Uncharged and charged anisotropic like--Durgapal stellar model with vanishing complexity

In this work we use the vanishing complexity factor as a supplementary condition to construct uncharged and charged like--Durgapal models. We provide the $g_{tt}$ component of the metric of the well-known Durgapal IV and V solutions and a particular form for the anisotropy, related to the electric charge, to close the system of differential equations. The physical acceptance of the models is discussed.


I. INTRODUCTION
The use of a great variety of physical conditions in the construction of real compact configurations satisfying Einstein field equations remains a great challenge. In some cases the material content filling the interior of the stellar object is composed by an isotropic fluid distribution where P r = P ⊥ [1]. Although locally isotropic models constitute a very common assumption in the study of compact objects, there is strong evidence suggesting that for certain density ranges a large variety of physical phenomena of the kind we expect to find in compact objects can cause local anisotropy (see Refs. [2][3][4][5][6][7][8][9][10][11], for an extensive discussion on this point). One possible source is related to the intense magnetic fields observed in compact objects such as white dwarfs, neutron stars or magnetized strange quark stars [12][13][14][15][16]. Another source is the high viscosity rates expected to be present in neutron stars, in highly dense matter produced by the opacity of matter to neutrinos in the collapse of compact objects [17][18][19][20][21][22], and the superposition of two isotropic fluids, that allows one to evaluate the fractional anisotropy in a neutron star due to the contamination of electrons and protons required to stabilize neutrons against β decay [23], just to name a few. It is important to remark that, although the degree of anisotropy may be small, the effects produced in compact stellar objects may be appreciable [24]. For these reasons, the assumption of isotropic pressure is a very restrictive condition, especially in a situation where the compact object is modeled as a structure with high * econtreras@usfq.edu.ec † ernesto.fuenmayor@ciens.ucv.ve ‡ gabriel.abellan@ciens.ucv.ve density (such as neutron stars, for example). Besides, the isotropic pressure condition is rendered unstable by the presence of physical factors such as dissipation, energy density inhomogeneity and shear [25]. For these reasons, relaxing the stringent isotropic condition to allow local anisotropy within stellar matter constitutes a more realistic situation from the astrophysical point of view. In addition to the anisotropy, we may consider other possible characteristics in the internal composition of the stellar fluid as for example an electric charge. The first reported solution for charged interior configurations was given by Bonnor [26] and subsequently the effects of electrical properties on compact objects have continued to be studied [27][28][29][30][31][32][33][34][35][36][37][38][39][40]. Even though it has been said that astrophysical systems are expected to be globally charge-neutral it is expected that at certain evolutionary stages a charged astrophysical object could arise especially in transient dynamic processes.
From a more technical point of view, self-gravitating fluids are characterized by a number of physical variables that exceeds the number of equations provided by the theory so that additional conditions must be given in order to close the system and solve the Einstein's field equations. For instance, in the case of a static spherically symmetric anisotropic fluid, we have a set of three ordinary differential equations for the five unknown functions (metric and thermodynamic functions) so two extra conditions must be given in terms of an equation of state or an heuristic assumption involving metric and/or physical variables. In the case of considering stellar objects that have a net electric charge, obviously, we have another degree of freedom or indeterminacy that must be satisfied. In this regard, we need to provide (in addition to the boundary or initial conditions) extra information related to the local physics or restrictions on the metric variables such as the conformally flat [41,42] or the Karmarkar condition which allow to choose one of the metric functions as generator of the total solution [43] (see, also [44][45][46][47][48][49]). In the same direction, a relevant and well known concept in physics as is the complexity, applied to the scope of General Relativity, can perfectly be considered to provide the extra necessary information.
