Decoupling gravitational sources by MGD approach in Rastall gravity
aa r X i v : . [ phy s i c s . g e n - ph ] J un Decoupling gravitational sources by MGD approach in Rastall gravity
S. K. Maurya ∗ and Francisco Tello-Ortiz † Department of Mathematics and Physical Science,College of Arts and Science, University of Nizwa, Nizwa, Sultanate of Oman Departamento de F´ısica, Facultad de ciencias b´asicas,Universidad de Antofagasta, Casilla 170, Antofagasta, Chile
In the present work, we investigate the possibility of obtaining stellar interiors for static self-gravitating systems describing an anisotropic matter distribution in the framework of Rastall gravitythrough gravitational decoupling by means of minimal geometric deformation approach. Due toRastall gravity breaks down the minimal coupling matter principle, we have provided an exhaustiveexplanation about how Israel-Darmois junction conditions work in this scenario. Furthermore, toobtain the deformed space-time, the mimic constraint procedure has been used. In order to checkthe viability of this proposal, we have applied it to the well known Tolman IV solution. A completethermodynamic description of the effects introduced by the additional source is given. Additionally,the results have been compared with their similes in the picture of pure general relativity, pureRastall gravity and within the framework of general relativity including gravitational decoupling.To perform the mathematical and graphical analysis we have taken the gravitational decouplingconstant α and the Rastall’s parameter λ as free parameters and the compactness factor describingthe general relativity sector to be 0 .
2. Besides, to provide a more realistic picture we have boundedboth parameters α and λ by using real observational data to explore the limits of the theory underthis particular model. Applications to study neutron or quark stars are suggested by using thismethodology. PACS numbers:
I. INTRODUCTION
Put forward by P. Rastall in 1972 [1], the so-calledRastall gravity theory can be seen as a modified gravitytheory or as a generalization of Einstein gravity theory ı.eGeneral Relativity (GR from now on). Rastall’s proposalis based on the argument that the energy-momentum ten-sor, which fulfills the conservation law (null divergence)in a flat space-time (Minkowskian space-time), does notnecessarily fulfill it in a curved background. The viola-tion of Bianchi’s identity occurs through the introductionof a covariant term in the Einstein field equations througha dimensionless coupling constant λ . Specifically, thisterm corresponds to the Ricci’s scalar curvature R. Al-though the field equations given by Rastall do not havean associated Lagrangian density from which they canbe obtained, these as a generalization of the GR equa-tions respect the symmetries of the theory ı.e, the generalcoordinate transformation.Despite that the new term is introduced by hand, itsprevalence modifies not only the field equations but alsothe way of coupling material fields to the gravitationalinteraction. Clearly, the principle of minimal couplingmatter breaks down. However, this brings with it newand intriguing contributions which can be useful to un-derstand certain commonly studied phenomena such ascosmological issues, collapsed structures (black holes),stellar structures, gravitational waves, etc. In this di- ∗ Electronic address: [email protected] † Electronic address: [email protected] rection, Rastall gravity is as competitive as other mod-ified gravity theories such as f(R) and f(R, T ) theories.As it is well known f(R) gravity theory was developed toaddress inflationary cosmological problems [2, 3]. Nowa-days, f(R) gravity has been used in a wide context byaddressing cosmological problems such as the existenceof dark components (dark energy and dark matter), stel-lar structures, among others. An incomplete list of recentworks concerning these issues can be found at the follow-ing references [4–22] (and references contained therein).Furthermore, to face related open questions in the cos-mological scenario f(R, T ) gravity is also a promising ap-proach [23]. In this respect f(R, T ) gravity theory can beseen as an extension of f(R) gravity. Where T standsfor the trace of the energy-momentum tensor. The cor-rections coming from the trace of the energy-momentumtensor can be attributed to quantum effects [23]. A widevariety of works available in the literature devoted totackle the accelerated expansion of the