1 Introduction

Despite research over several years by numerous researchers, the exact nature of the compact stars is still somewhat unknown. With the rapid advancement in observational techniques, theoretically modeling these objects have been ever so important. Speaking of observational techniques, in recent times, fruitful insights about compact stars have been found by studying the accreting systems, where the matter is supplied to the compact star by a donor star, and the observed radiation is generated by accreted matter. The temperature increases gradually as the matter comes near the compact star. Thus the inner parts of this flow emit high-energy X-rays and UV radiation is produced from the outer parts. Alongside this, one can think of the recent detection of gravitational waves as well. For example, the properties of some nuclear parameters (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]) have been constrained by observations by NICER [18, 19] of X-ray emissions coming out of hot spots on the surface of rotating neutron stars, observation of pulsars having mass \(2M_{\odot }\) [20], LIGO and VIRGO [21] observations of gravitational waves coming out of two colliding neutron stars. For probing strong-field gravity, these neutron stars are great sources of their compactness. Several modifications to GR have been influenced by observations of binary neutron star merger GW170817 [22,23,24,25,26,27] and binary pulsars [28,29,30,31,32,33,34,35,36]. It’s particularly fascinating to think about what goes on inside these objects, as they have nuclear matter in tremendously high densities and pressure. So how the fluid distribution may behave is a very important question to answer. Some of these compact objects have very strong magnetic fields, for example, pulsars and several other spinning stars. It can be assumed that these objects consist of elementary particles such as baryons, mesons, and leptons. It must be noted that the simultaneous and precise measurement of the mass and radius of a neutron star is quite difficult. The maximal mass value of a neutron star is still an open topic of research. Some observational studies estimated this limiting value as \(\approx 2M_{\odot }\). Some recent studies have found neutron stars having different masses such as PSR J1614-2230 (\( \sim 1.97 M_{\odot }\) [37]), Vela X-1 (\( \sim 1.8 M_{\odot }\) [38]), and 4U 1822-371 (\( \sim 2M_{\odot }\) [39]). However, LIGO/VIRGO collaboration reported a gravitational wave signal originating from the merging of a black hole and a compact object, whose nature is unknown. This is known as the GW190814 event which invoked debate among the scientific community about its origin. The identification of such an object can be done in several ways, such as, by looking at the mass, checking its tidal deformability or, analyzing the electromagnetic signal originating from the same binary responsible for the gravitational waves. However, for the GW190814 event tidal deformability could not be measured because of inadequate information, and the electromagnetic counterpart was also absent. The mass was found to be 2.5\(-\)2.67 \(M_{\odot }\), which is a bit higher than the expected maximum mass limit of a neutron star. Although it was argued by several researchers that this object can be a black hole, in a recent paper of 2020, it was demonstrated that this secondary compact star is likely to be a neutron star [40]. Later it will be shown that in the current article, the maximum value of mass was found to be \(\sim 2.64 M_{\odot }\), which backs up the theory that the unknown object of the GW190814 event can actually be a neutron star. In astrophysics, one of the main objectives is to find out the configuration and geometry of inshore and intramural substances of compact stars. Another fascinating research area is the formation of compact stars and the phenomenon of gravitational collapse. The first exact interior solution for symmetric stars was obtained by Schwarzschild in 1916 [41]. The gravitational collapse was explored by Oppenheimer and Snyder in 1939 [42]. So far, there are only a handful of interior solutions of Einstein’s field equations, considering perfect fluid distribution. This shows how difficult it is, to find interior solutions to Einstein’s field equations because of high non-linearity. But considering anisotropy often makes it more convenient to find interior solutions. But the question is, can the matter distribution at all be anisotropic inside a compact star? In this context, there has been a detailed analysis in a paper by Herrera and Santos [43], where it was shown that the matter distribution in compact stars is most likely anisotropic, and considering anisotropy is more realistic. In fact, even if the matter distribution was initially isotropic, eventually it may become anisotropic [44]. The idea of anisotropy in compact stars was initially proposed by Ruderman [45] in a review article on the dynamics of pulsar’s structure. Anisotropy in compact stars can generate for various reasons, such as very high-density core, different condensate states such as pion and meson condensate states, rotational motion, relativistic particles, superfluid 3A, different fluids intermixed and so on. Gravitational tidal effects also seem to be resulting in anisotropy, as it causes deformation and hence results in anisotropy. So it can be said that in extremely dense matter distribution, the pressure breaks into radial and tangential components, resulting in anisotropy \((\Delta =p_r -p_t)\). Several researchers have been studying compact stars considering different matter distributions including anisotropy [46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61].

As it was stated earlier, obtaining solutions to Einstein’s field equations is a very difficult task, researchers have adopted several techniques to make the problems easier to solve mathematically. In recent times, one particular methodology has been very popular amongst researchers trying to find solutions to field equations. This is known as the gravitational decoupling method. It was first incorporated by Ovalle [62, 63], in form of minimal gravitational deformation (MGD). MGD technique involves deformation along the radial component of the metric tensor. However, the MGD was further extended to incorporate transformation along both radial and temporal components of the metric tensor. The extended version of MGD is known as CGD. One great advantage of gravitational decoupling (GD) is that it allows the generalization of a simple case, and it can also be used to simplify a relatively complex problem. For example, a simple problem can be generalized by introducing an arbitrary source that can include many physical features including anisotropy, charged distribution, and so on. The first system is the Einstein system and the second system is the quasi-Einstein system. Then the field equations for both systems are then solved separately and the linear combination of the solutions yields the final solution. In order to solve the quasi-Einstein part, several more approaches such as the mimic approach, considering some well-acknowledged solution, and so on can be incorporated, which may make the problem more interesting. In a similar fashion, a complex system can be split into a combination of two sources, one being the Einstein system and another being the quasi-Einstein system and then the same methodology can be followed. The reader can go through several works by various researchers where GD has been effectively used in different contexts to obtain interior solutions for compact stars [64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93].

In this paper, another concept called the vanishing complexity condition has been used. For this, let us shed light on what is meant by “complexity” in the first place. The understanding of complexity can appear very arbitrary and subjective. This is the most challenging factor to quantify the concept such as complexity. For example, if we think in terms of information, a perfect crystal has minimum information so it can be thought of as the least complex system. On contrary, ideal gas has random motion and has maximum information. But, if we think about minimum disequilibrium, ideal gas has minimum disequilibrium whereas, disequilibrium is predominant for the perfect crystal. So one way of overcoming this issue is to express complexity as the product of information and disequilibrium. However, for the self-gravitating system, the much accepted and used definition of complexity was given by Herrera [94] where the complexity factor for static spherically symmetric fluid distributions was proposed. The notion of this concept of complexity is to take into account the complexity that may arise in a fluid’s structural properties as well as its pattern of evolution. The definition of complexity factor given by Herrera [95] is dependent on active gravitational mass or, Tolman mass. This scalar quantity named the “complexity factor” depends on pressure anisotropy and energy inhomogeneity. So, the vanishing complexity can be achieved in two scenarios, such as, when both pressure anisotropy and energy inhomogeneity vanishes, which is the most basic scenario. Another scenario can be when pressure anisotropy and energy inhomogeneity cancel one another. So the vanishing complexity condition is very much dependent on the properties of the system itself. Several researchers have used the definition given by Herrera in various circumstances [96,97,98,99,100,101,102,103].

