu^{\prime}\right)+r^{2}\left(2\mu\nu^{\prime\prime}+\mu\nu^{\prime 2}+\mu^{\prime}\nu^{\prime}\right)=0.\end{split}$$

(36)

So from this Eq. (36) we notice that all the solutions obtained in the general relativity case are also present in the modified $f(R,T)$ gravity theory case. Moreover, it is clear that, despite the qualitative subtlety between the solutions for Einstein’s theory of gravity and the modified $f(R,T)$ theory of gravity, there is a quantitative difference between the two models when considering the geometric and material content. In this regard the two models share only the geometrical content but not the material one, is in this situation that any solution to Einstein theory of gravity can be viewed as a solution in the gravitational $f(R,T)$ system. The standard expression for the mass takes the new form as,

	$\displaystyle m(r)$	$\displaystyle=$	$\displaystyle 4\pi\int_{0}^{r}{\rho(r)\,r^{2}\,dr}$		(37)
	$\displaystyle m(r)$	$\displaystyle=$	$\displaystyle\frac{4\pi}{[(8\pi+3\chi)^{2}-\chi^{2}]}\,\int^{r}_{0}\Big{\{}2\,(4\pi+\chi)\,(1-e^{-\lambda})$		(38)
		$\displaystyle+$	$\displaystyle(8\pi+3\chi)\lambda^{\prime}\,r\,e^{-\lambda}+\chi\,\nu^{\prime}\,r\,e^{-\lambda}\Big{\}}\,dr,$

Here $\chi$ is a dimensionless coupling constant, which plays a fundamental role for determining the astrophysical configuration. At this regard, it is essential to point out that when the coupling constant $\chi$ is zero, the modified $f(R,T)$ gravity recovers the same physical behavior as the general relativity. It should be noted that we have considered the Eq. (36), as the main differential equation in this study contains two fundamental integration constants. In any case, we can adapt the constants regarding the mass $M$ and radius $R$ of the distribution by solving the linear corresponding equation.

As previously mentioned, the quantities $\rho$ and $p$ after some algebraic handling in their proper formulas contain the coupling constant as anticipated. Moreover, using the equations (34)-(35) with respect to $\chi=0$ we find the standard GR field equations for distributions of the isotropic matter. additionally, combining Eqs (34) and (35), we obtain the usual inertial mass density $\rho+p$ in the following form

$$\rho+p=\frac{\mu\nu^{\prime}-\mu^{\prime}}{r}.$$

(39)

On the other hand, the second equations system reads

	$\displaystyle-\frac{f{{}^{\prime}}}{r}-\frac{f}{r^{2}}$	$\displaystyle=$	$\displaystyle 8\pi\theta^{t}_{t},$		(40)
	$\displaystyle-f\left({\frac{{\nu}^{{\prime}}}{r}}+\frac{1}{r^{2}}\right)$	$\displaystyle=$	$\displaystyle 8\pi\theta^{r}_{r}$		(41)
	$\displaystyle-\frac{f}{4}\left({2\nu}^{{\prime\prime}}+{\nu}^{{\prime\,2}}+2\frac{{\nu}^{{\prime}}}{r}\right)-\frac{f^{\prime}}{4}\left({\nu}^{{\prime}}+\frac{2}{r}\right)$	$\displaystyle=$	$\displaystyle 8\pi\theta^{\varphi}_{\varphi},$		(42)

and the conservation equations associated with the equations ensemble (34)-(35) and (38)-(40) are

	$\displaystyle\frac{{\nu^{\prime}}}{2}\,({\rho}+{p})+\frac{d{p}}{dr}-{\frac{\chi}{2(4\pi+\chi)}}\frac{d}{dr}\left(\rho-p\right)=0,$		(43)
	$\displaystyle-\frac{{\nu^{\prime}}}{2}\,(\theta^{t}_{t}-\theta^{r}_{r})+\frac{d\theta^{r}_{r}}{dr}-\frac{2}{r}\,(\theta^{\varphi}_{\varphi}-\theta^{r}_{r})=0.$		(44)

Furthermore, we remark that the linear combination of conservation equations (43) and (44) with respect to coupling constant $\alpha$ gives the conservation equation for the total stress-energy tensor ${T}^{\mu{(\text{eff})}}_{\nu}={\bar{T}}^{\mu}_{\nu}+\alpha\theta^{\mu}_{\nu}$, in the form