Universe, energyconditions, stellar interiors and so on, are found at thefollowing references [24–48]. In comparing f(R, T ) gravitywith Rastall theory, both theories break the minimal cou-pling matter scheme. However, the former smashes theminimal coupling matter principle by introducing mat-ter and geometric terms (curvature invariants) while thesecond one by inserting only geometric objects, preciselythe Ricci’s scalar. The effects introduced by the addi-tional term have been extensively studied on differentfronts to test at least theoretically how close the resultsare in comparison with the broad support that GR has.In this respect, the well known Tolman solutions havebeen extended into the Rastall gravity arena [49] in or-der to contrast the behavior of the main salient featureswith the corresponding GR ones. Furthermore, to re-inforce the study of stellar interiors in the Rastall sce-nario an anisotropic model in the background of Krori-Barua (KB) space-time was done in [50] (in [51] the sameauthors considered KB space-time supplemented by anequation of state, specifically a quintessence model) andan isotropic compact object using conformal Killing vec-tor technique was reported in [52]. Regarding the stronggravitational regime, remarkable solutions of black holeswere presented in [53, 54]. These works motivate the in-vestigation of the most exciting features and propertiesof these fascinating objects. For example the thermo-dynamic was studied in [55–57] and rotating black holeswere addressed at references [58, 59]. It is worth mention-ing that in the context of black holes solutions GR andRastall gravity share the same vacuum solution [1]. Thisis so because when the Rastall parameter goes to zero( λ → λ → θ µν to the energy-momentum tensorof the seed solution via a dimensionless coupling con-stant α . The presence of this extra piece in conjunctionwith a spherically symmetric and static space-time (inSchwarzschild like-coordinates) leads to an intricate sys-tem of equations with seven unknown functions (if theseed solution is taking to be isotropic). To solve this com-plicated system and translate the isotropic fluid distribu-tion to an anisotropic scenario ii) perform the MGD onone of the metric potentials (usually on the radial metriccomponent e λ ). With this deformed potential in hand,the tangled system is separated into two simpler sets.The first one is the usual Einstein system and the secondone contains the θ -sector and the decoupler function f ( r )introduced in the MGD to split the system of equations.However, the latter one contains four unknown and onlythree equations. Then this system must be supplementedwith extra information in order to close the problem. Atthis stage, a couple of comments are in order. First,after decoupling the systems, the resulting ones fulfillBianchi’s identities (the conservation law of their corre-sponding energy-momentum tensors), meaning that theoriginal source and the extra one only interact gravita-tionally. Second, the additional term could represent ascalar, vector or tensor field. Moreover, in principle, this new sector could not necessarily be described by GR. Fora more detailed discussion of how this machinery workssee sections III and IV and the following references [66–72].This last two years gravitational decoupling usingMGD has attracted many adepts. In this respect somewell known solutions (uncharged, charged) of the Ein-stein field equations have been extended by using MGD[73–85]. Also, Black holes in 3 + 1 (Schwarzschild outerspace-time) [86] and 2 + 1 (BTZ black hole) [87] dimen-sions were extended, the (anti) de Sitter space-time wasworked in [88] and also the inverse problem was addressedin 3 + 1 dimensions [89] and 2 + 1 dimensions includingcosmological constant [90]. Regarding another branchesthe method has been employed in cloud of strings [91],Klein-Gordon scalar fields as an extra matter content[92], extended to isotropic coordinates [93] and ultra com-pact Schwarzschild stars, or gravastars [94]. Moreover,the method was widespread to include deformation onboth metric potentials, it was called the extended-MGD[95–97]. More recently gravitational decoupling was usedto investigate higher dimensional compact structures [98]and spread out to the context of Lovelock [99] and f(R, T )[100] gravity theories and in the cosmological scenario[101].So, as we pointed out above our main goal is to in-troduce gravitational decoupling by means of MGD inthe Rastall gravity picture. This will bring