In the current work, anisotropic solutions of Einstein’s field equations describing embedding Class I compact stars have been obtained by using a vanishing complexity factor condition in the context of the gravitational decoupling method. The Finch–Skea model along with mimic to mass function approach [104] is applied to solve the system of equations. In this regard, some works on gravitational decoupling in the context of the complexity domain can be seen in the following Refs. [105,106,107,108,109,110,111].

The paper is organized as follows: In Sect. 2, a brief review of Einstein’s field equations under GD and complexity in self-gravitating systems have been discussed. Section 3 deals with complexity-free embedding Class-I solution via MGD. The energy exchange between the relativistic fluids for the sources \(\theta _{i j}\) and \(T_{i j}\) have been analyzed in Sect. 4. Section 5 deals with the junction condition and in Sect. 6, a detailed physical analysis has been done. The concluding remarks are given in the Sect. 7.

2 Brief review of Einstein’s field equations under gravitational decoupling and complexity in self-gravitating system

Let us write Einstein’s field equations by including of extra source through a coupling constant \(\beta \)

$$\begin{aligned} R_{j}^{i}-\frac{1}{2}\delta _{j}^{i}R=-8\pi ({T}^i_j+\beta \theta ^{i}_{j}), \end{aligned}$$
(1)

by assuming geometrical units \(G=c=1\). The line element for describing the spacetime geometry for the internal structure of the stellar model can be given by static spherically symmetric metric as

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

where \(\nu (r)\) and \(\lambda (r)\) are functions depending only on the radial coordinate r. Then we assume that the internal structure of the fluid model contains the anisotropic matter distribution governing by the energy–momentum tensor (\(T^i_{j}\)) given by,

$$\begin{aligned} T^i_j = (\rho + p_t) u^i u_j-p_t \delta ^i_j + (p_r - p_t) \chi ^i \chi _j, \end{aligned}$$
(3)

where the index \(i = 0, 1, 2, 3\). The tensor component \(u^i\) is called the fluid’s four-velocity vector while \(\chi ^i\) denotes the radial unit space-like vector defined as

$$\begin{aligned} u^i = (e^{-\nu /2}, 0, 0, 0) ~~ \text {and} ~~~\chi ^i = (0, e^{-\lambda /2}, 0, 0), \end{aligned}$$
(4)

for which \(\chi ^i u_j = 0\) and \(\chi ^i \chi _j = -1\).

Hence, the Einstein field Eq. (1) for a spherically symmetric line element (2) for anisotropic matter distribution is given by

$$\begin{aligned} \rho +\beta \theta ^0_0{} & {} = \frac{1}{8\pi }\Bigg [\frac{1}{r^2} - e^{-\lambda }\Bigg (\frac{1}{r^2} - \frac{\lambda ^{\prime }}{r} \Bigg ) \Bigg ], \end{aligned}$$
(5)
$$\begin{aligned} p_r-\beta \theta ^1_1{} & {} = \frac{1}{8\pi }\Bigg [-\frac{1}{r^2} + e^{-\lambda }\Bigg (\frac{1}{r^2} + \frac{\nu '}{r} \Bigg ) \Bigg ], \end{aligned}$$
(6)
$$\begin{aligned} p_t-\beta \theta ^1_1{} & {} = \frac{1}{8\pi } \Bigg [\frac{e^-{\lambda }}{4}\Bigg (2\nu '' +\nu '^2 -\lambda ' \nu ' + 2\frac{\nu '-\lambda '}{r}\Bigg )\Bigg ],~~ \nonumber \\ \end{aligned}$$
(7)

and Bianchi identity gives the conservation equation,

$$\begin{aligned}{} & {} -\frac{dp_r}{dr} -\beta \Bigg [\frac{\nu '}{2} (\theta _0^0 - \theta _1^1) - \frac{d\theta _1^1}{dr} + \frac{2}{r}\big (\theta _2^2 - \theta _1^1\big )\Bigg ]\nonumber \\{} & {} \quad -\frac{\nu '}{2}\big (\rho + p_r\big ) + \frac{2\big (p_t - p_r\big )}{r} = 0, \end{aligned}$$
(8)

The mass function m(r) can be determined as,

$$\begin{aligned} m(r) = 4\pi ~\int _0^r ~ (\rho +\theta ^0_0) \, x^2 dx. \end{aligned}$$
(9)

Now our primary objective is to solve the system of Eqs. (5)–(7) containing eight unknowns, which is not an easy task due to its non-linear nature. Therefore, we apply a condition for the vanishing complexity factor given by Herrera [95] for the field Eqs. (5)–(7),

$$\begin{aligned} Y_{TF}{} & {} = 8 \pi \left( p_r - p_t\right) + 8 \pi \left( \theta ^2_2 - \theta ^1_1\right) \nonumber \\{} & {} \quad - \frac{8\pi }{2r^3} \int _0^r x^3 [\rho +\beta \theta ^0_0] '\left( x\right) dx. \end{aligned}$$
(10)

On inserting pressure and density components in \(Y_{TF}\), we get

$$\begin{aligned} Y_{TF} = \frac{\nu '\big [r(\lambda ' - \nu ')+2\big ] - 2r\nu '' }{4re^\lambda }, \end{aligned}$$
(11)

Then \(Y_{TF} = 0\) yields,

$$\begin{aligned} \nu '\big [r(\lambda ' - \nu ') +2 \big ] - 2r\nu '' = 0, \end{aligned}$$
(12)

After integration and some manipulation, we obtain the following relation [104],

$$\begin{aligned} e^{\nu }~ = {~\Bigg (A~\int ~ re^{\lambda /2}dr + ~B\Bigg )}^2, \end{aligned}$$
(13)

where A and B are integration constants. Also, we split the system of equations into two systems by using transformations in order to reduce the unknowns in the equations,

$$\begin{aligned}{} & {} \nu \longrightarrow X +\beta ~ f(r), \end{aligned}$$
(14)
$$\begin{aligned}{} & {} e^{-\lambda } \longrightarrow Y + \beta ~g(r), \end{aligned}$$
(15)

where \(f(r)\ne 0\) and \(g(r)\ne 0\) are called deformation functions. After plugging the above transformations into the Eqs. (5)–(7), we get the following governing equations,

$$\begin{aligned} 8\pi \rho (r){} & {} = \frac{1}{r^2} - Y \Bigg (\frac{1}{r^2} + \frac{Y'}{Y_r} \Bigg ),~~ \end{aligned}$$
(16)
$$\begin{aligned} 8\pi p_r(r){} & {} = -\frac{1}{r^2} + Y\Bigg (\frac{1}{r^2} + \frac{X'}{X_r} \Bigg ),~~ \end{aligned}$$
(17)
$$\begin{aligned} 8\pi p_t(r){} & {} = \frac{Y}{4} \Bigg [2\frac{X''}{X} - \frac{X'^2}{X^2} + \frac{X'Y'}{XY} + \frac{2}{r} \Bigg (\frac{X'}{X} + \frac{Y'}{Y} \Bigg )\Bigg ], ~~~ \nonumber \\ \end{aligned}$$
(18)

and

$$\begin{aligned} 8\pi \theta _0^0{} & {} = -\frac{g}{r^2} - \frac{g'}{r}, \end{aligned}$$
(19)
$$\begin{aligned} 8\pi \theta _1^1{} & {} = -g\Bigg (\frac{1}{r^2} + \frac{X'}{X_r}\Bigg )- \frac{Y\,f^\prime }{r}, \nonumber \\ 8\pi \theta _2^2{} & {} = -\frac{g}{4}\Bigg (\frac{2X''}{X} - \frac{X'^2}{X^2} + \frac{2X'}{X_r}\Bigg )-\frac{g'}{4}\Bigg (\frac{X'}{X} + \frac{2}{r}\Bigg ), \end{aligned}$$
(20)
$$\begin{aligned}{} & {} \quad - \frac{Y}{4}\Bigg (2 f^{\prime \prime } + \beta f^{\prime 2} + \frac{2~f^\prime }{r} + 2 X^\prime f^\prime \Bigg ) + \frac{Y^\prime ~f^\prime }{4}, ~~~~~\nonumber \\ \end{aligned}$$
(21)