	$\displaystyle-\frac{d{p}}{dr}-\alpha\left[\frac{{\nu^{\prime}}}{2}\,(\theta^{t}_{t}-\theta^{r}_{r})-\frac{d\theta^{r}_{r}}{dr}+\frac{2}{r}\,(\theta^{\varphi}_{\varphi}-\theta^{r}_{r})\right]-\frac{{\nu^{\prime}}}{2}\,({\rho}+{p})$
	$\displaystyle\hskip 110.96556pt+{\frac{\chi}{2(4\pi+\chi)}}\frac{d}{dr}\left(\rho-p\right)=0.$		(45)

So from this Eq. (IV), we notice that the last factor is an additional term which is coupled via $\chi$ parameter in the $f(R,T)$ gravity arena. Furthermore, this additional term could on a fundamental level be attractive or repulsive in nature, due to its dynamical behavior of the physical parameters like energy density and pressure depends on the behavior of the free parameter $\chi$. In the limiting case of $\chi=0$, Eq. (IV) is similar to the explicit form of conservation law $\nabla^{\mu}T^{(eff)}_{\mu\nu}=0$ given in Eq. (III).

Now it is important to remark that from now onward we will characterize the complete physical parameters for the energy density, radial pressure and tangential pressure as follows :

	$\displaystyle\rho^{\text{(}tot)}(r)$	$\displaystyle=$	$\displaystyle\rho(r)+\alpha~{}\theta^{t}_{t}(r),$		(46)
	$\displaystyle p_{r}^{\text{(}tot)}(r)$	$\displaystyle=$	$\displaystyle p(r)-\alpha~{}\theta^{r}_{r}(r),$		(47)
	$\displaystyle p_{t}^{\text{(}tot)}(r)$	$\displaystyle=$	$\displaystyle p(r)-\alpha~{}\theta^{\varphi}_{\varphi}(r),$		(48)

where the quantities $\rho$ and $p$ are specified by Eqs. (34) and (35), respectively. On the other hand, we can see that the Eqs. (34)-(35) depends on the extra geometric term which is coupled by means of $\chi$ parameter. For this purpose, we can conclude that the physical quantities such as, the total anisotropy factor ($\Delta^{\text{tot}}$) and the effective anisotropy factor ($\Delta^{\text{eff}}$) are characterized by the Eqs. (47)-(48) and Eq. (26) respectively, are indistinguishable. Now the inner stellar geometry for the present MGD model can be given by the following line element,

$$ds^{2}=-{{e}^{\nu(r)}}{{dt}^{2}}+\left[1-\frac{2\,\bar{m}(r)}{r}\right]^{-1}{{dr}^{2}}+{r}^{2}({{d\theta}^{2}}+{{\sin}^{2}}\theta{{d\phi}^{2}}),$$

(49)

where $\bar{m}(r)$ defines the internal mass of the MGD model which can be given as,

$$\bar{m}(r)=\frac{r}{2}\left[\frac{2\,m(r)}{r}-{\alpha}\,f(r)\right].$$

(50)

To ensure a well behaved compact structure i.e., a confined and finite matter distribution with well posed mass $M$ and radius $R$, it is necessary to join the inner geometry $\mathcal{M^{-}}$ at the surface $\Sigma=r=R$ with the exterior space-time $\mathcal{M^{+}}$ surrounding the configuration. In this respect, in the case of general relativity framework the outer manifold is well known, precisely it corresponds to Schwarzschild vacuum space-time (in considering uncharged, non-radiating and static compact object). However, in the context of $f(R,{T})$ gravity the outer manifold surrounding the fluid sphere does not necessarily coincide with the Schwarzschild solution, what is more this exterior space-time could in principle receive contributions from the material sector given by the trace of the energy-momentum tensor due to the breakdown of the minimum coupling matter principle between the gravitational and the material sector. Thus, the usual junction conditions applicable in general relativity i.e., Israel-Darmois (ID from now on) [143, 144] matching conditions could not work in this context any more or should be appropriate redefined. So, a simple way to see how the contributions coming from the $f(R,{T})$ function could affect the exterior space-time, is re-written the field equations (II) as follows (putting $\alpha=0$),

$$\begin{split}G_{\mu\nu}\equiv R_{\mu\nu}-\frac{R}{2}g_{\mu\nu}=\frac{1}{f_{R}}\bigg{[}8\pi T_{\mu\nu}+\frac{f}{2}g_{\mu\nu}-\frac{R}{2}f_{R}~{}g_{\mu\nu}&\\ -\left(T_{\mu\nu}+\Theta_{\mu\nu}\right)f_{T}-\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)f_{R}\bigg{]},\end{split}$$