new insightson how compact objects behave by the inclusion of lo-cal anisotropies in the light of Rastall theory. Moreover,the possibility of comparing with other modified grav-ity theories and GR results in order to check the via-bility from both theoretical and experimental point ofview. To do so, we have followed the same proceduregiven in [61]. The approach proposed in [61] in orderto solve the θ -sector is to impose some suitable restric-tions on the thermodynamic quantities that character-ize the seed solution (the isotropic pressure p and theenergy-density ρ ) and the components of the extra source θ µν . These restraints are referred as mimic constraints.These mimic constraints yield to an algebraic or differen-tial equation that allows to obtain the deformation func-tion f ( r ). Each mimic constraint leads to a differentanisotropic solution, for example, the two common mimicconstraints worked in the literature are: i) p = θ rr andii) ρ = θ tt . This means that the radial component of the θ -sector mimics the isotropic pressure the temporal onemimics the energy-density of the seed solution. Althoughthere is not a physical foundation to support the afore-mentioned choices, until now they have not presented anyphysical or mathematical inconsistency or any behaviorthat is detrimental to what is reported within the frame-work of general relativity. Moreover, studies conductedin general relativity concerning to interior solutions withanisotropic component have been favored (or reinforced)in a certain way when the described mechanism has beenconsidered. However, in favor of these considerations,it should be noted that the mimic constraints have beenimposed at the level of the equations of motion, which en-sures the closure of the system of equations to be solvedand also a correct physical and mathematical behaviorof the θ -sector components. Therefore, one ends with awell-behaved solution. In addition, the virtues of eachrestriction are unique. For example, when the r − r com-ponent of the θ -sector mimics the seed pressure p ( r ), thetotal mass of the structure does not change, it is onlyredistributed inside the object. On the other hand, theelection ρ = θ tt changes the total mass of the compactobject. Consequently, the first case does not distinguishbetween an isotropic and an anisotropic object of similarcharacteristics (mass and radius). This is so because thesurface gravitational red–shift z s is the same in magni-tude. Nevertheless, in the second case, one can differenti-ate between a distribution with isotropic material contentfrom another with anisotropic content, since in the latter z s varies as expected with respect to its isotropic coun-terpart, changing its magnitude. It is worth mentioningthat another way to face the problem is imposing andadequate form of the decoupler function f ( r ) respectingall the physical and mathematical requirements as wasdone in [73, 74, 98].In the present work, we have considered both mimicconstraints. Due to Rastall gravity departs from GR onlyby the presence of the Ricci scalar coupled to the theorythrough the so-called Rastall’s parameter λ , then thereare not higher-order derivative terms of the metric poten-tials (no more than two spatial derivatives) in the theory.This feature facilitates the gravitational decoupling andwe have done it in such a way that the deformation func-tion and the resulting components of θ -sector contain theeffects of Rastall gravity. This allows the new contribu-tions to be compared exhaustively with respect to whatis obtained in pure Rastall gravity and the results of RGand RG + MGD. In addition, to contrast our results wehave taken the geometry of the inner space-time to beTolman IV. This solution which describes a sphericallysymmetric and static object whose material content re-spects a perfect fluid distribution has already been ex-tended to anisotropic domains by MGD [61] and has alsobeen studied in the Rastall gravity picture [49]. One ofthe most notable features of Rastall’s theory of gravityis that any perfect fluid solution of Einstein’s equationsis also a solution of it. Obviously, this is from the ge-ometrical point of view because the material content isdifferent due to Rastall contribution. Furthermore, tohandle with the numerical part we have taking as freeparameters the gravitational decoupling constant α andthe Rastall’s parameter λ and considering a compact-ness factor within the allowed range for compact stars tobe u = m GR ( R ) /R = 0 .