Complexity, rather than an intuitive notion, is a physical concept that deeply relates the fundamental structure of nature and has attracted a broad spectrum of researchers in various branches of science [50][51][52][53][54][55]. So far, although there is not an unifying definition applied to all scenarios, several efforts have been made to provide a definition of complexity associated with the concepts of information and entropy [56][57][58][59][60][61]. However, in Ref. [62] it is proposed for the first time a complexity factor for self-gravitating spheres defined by means of a structure scalar arising from the orthogonal splitting of the Riemann tensor [63][64][65], that manifestly exhibits that the complexity of a gravitational system is closely related to the internal structure of the object. This new approach of complexity, defined for static spherically symmetric relativistic fluid distributions , stems from the basic assumption that one of the less complex systems corresponds to a homogeneous (in the energy density) fluid distribution with isotropic pressure, assigning a zero value of the complexity factor for such a configuration. As we shall see, the vanishing complexity can be achieved also for inhomogeneous and anisotropic self-gravitating spheres so that this condition may be regarded as a non-local equation of state [62], that can be used to obtain non-trivial configurations with zero complexity. This paper is organized as follows. We dedicate the next section to introduce Einstein's equations for a spherically symmetric static and anisotropic fluid. Also some useful definitions and conventions are introduced. In section III we summarize the basics of defining the complexity factor and the vanishing complexity condition. In Sect. IV, we introduce all the variables, conventions and the equations for charged systems as well as the coupling conditions we will employ. Sects. V and VI are devoted to the study of the Durgapal IV model both uncharged and charged respectively. Sects. VII and VIII are dedicated to the study the uncharged and charged Durgapal V model. Finally, the last section is devoted to some discussions and concluding remarks.

II. RELEVANT EQUATIONS AND CONVENTIONS
We consider a spherically symmetric distribution of static fluid, which is assumed to be locally anisotropic and bounded by a spherical surface Σ. The line element is given in Schwarzschild like coordinates by, where ν and λ are functions of the radial coordinate only. The metric (1) satisfy Einstein field equations given by, The matter content of the sphere is described by the energy-momentum tensor where, is the four velocity of the fluid and s µ is defined as with the properties s µ u µ = 0, s µ s µ = −1 (we are assuming geometric units c = G = 1). The metric (1), has to satisfy the Einstein field equations (2), which are given by where primes denote derivative with respect to r. From the conservation of the energy momentum tensor it is a simple matter to find the Tolman-Oppenheimer-Volkoff (hydrostatic equilibrium) equation for anisotropic matter, which reads Alternatively, using which follows from the field equations, we may write (9) as, where m is the mass function defined by or, equivalently and Π is the local pressure anisotropy The exterior description of the spacetime is given by the Schwarzschild exterior solution [66], (15) Moreover, the matching conditions require the continuity of the first and the second fundamental form across the boundary r = r Σ = constant, implying where the subscript Σ indicates that the quantity is evaluated on the boundary surface Σ. Eqs. (16), (17) and (18) are the necessary and sufficient conditions for a smooth matching of the two metrics (1) and (15) on Σ.

III. THE COMPLEXITY FACTOR
This section is dedicated to summarizing the essential aspects of the definition for the complexity factor introduced in [62].
The Riemann tensor can be expressed through the Weyl tensor C ν αβµ , the Ricci tensor R µν and the curvature scalar R, In the spherically symmetric case, the magnetic part of the Weyl tensor vanishes so we express it only in terms of its electric part: Note that E µν may also be written as [64], with E µν satisfies the following properties: Now, it can be shown [65] that the Riemann tensor may be expressed through the tensors in what is called the orthogonal splitting of the Riemann tensor [63]. Here * denotes the dual tensor, i.e. R * µνγδ = 1 2 η σγδ R σ µν and η µνλρ corresponds to the Levi-Civita tensor.