Moreover, we consider spacetime describing the solution of the system of Eqs. (16)–(18) as,

$$\begin{aligned} ds^2 = X(r)dt^2 - \frac{1}{Y(r)} dr^2-r^2 \big (d\theta ^2 + \sin ^2{\theta d\phi ^2}\big ),~~~~ \end{aligned}$$
(22)

whose internal mass can be obtained from,

$$\begin{aligned} m_{GR}(r) = 4\pi ~\int _0^r ~ \rho (x) x^2 dx=\frac{r}{2} (1-Y), \end{aligned}$$
(23)

Then from Eq. (9), we can write

$$\begin{aligned} m(r) = m_{GR}+\beta \,m_{\theta }, \end{aligned}$$
(24)

where,

$$\begin{aligned} m_{\theta } = 4\pi ~\int _0^r ~ \theta ^0_0 (x)\, x^2\, dx =-\frac{r\,g}{2}, \end{aligned}$$
(25)

Now we solve both systems individually. The first system contains five unknowns which will be solved by using the embedding Class I condition. This condition can be given by Karmarkar condition [112],

$$\begin{aligned} R_{1010}R_{2323} =R_{1212}R_{3030} + R_{2102}R_{3103}, \end{aligned}$$
(26)

subject to \(R_{2323}\ne 0\) [113], where the quantities given in Eq. (26) are called the Riemann components which will be calculated under the metric (22). The following differential equation can be derived in terms of the metric function X(r) and Y(r) from the Karmarker condition (26) as,

$$\begin{aligned}{} & {} 2\frac{X^{\prime \prime }}{X^{\prime }}+X^{\prime }=\frac{Y^{\prime }}{Y-1}, \end{aligned}$$
(27)

with \(Y\ne 1\).

After integrating, we get the relation between X(r) and Y(r)

$$\begin{aligned} e^X=\Bigg (C+\hat{D} \int \sqrt{\frac{1-Y}{Y}}\, dr\Bigg )^2, \end{aligned}$$
(28)

where C and \(\hat{D}\) are constants of integration the integration constant.

Fig. 1
figure 1

This diagram shows how to generate the anisotropic solution from perfect fluid distribution using complexity factor and gravitational decoupling in GR

Full size image

3 Complexity-free embedding class I solution via MGD

In this section, we discuss the solution of the equation of motions (5)–(7) under the gravitational decoupling satisfying the complexity factor condition. For this purpose, we must solve both systems of Eqs. (16)–(18) and (19)-(21). In order to find the solution for the system of Eqs. (16)–(18), we need two auxiliary conditions to close the system. For this purpose, a well-behaved metric function corresponding to the Finch–Skea solution,

$$\begin{aligned} Y = \frac{1}{1 +Lr^2}, \end{aligned}$$
(29)

where L denotes constant with dimension \({length}^{-2}\) along with the mimic constraint for the mass function proposed by Contreras and Stuchlik [104],

$$\begin{aligned} m_{GR}(r)= m_{\theta } (r), \end{aligned}$$
(30)

have been employed. Now substituting Y in condition (28), we get

$$\begin{aligned} e^X = (C + D r^2 )^2, ~~~~\text {where}~~ D=\frac{\hat{D}\sqrt{L}}{2}, \end{aligned}$$
(31)

while the condition (30) gives

$$\begin{aligned} \frac{r}{2} (1-Y) = - \frac{r}{2} g, \end{aligned}$$
(32)

which derives the expression for g(r) as,

$$\begin{aligned} g(r) = -\frac{Lr^2}{1+Lr^2}, \end{aligned}$$
(33)

Then deformed metric function \(e^{-\lambda (r)}\) is calculated as

$$\begin{aligned} e^{-\lambda (r)} = \frac{1-\beta Lr^2}{1+Lr^2}, \end{aligned}$$
(34)

and the gravitational mass of a deformed star is

$$\begin{aligned} m (r)= (1+\beta )\, m_{GR} (r) \end{aligned}$$
(35)

After substituting potential \(\lambda (r)\) into the condition (13), we obtain

$$\begin{aligned} e^{\nu (r)}{} & {} = \Bigg [B-\frac{A}{2 \beta ^{3/2} L \sqrt{L r^2+1}} \Bigg \{\sqrt{\beta } \left( L r^2+1\right) \sqrt{1-\beta L r^2} \nonumber \\{} & {} \quad +(\beta +1) \sqrt{L r^2+1}~ \tan ^{-1}\left( \frac{\sqrt{1-\beta L r^2}}{\sqrt{\beta } \sqrt{L r^2+1}}\right) \Bigg \} \Bigg ]^2,~~ \nonumber \\ \end{aligned}$$
(36)

Hence, the other deformation function f(r) is determined by using the transformation (14) as,

$$\begin{aligned} f(r) = \frac{1}{\beta }\, \big [\nu (r)-X(r) \big ], \end{aligned}$$
(37)

After substituting the expression for \(\nu (r)\) and X(r) in above Eq. (37), we obtain

$$\begin{aligned} f(r){} & {} = \frac{1}{\beta }\, \Bigg [ 2 \ln \Bigg \{B-\frac{A}{2 \beta ^{3/2} L \sqrt{L r^2+1}} \Bigg (\sqrt{\beta } \left( L r^2+1\right) \nonumber \\{} & {} \quad \times \sqrt{1-\beta L r^2} +(\beta +1) \sqrt{L r^2+1}\, \nonumber \\{} & {} \quad \times \tan ^{-1}\Bigg (\frac{\sqrt{1-\beta L r^2}}{\sqrt{\beta } \sqrt{L r^2+1}}\Bigg )\Bigg ) \Bigg \}- 2\ln (C + D r^2 )\Bigg ],~~~~ \nonumber \\ \end{aligned}$$
(38)

and corresponding deformed spacetime is given by,

$$\begin{aligned} ds^2=e^{\nu (r)}dt^2- \frac{1+Lr^2}{1-\beta Lr^2} dr^2-r^2(\sin ^2\theta d\phi ^2+d\theta ^2), \nonumber \\ \end{aligned}$$
(39)

where, \(e^\nu (r)\) is given by Eq. (37). Now we define the effective components of density and pressures as,

$$\begin{aligned} \rho ^{\text {eff}} = \rho +\beta \, \theta ^0_0,~p^{\text {eff}}_r = p_r-\beta \, \theta ^1_1,~~ p^{\text {eff}}_t = p_t-\beta \, \theta ^2_2, \nonumber \\ \end{aligned}$$
(40)