2. In this respect, it is worthmentioning that the coupling constant α plays an im-portant role in the behavior on the main salient featuresand has a relevant incidence on the mass of the compactstructure. Although we have only considered positivevalues for α , in order to obtain a physically acceptablesolution. Notwithstanding, α < ρ = θ tt , for some solu-tions such as Heintzmann IIa [75] and Durgapal-Fuloria[77], requires that α takes negative values to avoid non-physical behaviors. In our case, α < α and λ by using real observational data.Finally, we want to mention that it is the first time thatgravitational decoupling by MGD is used in the Rastallgravity scenario.So, the article is organized as follows: In Sec. II werevisited in brief the main ingredients of Rastall gravitytheory and its comparison with other theories. In Sec.III the field equations for multiple sources are presented.Sec. IV discuss the gravitational decoupling via mini-mal geometric deformation scheme in the Rastall gravityframework. Next, in Sec. V the matching conditions areanalyzed and extensively discussed as well as the bind-ing energy in present context. In Sec. VI Tolman IVspace-time is projected into the anisotropic domain us-ing the methodology previously discussed. Furthermore,the problem is faced by employing the mimic constraintapproach, in order to determine the decoupler function f ( r ) and the new sector, the θ µν one. Sec. VII talksabout the physical implications of the mimic constraintsprocedure on the principal macro physical observables ofthe model as well as the maximum and minimum orderof magnitude of the free parameters α and λ allowingto contrast with real values of compact objects. In thefollowing section, Sec. VIII the principal physical, obser-vational and mathematical implications of the obtainedmodel are highlighted. Finally, Sec. IX provides someremarks of the study reported in this article. II. REVISITING IN SHORT: RASTALLGRAVITY THEORY
The main idea behind Rastall’s proposal [1] is to aban-don the free divergence of energy-momentum tensor in acurved space-time. Explicitly it reads ∇ µ T µν = 0 . (1)So, this non-conserved stress-energy tensor introduces anunusual non-minimal coupling between matter and geom-etry. Specifically, this non-minimal coupling is carriedout into the theory by the following assumption on thedivergence of the energy-momentum tensor ∇ µ T µν = λ ∇ ν R. (2)In the above expression R ≡ g µν R µν stands for theRicci’s scalar and λ is the so-called Rastall’s parame-ter which is used to depict the distraction from GR andmeasures the affinity of the space-time geometry to cou-ple with matter field in a non-minimal fashion. The as-sumption given by Eq. (2) is completely consistent withthe following field equations R µν − Rg µν = κ R ( T µν − λRg µν ) , (3)where κ R is the Rastall gravitational constant. It isworth mentioning that in the limit λ → G µν = κ R T (eff) µν , (4)and in some sense one regains the standard result ∇ µ T (eff) µν = 0. Now, by taking the trace of Eqs. (3)the Ricci scalar can be written as R = κ R T λκ R − , (5)then the effective stress-energy tensor reads T (eff) µν = T µν − γT γ − g µν , (6)where γ = κ R λ . From now on we shall assume κ R = 1,then γ = λ . From Eq. (6) one can infer several con-straints. Again taking λ = 0 the Rastall sector disap-pears and one recast GR. If a traceless energy-momentumtensor is considered, such as the electromagnetic one,Rastall contribution is totally vanished because T = 0.Additionally, λ = 1 / T µν to be aperfect fluid matter distribution, which is given by T µν = ( ρ + p ) u µ u ν − pg µν . (7)We utilize a comoving fluid 4-velocity u σ = e − ν/ δ σt ,and ρ and p are representing the energy-density and theisotropic pressure respectively. So, the components of T (eff) µν are given by T t (eff) t = ρ (eff) = (3 λ − ρ + 3 λp λ − , (8) T r (eff) r = T ϕ (eff) ϕ = T φ (eff) φ = − p (eff) = − ( λ − p + λρ λ − . (9)In obtaining the expressions (8) and (9) we have used T = ρ − p . Moreover, as before λ = 0 yields us to Eq.