(51)

then by taking the trace of the above equation one obtains

$$\begin{split}R=\frac{1}{f_{R}}\bigg{[}8\pi T+2f-\left(T+\Theta\right)f_{T}-3\,\Box f_{R}\bigg{]}.\end{split}$$

(52)

So, in considering an empty matter field i.e, $T_{\mu\nu}=0\rightarrow T=0$ Eq. (52) yields to

$$R=\frac{1}{f_{R}}\left[2f_{1}-3\,\Box f_{R}\right],$$

(53)

where $f_{1}$ represents the geometrical part of the $f(R,{T})$ function that is $f_{1}\equiv f_{1}(R)$. This is so because the $f(R,{T})$ function can be seen as $f(R,{T})=f_{1}(R)+f_{2}(T)$. Therefore, it is clear that a vanishing energy-momentum tensor in the framework of $f(R,{T})$ gravity theory does not mean a null Ricci scalar like in general relativity where for $T_{\mu\nu}=0\rightarrow R=0\rightarrow R_{\mu\nu}=0$ which represents a vacuum space-time. What is more, $T_{\mu\nu}=0$ does not imply $f_{2}=0$ at all, of course this term could contributes with a constant term for example. However, if $T_{\mu\nu}=0$ leads to $f_{2}=0$, the outer manifold is affected by the geometric terms encoded in $f_{1}$ and $f_{R}$. This last situation also happens in $f(R)$ gravity theory, where $R^{2}$, $R^{3}$ and so on, modify the interface between the inner geometry and the exterior one [83].

At this point it is clear that the ID conditions will work as they do in general relativity, if and only if one ensures that the external solution matches those of general relativity. Obviously it is not an easy task to achieve it, since the function $f(R,T)$ can be as complex as one wants. As regards this, equation (13) ensures that the contributions of the geometric and material sector remain confined within the range $0\leq r\leq R$. Moreover, the fact that the function $f(R,T)$ is linear in $R$ plus a linear coupling in $T$ through an adjustable parameter $\chi$, can be seen as a general relativity model coupled to a variable cosmological constant which also breaks the minimal coupling matter with the gravitational sector. Concretely, by inserting (13) into Eq. (53) and solving for $R$ provides $R=0$ and hence $R_{\mu\nu}=0$ (in Eq. (51)) which represents a vacuum exterior space-time ı.e, the Schwarzschild geometry surrounding the fluid sphere. On the other hand, only the potential contributions coming from the $f(R,T)$ model was analyzed, however one needs to study the behaviour of the extra piece of the energy-momentum tensor i.e., the $\theta$-sector. This new term could also, in principle, modify the material content of the external space-time. So, the specific geometry describing this external manifold will be given by

$$ds^{2}=-\left[1-\frac{2{M}}{r}\right]dt^{2}+\frac{dr^{2}}{1-\frac{2{M}}{r}+\alpha\,g(r)}+r^{2}d\Omega^{2},$$

(54)

where $g(r)$ is the geometric deformation for the exterior Schwarzschild space-time due to $\theta_{\mu\nu}$ source. The metric (54) represents a deformed Schwarzschild space-time which is not vacuum any more. However, without loss of generality and for the sake of simplicity one can render $g(r)$ to be null recovering the usual outer vacuum space-time.

So, to join in a smoothly way the internal configuration with the exterior one, the ID junction conditions involve the first and second fundamental forms. The first fundamental form express the continuity of the metric potentials across the boundary $\Sigma$. More specifically, the metric potentials describe the intrinsic geometry of the manifolds. So, the first fundamental form reads

$$\left[ds^{2}\right]_{\Sigma}=0,$$

(55)

concisely

$$e^{\nu^{-}(r)}|_{r=R}=e^{\nu^{+}(r)}|_{r=R},$$

(56)

and

$$e^{\eta^{-}(r)}|_{r=R}=e^{\eta^{+}(r)}|_{r=R},$$

(57)

standing $"-"$ and $"+"$ the inner and external geometry respectively. The second fundamental form is related with the continuity of the extrinsic curvature $K_{\mu\nu}$ induced by $\mathcal{M}^{-}$ and $\mathcal{M}^{+}$ on $\Sigma$. The continuity of $K_{rr}$ component across $\Sigma$ yields to

$$\left[p^{(\text{tot})}_{r}(r)\right]_{\Sigma}=\left[p(r)-\alpha\,\theta^{r}_{r}(r)\right]_{\Sigma}=0.$$