whose expressions are given as,

$$\begin{aligned}{} & {} 8 \pi \rho ^{\text {eff}}= \frac{(\beta +1) L \left( L r^2+3\right) }{\left( L r^2+1\right) ^2}, ~~ \end{aligned}$$
(41)
$$\begin{aligned}{} & {} 8 \pi p^{\text {eff}}_r =-\frac{4 A \beta ^{3/2} L \sqrt{1-\beta L r^2}}{\sqrt{\beta } \left( A \left( L r^2+1\right) \sqrt{1-\beta L r^2}-2 \beta B L \sqrt{L r^2+1}\right) +A (\beta +1) \sqrt{L r^2+1} \tan ^{-1}\left( \frac{\sqrt{1-\beta L r^2}}{\sqrt{\beta } \sqrt{L r^2+1}}\right) }\nonumber \\{} & {} \qquad \qquad \quad -\frac{(\beta +1) L}{L r^2+1}, \end{aligned}$$
(42)
$$\begin{aligned}{} & {} 8 \pi p^{\text {eff}}_t =\frac{L}{p_{t1(r)}} \Bigg [\beta \Bigg (A^2 \left( L r^2+1\right) \Bigg [\beta ^2 L r^2 \left( 4 L r^2+5\right) -\beta \left( 3 L r^2+5\right) -1\Bigg ] \nonumber \\{} & {} \qquad \qquad \quad \times \sqrt{1-\beta L r^2}-4 A \beta B L \sqrt{L r^2+1} \Bigg [\beta ^2 L r^2 \left( 2 L r^2+3\right) -\beta \left( L r^2+3\right) -1\Bigg ]\nonumber \\{} & {} \qquad \qquad \quad -4 \beta ^2 (\beta +1) B^2 L^2 \sqrt{1-\beta L r^2}\Bigg )-A^2 (\beta +1)^3 \sqrt{1-\beta L r^2} ~\tan ^{-1}\left( \frac{\sqrt{1-\beta L r^2}}{\sqrt{\beta } \sqrt{L r^2+1}}\right) ^2 \nonumber \\{} & {} \qquad \qquad \quad +2 A \sqrt{\beta } (\beta +1)~~ \tan ^{-1}\left( \frac{\sqrt{1-\beta L r^2}}{\sqrt{\beta } \sqrt{L r^2+1}}\right) \nonumber \\{} & {} \qquad \qquad \quad \times \Bigg [A \sqrt{L r^2+1} \left( \beta ^2 L r^2 \left( 2 L r^2+3\right) -\beta \left( L r^2+3\right) -1\right) +2 \beta (\beta +1) B L \sqrt{1-\beta L r^2}\Bigg ]\Bigg ], \end{aligned}$$
(43)

where,

$$\begin{aligned} p_{t1} (r){} & {} =\left( L r^2+1\right) \sqrt{1-\beta L r^2} \\{} & {} \quad \times \Bigg [\sqrt{\beta } \left( A \left( L r^2+1\right) \sqrt{1-\beta L r^2}-2 \beta B L \sqrt{L r^2+1}\right) \\{} & {} \quad +A (\beta +1) \sqrt{L r^2+1} \tan ^{-1}\left( \frac{\sqrt{1-\beta L r^2}}{\sqrt{\beta } \sqrt{L r^2+1}}\right) \Bigg ]^2. \end{aligned}$$

A systematic diagram for generating the anisotropic solution from perfect fluid distribution using complexity factor and gravitational decoupling in GR is shown in Fig. 1.

Before moving next section, it is necessary to mention an important development given by Herrera et al. [114] in connection to a solution-generating technique to construct all possible types of static spherically symmetric solutions to the Einstein field equations for the anisotropic fluids distributions. They proved that this can be given in terms of two generating functions. By inspiring that methodology, we discover the above two generating functions for the present gravitationally decoupled complexity-free solution. To obtain this, let us analyze by using Eqs. (6) and (7), one finds

$$\begin{aligned} 8\,\pi \, (p^{\text {eff}}_{r} -\, p^{\text {eff}}_{t} )= & {} e^{-\lambda } \Bigg [\frac{1}{r^{2} }-\frac{\nu ''}{2} +\frac{\lambda ' \nu '}{4} -\frac{{\nu '}^{2} }{4} +\frac{\nu '+\lambda '}{2r} \Bigg ]\nonumber \\{} & {} -\frac{1}{r^{2} }, \end{aligned}$$
(44)

To simplify the calculations, we present new variables as

$$\begin{aligned} \nu (r)=\left( \int {2\,\Psi (r)-\frac{2}{r}}\right) dr~~~and ~~~e^{-\lambda (r)}=\Phi (r) \end{aligned}$$
(45)

Substituting Eqs. (45) into the Eq. (44), we get

$$\begin{aligned} \Phi ^{\prime }+\Phi \,\left[ \frac{2\Psi ^{\prime }}{\Psi }+2\,\Psi -\frac{6}{r}+\frac{4}{r^2\,\Psi }\right] =-\frac{2}{\Psi }\,\left[ \frac{1}{r^2}+\Pi (r)\right] , ~~ \end{aligned}$$
(46)

where \(\Pi (r) = 8\,\pi \,(p^{\text {eff}}_{r} -\, p^{\text {eff}}_{t} )\). Integrating Eq. (46) we obtain \(\Phi (r)\) as

$$\begin{aligned} \Phi (r)=\frac{r^6\left[ -2\,\int {\frac{\Psi (r)\,\left[ 1+\Pi (r)\,r^2\right] \,e^{\int {\left[ \frac{4+2\,r^2\,\Psi ^2(r)}{r^2\,\Psi (r)}\right] dr}}}{r^8}\,dr}+D\right] }{\Psi ^2\,e^{\int {\left[ \frac{4+2\,r^2\,\Psi ^2(r)}{r^2\,\Psi (r)}\right] dr}}}, \nonumber \\ \end{aligned}$$
(47)

This general formula was given by Herrera et al. [114] to find all possible solutions in terms of two generating functions \(\Psi (r)\) and \(\Pi (r)\). Then, the general spacetime can be written using (45) and (47) as,

$$\begin{aligned} ds^2{} & {} = -\frac{\Psi ^2\,e^{\int {\left[ \frac{4+2\,r^2\,\Psi ^2(r)}{r^2\,\Psi (r)}\right] dr}}}{r^6\left[ -2\,\int {\frac{\Psi (r)\,\left[ 1+\Pi (r)\,r^2\right] \,e^{\int {\left[ \frac{4+2\,r^2\,\Psi ^2(r)}{r^2\,\Psi (r)}\right] dr}}}{r^8}\,dr}+D\right] } dr^2\nonumber \\{} & {} \quad -r^2 \big (d\theta ^2 + \sin ^2{\theta d\phi ^2}\big )+e^{\left( \int {2\,\Psi (r)-\frac{2}{r}}\right) } dt^2, \end{aligned}$$
(48)

Therefore, in our present case the Eq. (45) turns out to be:

$$\begin{aligned} \Psi (r)=\left[ \frac{\nu ^{\prime }(r)}{2}+\frac{1}{r}\right] \end{aligned}$$
(49)

because of the resulting algorithm, the generating functions for the present complexity-free anisotropic fluid solution under the gravitational decoupling solution are as follows [using the Eqs. (42) and (43) along with Eq. (36)]:

$$\begin{aligned}{} & {} \Pi (r) =-\Bigg [(\beta +1) L^2 r^2 \Bigg (\beta \Bigg \{A^2 \left( L r^2+1\right) ^{3/2} \left( 1-\beta L r^2\right) ^{3/2}+4 A \beta B L \left( L r^2+1\right) \left( \beta L r^2-1\right) +4 \beta ^2 B^2 L^2 \sqrt{L r^2+1} \sqrt{1-\beta L r^2}\Bigg \}\nonumber \\{} & {} \quad +A^2 (\beta +1)^2 \sqrt{L r^2+1} \sqrt{1-\beta L r^2} \tan ^{-1} \times \left( \frac{\sqrt{1-\beta L r^2}}{\sqrt{\beta } \sqrt{L r^2+1}}\right) ^2 {}-2 A \sqrt{\beta } (\beta +1) \tan ^{-1}\left( \frac{\sqrt{1-\beta L r^2}}{\sqrt{\beta } \sqrt{L r^2+1}}\right) \left( A \left( L r^2+1\right) \left( \beta L r^2-1\right) \right. \nonumber \\{} & {} \quad \left. +2 \beta B L \sqrt{L r^2+1} \sqrt{1-\beta L r^2}\right) \Bigg )\Bigg ]\Bigg / \Bigg [\left( L r^2+1\right) ^{3/2} \sqrt{1-\beta L r^2} \Bigg (\sqrt{\beta } \Bigg \{2 \beta B L \sqrt{L r^2+1} -A \left( L r^2+1\right) \nonumber \\{} & {} \quad \times \sqrt{1-\beta L r^2}\Bigg \}-A (\beta +1) \sqrt{L r^2+1} \tan ^{-1}\left( \frac{\sqrt{1-\beta L r^2}}{\sqrt{\beta } \sqrt{L r^2+1}}\right) \Bigg )^2 \Bigg ], \end{aligned}$$
(50)
$$\begin{aligned}{} & {} \Psi (r) =\frac{\sqrt{\beta } \left( -A \left( L r^2+1\right) \left( 3 \beta L r^2-1\right) -2 \beta B L \sqrt{L r^2+1} \sqrt{1-\beta L r^2}\right) +A (\beta +1) \sqrt{L r^2+1} \sqrt{1-\beta L r^2} \tan ^{-1}\left( \frac{\sqrt{1-\beta L r^2}}{\sqrt{\beta } \sqrt{L r^2+1}}\right) }{r \sqrt{1-\beta L r^2} \left( \sqrt{\beta } \left( A \left( L r^2+1\right) \sqrt{1-\beta L r^2}-2 \beta B L \sqrt{L r^2+1}\right) +A (\beta +1) \sqrt{L r^2+1} \tan ^{-1}\left( \frac{\sqrt{1-\beta L r^2}}{\sqrt{\beta } \sqrt{L r^2+1}}\right) \right) }. \end{aligned}$$
(51)

4 Energy exchange between the relativistic fluids for the sources \(T_{ij}\) and \(\theta _{ij}\)

The idea of the energy exchange concept under extended gravitational decoupling was discussed by Ovalle [63]. Later on, this concept was introduced by Ovalle et al. [124] and Contreras and Stuchlik [125] for the first time to discuss the flow of energy exchange between the sources. It was noted that a successful decoupling of both systems under the CGD approach requires the energy exchange (\(\Delta E\)) between both sources \({T}_{ij}\) and \(\theta _{ij}\). Therefore, it is necessary to discuss the exchange between both sources which can be determined by the formula by,

$$\begin{aligned} \Delta E=\frac{f^\prime }{2} \big (p + \rho \big ). \end{aligned}$$
(52)

Further details for the development of this formula can be seen in the above-mentioned references. On inserting the temporal deformation function along with the seed pressure and density in the above formula, we find an expression for \(\Delta E\) as,

$$\begin{aligned} \Delta E= & {} \frac{1}{\beta \left( L r^2+1\right) ^2 \left( 3 C+D L r^3\right) }\left[ L \big \{3 C+D r \left( 4 L r^2+3\right) \big \}\right. \nonumber \\{} & {} \left. \times \left( -\frac{4 A \beta ^{3/2} L r \left( L r^2+1\right) ^2}{p_{t1} (r)}-\frac{4 D r}{C+D r^2}\right) \right] , \end{aligned}$$
(53)

5 Junction condition

The essential boundary conditions will be discussed in this section in order to obtain the constants involved in the complexity-free anisotropic solution. In this regard, we match the spacetime for the interior solution i.e. complexity-free anisotropic solution with the Schwarzschild exterior spacetime

$$\begin{aligned} ds^2= & {} \left( 1 - \frac{2M}{r}\right) dt^2 - \frac{dr^2}{1 - \frac{2M}{r}}\nonumber \\{} & {} - r^2 (d\theta ^2 + \sin ^2\theta ~d\phi ^2).~~~ \end{aligned}$$
(54)

at the boundary of star \(r=R\). Here, M denotes the total mass of the object at \(r=R\) and then \(M=m(R)\). The necessary and sufficient boundary conditions are derived by Darmois-Israel boundary conditions [119, 120] which are given by,

$$\begin{aligned}{} & {} 1-\frac{2M}{R}=e^{\nu (R)},~~~ \end{aligned}$$
(55)
$$\begin{aligned}{} & {} 1-\frac{2M}{R}=e^{-\lambda (R)},~~~ \end{aligned}$$
(56)
$$\begin{aligned}{} & {} P_r(R)=0. \end{aligned}$$
(57)

We find the constants after utilizing of above boundary conditions as

$$\begin{aligned} \frac{A_1}{B1}{} & {} =\Big (2 \beta ^{3/2} (\beta +1) L \sqrt{L R^2+1}\Big )\Bigg \{(5 \beta +1) \sqrt{\beta } \nonumber \\{} & {} \quad \times \left( L R^2+1\right) \sqrt{1-\beta L R^2}+(\beta +1)^2 \sqrt{L R^2+1} \nonumber \\{} & {} \quad \times \tan ^{-1}\Bigg (\frac{\sqrt{1-\beta L R^2}}{\sqrt{\beta } \sqrt{L R^2+1}}\Bigg )\Bigg \}, \end{aligned}$$
(58)
$$\begin{aligned} M{} & {} =\frac{(\beta +1) L R^3}{2 L R^2+2}, \end{aligned}$$
(59)
$$\begin{aligned} B_1{} & {} = \sqrt{ \frac{1-\beta L R^2}{L R^2+1}} \Bigg [ 1-\frac{A_1}{B_1\,(2 \beta ^{3/2} L \sqrt{L R^2+1})}\nonumber \\{} & {} \quad \times \Bigg (\sqrt{\beta } \left( L R^2+1\right) \sqrt{1-\beta L R^2} +(\beta +1) \sqrt{L R^2+1}\nonumber \\{} & {} \quad \times \tan ^{-1}\left( \frac{\sqrt{1-\beta L R^2}}{\sqrt{\beta } \sqrt{L R^2+1}}\right) \Bigg )\Bigg \}\Bigg ]^{-1},~~~~ \end{aligned}$$
(60)

6 Physical analysis

6.1 Regularity conditions

It is already known that any physically viable regular solution must have finite central pressures and density in order to ensure a singularity-free solution at the center. In the present complexity-free model, the central pressures and density are given as:

$$\begin{aligned}{} & {} \quad p^{\text {eff}}_i(0)=\frac{L \left( F_1+\sqrt{\beta } (5 A \beta +A-2 \beta (\beta +1) B L)\right) }{8\pi \sqrt{\beta } (2 \beta B L-A)-A (\beta +1) \tan ^{-1}\left( \frac{1}{\sqrt{\beta }}\right) },~~~~ \nonumber \\ \end{aligned}$$
(61)
$$\begin{aligned}{} & {} \quad \rho ^{\text {eff}}(0) = \frac{3 (\beta +1) L}{8\pi } ~. \end{aligned}$$
(62)

where, \(i\equiv \{r, t\},~~~ F_1=A (\beta +1)^2 \tan ^{-1}\left( \frac{1}{\sqrt{\beta }}\right) \). It can be observed from Eqs. (61) and (62) that both pressures and density are free from the singularity. Furthermore, for modeling any realistic object, it is required that pressure and density must be positive everywhere inside the object and the pressure-density ratio must be less than one. Using these bounds on the pressures and density, we get the following constraints on the constant L,

$$\begin{aligned}{} & {} p^{\text {eff}}_i(0)\ge 0~\nonumber \\{} & {} \quad \Longrightarrow ~L\le \frac{A \left( (5 \beta +1) \sqrt{\beta }+(\beta +1)^2 \tan ^{-1}\left( \frac{1}{\sqrt{\beta }}\right) \right) }{2 \beta ^{3/2} (\beta +1) B}. \nonumber \\ \end{aligned}$$
(63)