Using Einstein's field equations (2) in (19) we obtain a decomposition of the Riemann tensor given as a function of the components of the energy-momentum tensor. In [62] Eq. (3) was expressed in a particularly useful way so that after some manipulations (see [64] for details), we can find explicit expressions for the tensors Y µν , Z µν and X µν in terms of the physical variables, namely and with Π µ ν = Π s µ s ν + From the tensors X µν and Y µν we can define four structure scalars functions [64] in terms of which these tensors may be written, these may be expressed as, and From the above it follows that local anisotropy of pressure is determined by X T F and Y T F by and a simple but instructive calculation performed in [62] allows us to express Y T F in terms of the inhomogeneity of the energy density and the local anisotropy of the system like, Also, this last result leads us to be able to write Tolman's mass [2,67] as, Then, it is assumed that at least one of the simplest systems is represented by a homogeneous energy density distribution with isotropic pressure. For such a system the structure scalar Y T F vanishes. Furthermore, this single scalar function, encompasses all the modifications produced by the energy density inhomogeneity and the anisotropy of the pressure, on the active gravitational (Tolman) mass so there is a solid argument to define the complexity factor by means of this structure scalar.
The complexity factor Y T F , not only vanishes for the homogeneous, isotropic fluid, where the two terms in (36) vanish identically, but also for all configurations where the two terms, in the same expression, cancel each other. According to (36), the vanishing complexity factor condition, reads: which may be regarded as a non-local equation of state (similar to the one proposed in [68]), so we can use it to impose a plausible condition on the physical variables when solving the Einstein equations. Accordingly, if we impose the condition Y T F = 0 we shall still need to provide more information in order to solve the system. Finally, we note that by using Einstein's equation, that (36) can be written as which is summarized in terms of a relationship between the metric functions, so this condition is joined with the rest of the equations in order to integrate the system.

IV. FIELD EQUATIONS FOR CHARGED SYSTEMS
We start with the line element (1) which is now associated with a static, spherically symmetric anisotropic and charged fluid distribution bounded by a surface Σ, described in Schwarzschild-like coordinates. If the fluid is charged, we must add the electromagnetic tensor S αβ which is given by, where F µν represents the skew-symmetric electromagnetic tensor defined as usual by, satisfying Maxwell equations, where A µ is the four potential and J µ the four current.
In the static and spherically symmetric case, they are expressed as, where Φ is the electric scalar potential and σ the charge density (both functions of the coordinate r). The electric charge interior to radius r has been defined by using the relativistic Gauss's law as follows [69], implying charge conservation. As we are dealing with a spherically symmetric, static configuration, this implies that the only non-vanishing components of the electromagnetic tensor F µν are F 01 , which correspond precisely to the electric field E(r) along the radial direction, namely From now on we consider that the energy distribution within the fluid corresponds to an anisotropic fluid, given in (3), coupled to an electromagnetic field (40). The metric must satisfy Einstein-Maxwell equations for the interior spacetime, that in this case may be written as It is clear that the uncharged solution is recovered when q = 0. Next, the Misner and Sharp mass function m(r) (12) can be generalized to include the electromagnetic contribution by [70], where T 0 0 is given by the left side of (47) in this case. So when considering the total mass of the compact structure, if the fluid distribution contains electric charge, it will increases by a certain amount, and this effect is provided by the electric field. This implies that the surface gravitational red-shift is also altered and one could try to check the models using some observational data.
Note that Eqs. (47), (48) and (49) corresponds to a system of three independent equations with six unknowns, namely {ν, λ, ρ, P r , P ⊥ , q} so we need to provide three conditions in order to solve the system. In this work we shall provide the g tt metric component of the well known Durgapal IV and Durgapal V solutions, the vanishing complexity condition and a particular profile for the anisotropy of non-vanishing charge solutions.
In order to find a complete and consistent stellar model, one needs to satisfy the matching conditions in a smoothly way at the surface interface Σ (that represents the compact object surface) with the external space-time which we assume as the Reissner-Nordström spacetime, where M and Q denote the total mass and charge respectively. Moreover, the matching conditions require [71] A null radial pressure at the boundary surface Σ is a necessary mechanism to confine the matter content inside a bound space-time region, which in turn determines the size of the configuration. It is opportune to clarify that the same expression presented for the complexity factor Y T F and the nonlocal state equation that defines the vanishing complexity of the system, for the non-trivial case (38), is perfectly valid in the general case of an electrically charged fluid distribution. The complexity factor for a charged (spherically symmetric) fluid is known, because the structure constants in that case have also been found (see [70] for details and discussion). The inclusion of electric charge results in defining "effective" thermodynamic matter variables that "absorb" the contribution of the electric charge. The effective variables are just the corresponding ordinary variables with all contributions (electric charge) included.