(58)

At this stage we would like to mention that, $p^{(\text{tot})}_{r}$ has taken a different form in comparison with the expression (24) since $\rho$ and $p$ were separated implying that the coupling parameter $\chi$ is involved. Now the terms coming from the coupling parameter are encoded in separate expressions for $\rho$ and $p$ given by Eqs. (34)-(35). From this point of view it is clear how $\chi$ contribution comes into the pressure equation. Now we shall denote the extra components for coupling parameter $\chi$ in the expression (35) as follows

$$F_{\chi}(r)=\chi\,F^{1}_{\mu\nu}(r)-h_{\chi}\,\sum_{n=0}^{\infty}(-1)^{n}(h_{\chi})^{n}\,[F^{2}_{\mu\nu}(r)+\chi\,F^{1}_{\mu\nu}(r)].$$

(59)

where,

	$\displaystyle F^{1}_{\mu\nu}(r)=\frac{3\mu}{8\pi}\bigg{(}\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\bigg{)}-\frac{1}{8\pi}\bigg{(}\frac{\mu^{\prime}}{r}+\frac{\mu}{r^{2}}-\frac{1}{r^{2}}\bigg{)},$
	$\displaystyle F^{2}_{\mu\nu}(r)=-\bigg{(}\frac{\mu^{\prime}}{r}+\frac{\mu}{r^{2}}\bigg{)}+\frac{1}{r^{2}},~{}~{}~{}~{}h_{\chi}=\bigg{(}\frac{\chi}{4\pi}+\frac{\chi^{2}}{8\pi^{2}}\bigg{)}.$

It should be noted that the form of $F_{\chi}$ depends on the choice of $T_{\mu\nu}$ which in our case is given by Eq. (11) describing a perfect fluid matter distribution. At this point a couple of comments are pertinent. First, $p(r)$ in Eq.(47) is given by Eq. (35). Second, in this way the $f(R,T)$ contribution will come into the $\theta$-sector through the decoupler function $f(r)$ (as we will see later) in order to see the effects on it. So Eq. (58) reads

$$\left[p(r)-\alpha\,\theta^{r}_{r}(r)\right]_{r=R^{-}}=\left[-F_{\chi}(r)-\alpha\,\theta^{r}_{r}(r)\right]_{r=R^{+}}.$$

(60)

Taking into account the previous discussion Eq. (60) becomes to

$$\left[p(r)-\alpha\,\theta^{r}_{r}(r)\right]_{r=R^{-}}=0.$$

(61)

Moreover, by using Eq. (41) in (58) we obtain

$$\begin{split}p(R)+\alpha f(R)\left(\frac{1}{R^{2}}+\frac{\nu^{\prime}(R)}{R}\right)=0.\end{split}$$

(62)

As said earlier, equation (62) is an important result since the compact object will therefore be in equilibrium in a true exterior space-time without material content (vacuum) only if the total radial pressure at the surface vanishes. The obtained condition (62) determines the size of the object ı.e the radius $R$ which means that the material content is confined within the region $0\leq r\leq R$. Furthermore, the continuity of the remaining components $K_{\theta\theta}$ and $K_{\phi\phi}$ leads to

$$\bar{m}(R)=\bar{M}.$$

(63)

Equation (63) refers to the total effective mass contained in the sphere.

It is interesting to consider the form adopted by the field equations (34)-(35) and (40)-(42). In this respect, we incorporate the gravitational decoupling technique (30), one obtain a few limitations imposed on these field equations, namely the mimic constraints. Furthermore, these options lead to new solutions are well behaved from the physical and mathematical point of view as well as free of geometrical singularities, violation of the causality condition, non-decreasing thermodynamic functions etc. Moreover, other choices can be fulfilled through some basic requirements in order to be an admissible model from the physical and mathematical point of view. In this regard, a direct and satisfactory portrayal for the geometric deformation function $f(r)$ [128, 129, 145, 130] should be fulfilled with these basic requirements or only concern the components of $\theta$-term with a barotropic, polytropic or linear state equation.