On the other hand, the \(\frac{p^{\text {eff}}(0)}{\rho ^{\text {eff}}(0)}\le 1\) provides

$$\begin{aligned}{} & {} L \ge \frac{A \left( (2 \beta +1) \sqrt{\beta }+(\beta +1)^2 \tan ^{-1}\left( \frac{1}{\sqrt{\beta }}\right) \right) }{2 \beta ^{3/2} (\beta +1) B}.~~~~~~~ \end{aligned}$$
(64)

After combining the inequalities (63) and (64), we get,

$$\begin{aligned}{} & {} \frac{A \left( (2 \beta +1) \sqrt{\beta }+(\beta +1)^2 \tan ^{-1}\left( \frac{1}{\sqrt{\beta }}\right) \right) }{2 \beta ^{3/2} (\beta +1) B} \le L \nonumber \\{} & {} \quad \le \frac{A \left( (5 \beta +1) \sqrt{\beta }+(\beta +1)^2 \tan ^{-1}\left( \frac{1}{\sqrt{\beta }}\right) \right) }{2 \beta ^{3/2} (\beta +1) B}. \end{aligned}$$
(65)

The regular behavior of the density, pressures and anisotropy within the compact objects can be observed via density plots (Figs. 2, 3, 4 and 5). The energy density (\(\rho ^{\text {eff}}\)) is seen to be maximum at the center and falls off gradually in the radially outward direction (Fig. 2). This means the core has more energy density than the surface. From Fig. 3 it can be seen that the radial pressure (\(p_r^{\text {eff}}\)) has the maximum value at the center and decreases towards the surface. It is also seen that the radial pressure approaches zero near the surface. The tangential pressure (\(p_t^{\text {eff}}\)) is also maximum at the center and decreases outwards and is minimum at the surface (Fig. 4). However, \(p_t^{\text {eff}}\) does not approach zero near the surface and keeps a non-zero positive value. So, the radial pressure approaches zero near the surface and tangential pressure remains non-zero positive value, which means the pressure distribution satisfies physically acceptable stellar objects. Also, as both the pressure components grow from the surface towards the center, it indicates positive hydrostatic force acting in the outward direction. The gravitational pull is balanced by the hydrostatic force and which in turn helps avoid gravitational collapse, which is one of the fundamental requirements of realistic compact objects. It must be noted, for \(\beta \approx 0.35\), the tangential pressure doesn’t monotonically decrease, instead, it slightly increases with r initially, then starts decreasing. This anomaly suggests, when gravitational decoupling is not in effect, the model is not completely well-behaved. The variation in the anisotropic factor shows that the anisotropic factor (\(\Delta ^{\text {eff}}\)) is zero at the center and it gradually increases and is maximum at the surface (Fig. 5). This means the anisotropy is positive and thus the repulsive anisotropic force helps maintain stability inside the star. Looking into the dependence of \(\beta \), it is seen that both \(p_r^{\text {eff}}\) and \(p_t^{\text {eff}}\) increase with the increase in \(\beta \), but the energy density (\(\rho ^{\text {eff}}\)) and anisotropic factor \(\Delta ^{\text {eff}}\) seem to be relatively less impacted by the influence of \(\beta \) as both \(\rho ^{\text {eff}}\) and \(\Delta ^{\text {eff}}\) slightly increases with the increase in \(\beta \).

Fig. 2
figure 2

The contour diagram for equi-energy density on the \(r-\beta \) plane for \(L=0.0095~\text {km}^{-2}\)

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Fig. 3
figure 3

The contour diagram for equi-radial pressure on the \(r-\beta \) plane for \(L=0.0095~\text {km}^{-2}\)

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Fig. 4
figure 4

The contour diagram for equi-tangential pressure on the \(r-\beta \) plane for \(L=0.0095~\text {km}^{-2}\)

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Fig. 5
figure 5

The contour diagram for equi-anisotropy on the \(r-\beta \) plane for \(L=0.0095~\text {km}^{-2}\)

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Fig. 6
figure 6

The contour diagram for equi-velocities of sound on the \(r-\beta \) plane for \(L=0.0095~\text {km}^{-2}\)

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6.2 Causality condition

For every model to be physically acceptable, it has to follow the causality condition, i.e. neither of the radial and tangential components of the velocity of sound should exceed the speed of light. If the speed of light is scaled to unity, the causality condition becomes \(0< \textrm{v}_r^2 < 1\) and \(0< \textrm{v}_t^2 < 1\) where \(\textrm{v}_r\) and \(\textrm{v}_t\) are the radial and tangential components of the velocity of sound respectively. The analysis of the causality condition has been done via density plots in \(r - \beta \) plane (Fig. 6). Here, it is noted that \(\textrm{v}^2_t\) becomes negative for \(\beta \approx 0.35\) in the range \(0 \le r \le 4\) which is not physically acceptable. So, this shows, for \(\beta \approx 0.35\), or, when gravitational decoupling is absent, the model is not physically well-behaved and the presence of gravitational decoupling stabilizes the model. Barring the above range, it can be seen that both \(\textrm{v}_r^2\) and \(\textrm{v}_t^2\) are well within the range \(0< \textrm{v}_r^2 < 1\) and \(0< \textrm{v}_t^2 < 1\), which means, the causality condition is satisfied everywhere else. Both radial and tangential velocities increase radially, i.e. they have a minimum value at the center and increase as one travels towards the boundary. Also, it is recorded, that \(\beta \) increases both \(\textrm{v}_r^2\) and \(\textrm{v}_t^2\). The maximum value of both \(\textrm{v}_r^2\) and \(\textrm{v}_t^2\) are recorded near the surface when \(\beta =0.60\).

6.3 Adiabatic index

Another key aspect to check while checking the stability of any model is to check the radial component of the adiabatic index (\(\gamma \)). This becomes even more significant when the fluid distribution has the presence of anisotropy. This is because the presence of anisotropy may result in a departure from the spherically symmetric distribution in order to escape gravitational collapse. The collapsing condition for perfect fluid distribution with non-relativistic distribution is \(\gamma < 4/3\) [115, 116]. However, when the matter distribution is anisotropic, this changes the form [117, 118] to,

$$\begin{aligned} \gamma < \frac{4}{3} + \Bigg [\frac{1}{3} \kappa \frac{\rho _0 p_{r0}}{|p^\prime _{r0}|} + \frac{4}{3} \frac{p_{t0}-p_{r0}}{|p^\prime _{r0}| r} \Bigg ]_{max} \end{aligned}$$
(66)

The second and third terms of the r.h.s represent correction due to relativistic consideration and anisotropy respectively. It is obvious from the above equation that for non-relativistic perfect fluid distribution, the previous relationship is retrieved. Moreover, as anisotropy increases, the instability decreases, so the stability condition further gets updated to \(\gamma > 4/3\) considering the anisotropic factor (\(\Delta =p_t-p_r\)). The Equi-adiabatic index, as seen in Fig. 7 contour diagram in \(r-\beta \) plane is seen to be slowly increasing with r up to some point and then it rapidly increases near the boundary. Also, the condition of \(\gamma > 4/3\) is always fulfilled everywhere within the model. Moreover, the adiabatic index (\(\gamma \)) is seen to be very slightly increasing with \(\beta \).