V. DURGAPAL IV SOLUTION USING VANISHING COMPLEXITY
Let us first consider the uncharged case. For this we set q = 0 in Eqs. (47), (48) and (49). We use the g tt metric coefficient as a seed known solution where A and c are constants. The vanishing complexity condition (38) enables us to find the radial metric coefficient Replacing both metric coefficients in Eqs. (47), (48) and (49) (with q = 0) we find ρ = c 6 + cr 2 cr 2 + 3 Using equations (59) and (60) or just evaluating (38) we obtain directly Now, we proceed with the matching conditions (52), (53) and (54) (with Q = 0) and find that It is worth noticing that from Eq. (63) the compactness parameter (M/R) is 0.4112 which corresponds to a very compact object. Besides, the matching conditions exhaust the free parameters of the model so we do not have any parameter that modulates the anisotropy of the system (in other words if R is given then also c is given).
In in figures (1) and (2) we show the metric functions e ν and e −λ as a function of the radial coordinate. Note that both are positive, finite and free of singularities, as expected. Moreover, at the centre, they satisfy e −λ(0) = 1 and e ν(0) = const. From this we can observe a monotonously increasing correct behavior for the mass function, this is consistent with the adequate behavior of the metric function λ [72][73][74]. Now, in figures 3, 4 and 5 we show the behavior of the matter sector (thermodynamic variables) plotted as a function of the radial coordinate r using R = 1. Note that the density is positive inside the star, reaches its maximum at the center and decreases monotonously outwards, as expected. Also, we observe that the radial pressure and tangential pressure are positive quantities inside the star, they reach their maximum at the center and then monotonously decrease outwards, representing an appropriate behavior. This, obviously translates into a suitable behavior for the anisotropy of the system as we verify by means of figure 6. Besides, for this model the dominant energy condition (DEC) is satisfied as it can be appreciated in both figures 7 and 8, where the density is greater than both pressures, radial and tangential.

VI. CHARGED LIKE DURGAPAL IV SOLUTION
We use as a seed, in order to solve the system, the g tt component of the Durgapal IV solution given by, but now we will consider the charged case where q = 0. As before, the vanishing complexity condition given by Eq. (38) leads us to a relationship between the metric functions that produces Replacing (65) and (66) in (47), (48) and (49) we obtain ρ = − q 2 8πr 4 + c 6 + cr 2 cr 2 + 3 Now, let us consider the following anisotropy [75,76] from where, introducing (68) and (69) we arrive at The matching conditions with the outside metric leads to whith Due to the choice of the specific function for the local anisotropy, which is proportional to the electric charge, increasing the charge (or field) value is equivalent to increasing the anisotropy of the system (note that it can also be modulated with the α parameter) as seen from equation (70). In addition, we observe from equations (67), (68) and (69) that, in effect, one could choose effective variables for the fluid in the sense that they incorporate the term associated with the electric field, which will be translated into the effective anisotropy of the system.
In figures 9 and 10 we show the behavior of the metric functions, with respect to the radial coordinate r for the Durgapal IV charged (q = 0) model. We observe that these functions are positive, finite and free of singularities, as they should be for a physical accepted solution. Furthermore, evaluated at zero, both reach the expected values, e −λ(0) = 1 and e ν(0) = const.