Fig. 7
figure 7

The contour diagram for equi-adiabatic index on the \(r-\beta \) plane for \(L=0.0095~\text {km}^{-2}\)

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Table 1 Numerical values of the physical quantities for different values of decoupling constant \(\beta \) with fixed \(L=0.0095~[\text {km}]^{-2}\) and \(R=10~\text {km}\)
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Due to the consideration of the relativistic matter, some instabilities may occur inside the compact stars [121, 122]. To resolve this issue, another condition was imposed by Moustakidis [123], where a critical value of \(\gamma \) was introduced (\(\gamma _{crit}\)). For a model to be stable, \(\gamma \ge \gamma _{crit}\) condition must be satisfied. The \(\gamma _{crit}\) is defined as, \(\gamma _{crit} = \frac{4}{3} + \frac{19 M}{21 R}\). From Table 1, it was found, that the condition of \(\gamma \ge \gamma _{crit}\) is not satisfied for lower values of \(\beta \), and from \(\beta \ge 0.55\), this condition is satisfied. So it suggests, that gravitational decoupling leads more stable model as compared to pure GR.

Fig. 8
figure 8

The contour diagram for equi-mass on the \(r-\beta \) plane for \(L=0.0095~\text {km}^{-2}\)

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Table 2 Predicted radii of 11 observed compact stars for different values of coupling constant \(\beta \) with \(L=0.0095~[\text {km}]^{-2}\)
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Table 3 Mass-radius ratio of 11 observed compact stars corresponding to Table 2 for different values of coupling constant \(\beta \) with \(L=0.0095~[\text {km}]^{-2}\)
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6.4 Mass function and compactness

The effect of gravitational decoupling on the mass m(r) is discussed here. For a detailed clarification, we start by analyzing the consequences of gravitational decoupling on Einstein’s gravity theory. As mentioned above, the mass of the object in the context of gravitational decoupling is the sum of seed mass (\(m_{GR}\)) and mass (\(m_{\theta }\)) for an extra source via decoupling constant \(\beta \). Since we find the deformation function using the condition (31), then the mass of the deformed spacetime becomes

$$\begin{aligned} m(r)=(1+ \beta )\, m_{GR}(r)=\frac{(1+\beta )\,r}{2}(1-\mu ), \end{aligned}$$
(67)

Long back Buchdahl [126] proposed the maximum allowable mass-radius ratio constraint \(\frac{{M}_{GR}}{R}\le \frac{4}{9}\) for isotropic matter distribution satisfying the decreasing density. Here, we check the effect of decoupling constant \(\beta \) on the total mass (M) and whether the object satisfies the Buchdahal limit under gravitational decoupling. It is obvious that the total mass of the object under gravitational decoupling will be \((1+\beta )\) times the total mass in the pure GR scenario i.e. for fixed radius R, \(M\ge M_{GR}\) will always hold true when \(\beta \ge 0\) and consequently the upper bound will change in presence of gravitational decoupling. The graphical analysis of mass is done by a contour diagram of equi-mass in \(r-\beta \) plane which shows how the mass is varying with \(\beta \) along the radius r. From Fig. 8 it is recorded that mass (in terms of \(M/M_{\odot }\)) is zero at the center and it increases as one moves towards the surface. This means the compact object has a kind of mass distribution where it is more concentrated towards the boundary and near the core has a lower mass. Also, it is noticed that \(M/M_{\odot }\) also increases with the increase in \(\beta \), which means, \(\beta \) allows the star to be more massive. The maximum value of \(M/M_{\odot }\) is attained at the surface when \(\beta =0.60\) which is \(2.64M_{\odot }\).

Moreover, we have also predicted the radii for 11 different compact objects for different values of decoupling constant \(\beta \) in order to predict the influence of the gravitational decoupling on the predicted radii. These predicted radii are given in Table 2. In this regard, Table 3 shows the mass-radius ratio (M/R) to the same 11 different compact objects for different \(\beta \). We can observe that the mass-radius ratio (compactness) increases when \(\beta \) increases but the obtained compactness values for each object lie within the range given the Buchdahal. On the other hand, the increasing value of compactness shows that the gravitational decoupling provides more compact objects as compared to the model in a pure GR scenario. and e each deformed compact object.

6.5 Surface redshift

Furthermore, the surface redshift \(\text {z}_s\) of the compact objects under the gravitational decoupling can be given from the following formula,

$$\begin{aligned}{} & {} \text {z}_s=\left( 1-2 {u}\right) ^{-1/2}-1 \nonumber \\{} & {} \hbox {where}, u\equiv \frac{M}{R}=\frac{(1+\beta )}{R} M_{GR}. \end{aligned}$$
(68)

The Eq. (68) clearly shows that the decoupling parameter \(\beta \) affects the surface red-shift value due to the change in compactness \(u=\frac{M}{R}\). This was well supported in the result found in Table 1. Here for fixed radius, it was found that \(\text {z}_s\) changes for different values of mass. So it essentially varies with \(\beta \) and as M/R ratio gets changed due to change in \(\beta \), \(\text {z}_s\) also varies.

Fig. 9
figure 9

The contour diagram for equi-surface redshift on the \(r-\beta \) plane for \(L=0.0095~\text {km}^{-2}\)

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Fig. 10
figure 10

The distribution of energy exchange (\(\Delta E\)) on (\(r-\beta \)) plane for \(A_1/B_1=1\) (left panel) and \(A_1/B_1=2\) (right panel)

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The surface redshift is studied via a contour diagram in \(r-\beta \) plane and it is observed that (Fig. 9) \(\text {z}_s\) increases as one moves from the center to the surface. Moreover, it doesn’t vary much with \(\beta \). However, it seems to be slightly increased with the increase in \(\beta \). The highest value of \(\text {z}_s\) is attained near the surface and at \(\beta =0.60\).

6.6 Energy exchange

Fig. 11
figure 11

The distribution of energy exchange (\(\Delta E\)) on (\(r-\beta \)) plane for \(A_1/B_1=-0.03\) (left panel) and \(A_1/B_1=-1\) (right panel)

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Here, we present the energy exchange between the relativistic fluids for sources \({\theta }_{ij}\) and \({T}_{ij}\) based on the discussion proposed in the works [124, 125]. They have done the analysis of the energy distribution between fluid distributions according to the positive and negative values of \(\Delta E\). The negative value of \(\Delta E\) shows the source \({\theta }_{ij}\) takes the energy from the environment (seed fluid) while \(\Delta E>0\) indicates that the new source gives the energy to the seed fluid or environment. In order to find the flow of energy exchange, we plotted the equi-contour diagram for energy exchange (\(\Delta E\)) on \(r-\beta \) plane against the free constants ratio A1/B1 governing the seed solution. The contour diagrams for \(\Delta E\) are plotted in Figs. 10 and 11 for \(A1/B1=1,~2~-0.03,~\text {and} ~-1\). The left panel (\(A1/B1=1\)) of Fig. 10 shows that \(\Delta E\) is negative on \(r-\beta \) plane within the object i.e. when \(r\in (0,10)\). This shows that the source \({\theta }_{ij}\) takes the energy from the environment while there is no energy exchange between the sources at the center and boundary of the star. It is highlighted that there is a huge amount of energy exchange near the core when \(\beta <0.40\). If we see the right panel (\(A1/B1=2\)), the same situation is happening as the left panel but the amount of energy exchange is less as compared to the left panel. This implies that when A1/B1 increases then the amount of energy exchange from the environment to the new source decreases. Furthermore, when \(0.35\le \beta \le 0.37\) and \(0.5\le r \le 1.5\), then the value of \(\Delta E=-1.0\) at \(A1/B1=0.1\) and the value of \(\Delta E\) is more negative when \(A1/B1 \rightarrow 0\). However, the value of \(\Delta E\) is ruled out for some range of r near the core, and this range shifts towards the boundary when \(A1/B1<0\), which can be observed from the left and right panels of Fig. 11. As we can see from the left panel of Fig. 11 when \(0.2< r \le 1.8\), then \(\Delta E\) is positive and \(\Delta E\) is negative for \(r > 2.5\) for all \(0.35\le \beta \le 0.60\) while the value of \(\Delta E\) is ruled out for \(1.8< r \le 2.5\). The right panel of Fig. 11 shows the same trend for different ranges of r. If \(0.0< r \le 6.2\), then \(\Delta E\) is positive while \(\Delta E\) is negative for \(r > 7\) for all \(0.35\le \beta \le 0.60\). However, the value of \(\Delta E\) is ruled out \(6.2< r \le 7\). By comparing of left and right panels of Fig. 11, we observe that if the value of A1/B1 moves left from 0, then \(\Delta E\) has both positive and negative values for different ranges of r which implies that the new source gives the energy to the environment and receives the energy from the environment.

7 Concluding remarks

In this paper, a new interior solution for compact stars has been obtained by solving Einstein’s field equations, using the gravitational decoupling method alongside the vanishing complexity condition, which provides a bridge equation between metric functions. To solve this equation, we use the Finch–Skea model along with the mimic-to-mass functions approach. After obtaining the solutions, a thorough physical analysis has been done including thermodynamic properties, causality, and stability analysis in terms of adiabatic index, mass, mass-radius ratio, surface redshift, and energy exchange. To begin with, the thermodynamic properties were studied. The energy density \(\rho ^{\text {eff}}\) has a high value at the core and then decreases along the radially outward direction (Fig. 2). It essentially means that the core has much more energy density than the outer portion of the compact star. The radial and tangential pressure was studied and it was found that both radial and tangential pressure have their maximum value at the center and then gradually decrease in the outward direction (Figs. 3 and 4). However, at the surface, the radial pressure approaches zero, while the tangential pressure remains non-zero and positive. The study of the anisotropic factor (\(\Delta ^{\text {eff}}\)) revealed that it is zero at the center and then keeps increasing as one travels toward the boundary (Fig. 5). So it can be said that the core has high energy density, low anisotropy, and the highest magnitude of pressure, and as one moves away from the core, the energy density and magnitude of pressure components decrease, and anisotropy increases. Now, let’s look into how \(\beta \) impacts the thermodynamic parameters, which would suggest the impact of gravitational decoupling. It is noted that increase in \(\beta \) increases both \(p_r^{\text {eff}}\) and \(p_t^{\text {eff}}\) significantly, and \(\rho ^{\text {eff}}\), \(\Delta ^{\text {eff}}\) mildly increase with the increase in \(\beta \). Also, for \(\beta \approx 0.35\), some anomaly in the nature of tangential pressure is found, as it doesn’t monotonically decrease along the radius, instead, it slightly increases up to some point and then starts to decrease again. Another anomaly for \(\beta \approx 0.35\) is found while studying the causality condition (Fig. 6). The tangential component of velocity (\(\textrm{v}^2_t\)) becomes negative for \(\beta \approx 0.35\) in the range \(0 \le r \le 4\), which is not physically relevant. This shows, when the gravitational decoupling is not present, the model is not well-behaved. And as gravitational decoupling takes place, the model stabilizes and becomes well-behaved. Apart from \(\beta \approx 0.35\) and in the \(0 \le r \le 4\), both radial and tangential velocity components satisfy the causality condition \(0< \textrm{v}_r^2 < 1\) and \(0< \textrm{v}_t^2 < 1\). If we look into the nature of the curves of \(\textrm{v}_r^2\) and \(\textrm{v}_t^2\), it can be noticed that both of them have minimum values at the center and increase in the outward direction. Also, the increase in \(\beta \) increases both radial and tangential velocities.

The stability in terms of the adiabatic index (\(\gamma \)) was studied and it was found that the stability condition \(\gamma > 4/3\) is maintained throughout the model (Fig. 7). This means, the model is stable in terms of the adiabatic index, which is a very essential thing to check, especially when the fluid distribution is anisotropic. Also, \(\gamma \) is seen to be slightly increasing with r up to some point, and then increases drastically towards the surface. \(\beta \) is seen to be slightly increasing the value of \(\gamma \). Table 1 showed that the stability condition based on the critical value of the adiabatic index is not satisfied for lower values of \(\beta \) and only gets satisfied for \(\beta \ge 0.55\) onwards. This essentially shows that the model is more stable at higher values of \(\beta \), i.e. when the effect of gravitational decoupling is predominantly more.

The study of mass via a contour diagram of equi-mass (Fig. 8) revealed that mass is minimum at the core and gradually increases towards the surface. So this essentially suggests that the value of mass is more on the surface and minimum in the center, where the energy density is maximum. Moreover, it was noticed that \(M/M_{\odot }\) increases with the increase in \(\beta \). The maximum value of mass was found to be \(2.6M_{\odot }\) and it is achieved near the surface for \(\beta =0.60\). Table 1 revealed that the mass-radius ratio (M/R) for 11 different compact stars increases with \(\beta \). This shows, the compactness increases with \(\beta \), so it’s evident that more compact objects are supported by gravitational decoupling as compared to pure GR. Even though the compactness increased with \(\beta \), it was always found to be staying within the given range by Buchdahal.

The surface redshift (\(\text {z}_s\)) was studied and it was found that \(\text {z}_s\) slightly increases with \(\beta \). This was also suggested by Eq. (68) that \(\text {z}_s\) gets affected by \(\beta \) for the changes in compactness \(u=\frac{M}{R}\). The variation of \(\text {z}_s\) shows, it increases with r and its highest value is obtained near the surface at \(\beta =0.60\) (Fig. 9).

The energy exchange (\(\Delta E\)) was studied in Figs. 10 and 11 for \(A1/B1=1,~2~-0.03,~\text {and} ~-1\). From the left panel (\(A1/B1=1\)) of Fig. 10, it was found out that energy is being taken from the environment by the source \({\theta }_{ij}\) for \(r\in (0,10)\), while at the center and boundary, the energy exchange between the sources is non-existent. Notably, it was found that when \(\beta <0.40\), there is a great amount of energy exchange near the core. For \(A1/B1=2\) plot, in the Fig. 10 right panel, it was seen that the scenario is quite similar to the one before, only here the energy exchange is lesser than the previous case. So, it suggests that the increase in A1/B1 decreases the energy exchange from the environment to the new source. If we look at Fig. 11, we observed different scenarios under the different range of r when \(A_1/B_1\) is negative. In this case, the new source gives the energy to the environment as well as takes energy from the environment for different ranges of r while some value of \(\Delta E\) is ruled out for some range of r.

Finally, we would like to mention here that the complexity-free conditions along with the gravitational decoupling approach open a new tool for generating the series of solutions which is not limited to GR only but also applicable to modified gravity theory.