In figures 11, 12 and 13 the matter sector and the fluid tensions are shown. Note that the density and pressure (radial and tangential) behave as expected: positive quantities inside the star, their maximum is at the centre and then decrease monotonously outwards. As a consequence of this, the anisotropy shown in the figure 14 will have the appropriate behavior. For this model we note that the α parameter controls the anisotropy present in the compact object, so that α = 0 represents the isotropic charged case. Also, setting q = 0 automatically returns us to the previous anisotropic un-charged Durgapal IV model. In the internal regions (up to values for the radial coordinate between 0.4−0.6) of the stellar object, all these variables decrease with the increasing anisotropy, as shown in the figure 14. For the same interval of values of r the curve of the anisotropy function is smoothed (reaches a maximum approximately for r = 0.6) so from there on the behavior of the matter sector is the opposite. Also, for this charged Durgapal IV anisotropic model, the energy condition (DEC) is also satisfied since ρ ≥ P r , and ρ ≥ P ⊥ , as can be seen in the figures 15 and 16.
Finally we show in figure 17 the behavior of the interior electric charge (the electric field will be proportional to the charge) for this model. Initially, in the center Gauss's law tells us that it must be zero and then grows monotonously to its full value at the surface q = q Σ = Q, as it should be. Although the behavior of the interior charge is linked to the anisotropy of the system, we can observe this fact from the expression (70), the existence of charge inside the stellar object is independent of whether or not there is anisotropy present, as we can see for the black line (α = 0) of the same figure. The internal charge decreases, throughout the range of values of r by increasing α and therefore the local anisotropy. This could have been read directly from the equation (71).

VII. DURGAPAL V SOLUTION USING VANISHING COMPLEXITY
We now proceed completely analogously with the Durgapal V model, both in the uncharged (q = 0) and charged (q = 0) cases. The objective of carrying out this other case is to be able to have a comparative framework with the previous results and verify their consistency. In addition, for completeness, we present a self-contained work. First we consider the uncharged Durgapal V solution. Setting q = 0 in Eqs. (47), (48) and (49) we use the g tt metric coefficient as a seed (a well-behaved known solution) that will allow us to obtain the complete solution of the problem considered. So, we start with the metric temporal function, The vanishing complexity condition (38), as before, translates into a relationship between the metric variables and enables us to find the radial metric coefficient, given by Replacing both metric coefficients in Eqs. (47), (48) and (49) (with q = 0) we find ρ = c 9 + 6cr 2 + 4c 2 r 4 + c 3 r 6 We note that in the previous equations we only have one free parameter, the constant c. Also, using the previ- ous equations (for P r and P ⊥ ) we arrive to the local anisotropy function of the solution, Now, the matching conditions (52), (53) and (54), with Q = 0 (which corresponds to the Schwarzschild vacuum solution), allow us to obtain the following expressions Note that only the parameter R remains free, so as before (un-charged Durgapal IV) this model presents the peculiarity of not having free parameters that allows controlling the anisotropy of the system. Also, note that for this model the compactness parameter (M/R) is 0.4047 that is close to the compactness obtained in the previous un-charged Durgapal IV model. In figures 18 and 19 we show the graphic behavior of the temporal and radial metric functions respectively, plotted as a function of r for the un-charged Durgapal V (vanishing complexity) solution with R = 1. Note that both metric functions present the appropriate behavior as it occurred in the case of the Durgapal IV. The temporal metric function decreases while the radial increases as long as the radius increases up to its value on the surface, as expected.
We continue showing the graphs corresponding to the matter sector, composed by its thermodynamic variables (ρ, P r and P ⊥ ) in figures 20, 21 and 22 respectively, for the un-charged Durgapal V (vanishing complexity) solution, as well as the behavior of the resulting anisotropy ∆ = P ⊥ − P r exposed in figure 23. With this done, it is straightforward to see that the dominant energy condition is trivially satisfied (since ρ ≥ P ⊥ and obviously ρ ≥ P r ) as shown in the respective figures 24 and 25.

VIII. CHARGED DURGAPAL V SOLUTION
Finally, to complete this work, we shall consider the charged Durgapal V solution in conjunction with the choice of a particular anisotropy function (proportional to q 2 ) and the condition for the vanishing of the complexity parameter Y T F = 0. Again, the metric seed g tt component is and the vanishing complexity condition (38) leads to Replacing (93) and (94) in (47), (48) and (49) we obtain ρ = − q 2 8πr 4 + c 9 + 6cr 2 + 4c 2 r 4 + c 3 r 6 8π (1 + cr 2 ) 4 , (95) From the previous equations, it is inferred that it is possible to absorb the electric charge in the thermodynamic variables of the matter sector by redefining effective variables, this constitutes a common procedure (see for example, [70]). Just as in the previous treatment (charged Durgapal IV model) we use the anisotropy given by Eq. (70). Then, introducing the expressions (96) and (97) for both pressures in the left side of (70) we find the interior charge function, given by q = cr 3 √ 6 + 4cr 2 + c 2 r 4 This function, like in the Durgapal charged IV model, is inversely proportional to α. This dependence of the internal electric field function with the parameter that modulates the local anisotropy has been reported before [76]. Finally we apply the matching conditions (52), (53), (54) and (55) with the exterior Reissner-Nordström solution and we find that .
Note that in order to fix a specific solution, values must be given to R and α. If we turn off the interior charge function q = 0, we obtain the un-charged anisotropic Durgapal V solution given in the previous case; but when there is indeed a presence of electric interior charge, the α parameter is used to control the anisotropy of the system, even to turn it off. The behavior for this anisotropiccharged Durgapal V model is in essence very similar, in practically all aspects, to that discussed for its Durgapal IV analog, so we proceed to summarize the most relevant issues.
In figures 26 and 27 its exposed the behavior of metric function variables with respect to the radial coordinate r for the anisotropic Durgapal V charged (q = 0) model, with R = 1 and different values for the parameter α (that implies different anisotropies). Both functions are positive, finite and free of singularities, as they should be for a physical accepted solution. Both metric functions increase in value with increasing anisotropy as determined by the parameter α.
In figures 28, 29 and 30 we show the thermodynamic variables that constitute our stellar compact fluid, the density and pressures behave as expected so the inherited anisotropy present in the system (figure 31) will do as well. In the deepest regions of the stellar body (r < 0.5) all quantities that make up the matter sector decrease in value with the increasing of the anisotropy via the α parameter. Then, when approaching the surface, the same figures show the opposite behavior (even all curves tend to converge). Consistent and similar results were found for the previous charged case (Durgapal IV). Furthermore, it is straightforward to observe that the energy conditions are correctly met as it is exposed in figures 32 and 33.
Finally, the electric charge (or electric field) trend against the radial coordinate r for the latter case, is shown in figure 34. As in the previous model containing electric charge, our results are totally consistent. Both electric charge q(r) (and electric field E(r)) must be strictly positive and increasing functions with radius, meaning that at the origin both must be null (Gauss's law) i.e., q(0) = E(0) = 0. On the surface of the stellar object it is equivalent to the total charge Q. As before, the same decrease for the interior electric charge with increasing anisotropy is observed.

IX. DISCUSSION
Einstein's system of field equations in the case of a spherically symmetric static, anisotropic fluid form a system of three differential equations for five unknown quantities, the two metric functions ν, λ and the three thermodynamic variables (that constitute the so-called material sector): the energy density ρ, radial P r and tangential pressure P ⊥ of the fluid used to model the interior of the relativistic object. If we also consider an electrically charged fluid, we will therefore have an additional unknown variable (the charge or electric field within the fluid distribution). Accordingly, we shall need to provide extra information to be able to build a consistent model. One way to achieve this, allows us to use a well known solution as a seed to generate new ones. Unlike many previous works, where the radial component of the met- ric (mass function) is used as the seed solution [62], we base the present work using the temporal component of the metric for Durgapal IV and V models, as a seed, and explore the possibility of finding solutions to Einstein's equations. If we consider the un-charged (charged) case we shall need still another (two more) condition (s) in order to solve the system. In this work, together with the Durgapal IV and V temporal metric as a seed solution we impose the vanishing of the complexity parameter Y T F = 0 (known as the vanishing complexity condition) that represents a non-local equation of state that allows to obtain a relationship between the metric functions. In general, setting some value for the complexity factor (for example, a system with vanishing complexity), works like an equation of state that leads to close the Einstein's field equations. Besides considering anisotropy as a quantity playing a fundamental role in relativistic compact stars, we explored the effect of this other interesting physical quantity that involves both the anisotropy and the gradients in the energy density (density contrast), namely the complexity factor.
The new concept of complexity for static spherically symmetric relativistic fluid distributions, arises from the basic assumption that the less complex systems corresponds to an homogeneous (in the energy density) fluid distribution with isotropic pressure [62]. In the case of fluid distributions the variable measuring the complexity of the fluid (the complexity factor Y T F ) appears in the trace-free part of the orthogonal splitting of the electric Riemann tensor (in vacuum the Riemann tensor and the Weyl tensor are the same) [63][64][65]. If we consider a spherically symmetric static distribution of fluid, the magnetic part of the Weyl tensor is zero, so the process starts by calculating the scalar functions defining the electric part of the Weyl tensor. The scalar func- tion Y T F contains contributions from the energy density inhomogeneity and the local pressure anisotropy, combined in a very specific way so it measures the departure from the value of the Tolman mass of a homogeneous and isotropic fluid and allows finding a great variety of models with vanishing complexity (Y T F = 0) different from the trivial case (perfect fluid). In the case that we have a electric charged fluid it is useful to introduce "effective" variables that are just the corresponding ordinary thermodynamic variables with the contributions of electric charge included (see [70]). The remarkable fact emerging from this is that the charge contribution is always absorbed into the effective variables. In the absence of electrical charge, the structure scalars are obtained just replacing the effective variables by the corresponding ordinary ones [70], question that allows us to define complexity in the same way.
In this work, the equation of state resulting from canceling the structure scalar Y T F represents the other useful piece of information necessary for the consolidation of the models (in the case of configurations with electrical charge, it is also necessary to choose an anisotropy func-tion). We analyzed the plausibility of each model based on the physical conditions established for the existence of anisotropic compact stars. All the considered cases behaved as expected in terms of metric potentials, matter sector, anisotropy function, internal charge of the system, energy conditions, redshift and stability (although in these latter conditions we did not present the graphs).
It is worth mentioning that the new definition of complexity for self-gravitating relativistic fluids, has been extended to consider the dynamic spherically symmetric situation (non-static case) [37,77] and cases where the spherical symmetry is broken: the axially symmetric static case [78] and applications for some particular cases of cylindrically symmetric fluid distributions may be also found [79,80]. In reference [37] was treated complexity for dynamical spherically symmetric dissipative self-gravitating fluid distributions and in [81] the complexity for the Bondi metric [82]. Besides the fact that the Bondi metric covers a vast numbers of spacetimes, Minkowski spacetime, the static Weyl metrics, nonradiative nonstatic metrics, it has, among other things, the virtue of providing a clear and precise criterion for the existence of gravitational radiation. Finally, electrically charged fluid configurations have also been included (see references [37,38] for details).
It is important to highlight that the role of charge distribution in the stability of such configurations can be clearly exhibited by equation (50). In particular it is worth stressing the fact that electric charge, unlike pressure, does not always produce a "regeneration effect" (does not always increase the Tolman mass). This fact together with the presence of the Coulomb term in the full set of the equations required for a description of physically meaningful models of collapsing charged spheres indicates the relevance of the electric charge in the process of collapse and therefore in the stability of compact spheres in hydrostatic equilibrium. Before concluding this work, it is worth mentioning that a detailed analysis of the solution by fixing the parameters to describe real stars is compulsory. However, as this is out of the scope of this paper we left this analysis to a future work.