(7). As we will see later the isotropic quantities ρ and p will be separated carrying out in their expressions thecorresponding Rastall contributions.Despite its attributes, recently was claimed by Visser[102] that Rastall gravity theory is equivalent to Ein-stein’s general relativity theory. The main concern ofVisser was that the defined energy-momentum-tensorprovided by Rastall was not right and that Rastall’s the-ory is just the rearrangement of the matter sector of GR. Of course as can be seen from Eq. (4) the fields equationgiven in the original Ratsall’s article [1] can be adjustedto recast the usual Einstein field equations. However, indistinction if the energy-momentum tensor is conservedor not, this rearrangement can be performed in any mod-ified gravity theory e.g. f ( R, T ) gravity [23], f ( R ) the-ory [103] among others, where the terms non conform-ing Einstein tensor are grouped giving rise to an effectiveenergy-momentum tensor. So, it does not imply that GRis equivalent to these theories. In fact, such equivalenceexists only in particular cases such as: Putting f ( R ) = R in f ( R ) gravity theory, or dropping out the T term andsetting f ( R ) = R in f ( R, T ) gravity [104]. Therefore, onecan conclude that Rastall’s proposal is not equivalenceto GR unless λ →
0. What is more, recently, from the f ( R, T ) Lagrangian formulation was obtained the corre-sponding Lagrangian functional associated with Rastall’stheory [105] and G¨odel-type solutions in the cosmologi-cal scenario were investigated [106]. At this point it isrelevant to compare Rastall’s theory with other modifiedtheories of gravity which violate the conservation of theenergy-momentum tensor. As it is well known the orig-inal f ( R ) gravity proposal respects Bianchi’s identities.However, this theory was extended by including extraterms which violate the minimal coupling matter prin-ciple. Then, the energy-momentum tensor associated tothis formulation in not conserved [107]. Specifically, themodified Einstein-Hilbert action reads S = Z (cid:26) f ( R ) + [1 + βf ( R )] L m (cid:27) √− gd x, (10)where f and f are arbitrary functions of the Ricci’sscalar and L m is the Lagrangian matter characterizedby the constant β . The field equations provided by theabove action (10) are given by ∇ µ T µν = βF βf [ g µν L m + T µν ] ∇ µ R. (11)As it is observed, Eqs. (2) and (11) are in some sensecomparable, since both shown a non-trivial coupling be-tween the gravitational and material sectors. In Eq. (11) T µν is representing the usual energy-momentum tensordescribing isotropic, anisotropic matter distribution, etc.If the term βf is taking to be constant, namely βf = K then F ≡ df /dR = 0, thus the usual conservation lawis regained. Also the familiar conservation equation isrecovered from (11) when β → λ → ν ′ ρ + p ) + dpdr − λ λ − ddr ( ρ − p ) = 0 , (12) ν ′ ρ + p ) + dpdr − βp dRdr = 0 . (13)It should be noted that the nature (attractive or repul-sive) of the force introduced by the extra term in theabove expressions in principle depends on the sign of theconstants λ and β respectively. Nevertheless, these pa-rameters must be constrained by solar system tests.In comparing the Newtonian limit reproduced byRastall gravity theory and (10) for a perfect fluid matterdistribution one has∆Φ − πBρ Φ = 4
Aπρ, (14)where Φ is the Newtonian gravitational potential, A ≡ (3 λ − / (6 λ −
1) and B ≡ − λ/ (6 λ − λ . It isworth mentioning that λ should be zero in order to obtainthe usual Poisson equation for weak gravitational fields.Otherwise for large mass and strong gravitational fieldregime, expression (14) could play a crucial role in thecosmological scenario [111]. Moreover, if ρ = constantEq. (14) transforms into Sileeger equation [112]. Onthe other hand, the Newtonian limit for the theory (10)provides the following gravitational potential Φ,Φ = −
12 [ ξ + Ln (1 − ξ )] , (15)where ξ = ξ ( r ). In this case the gravitational potentialdoes not coincide with Newtonian one, this is because anadditional logarithmic term appears. So, in both casesthe corresponding Newtonian limit goes beyond the clas-sical one. This suggests that such modifications could in-troduce new insights and implications. Another interest-ing point to be compared here, are the cosmological andcluster dynamic consequences. In this respect, Rastallgravity theory has been theoretically tested as a feasi-ble framework to explain the ΛCDM model issues. Inthis direction abandon exotic fluid such as Chaplygin gasto explain the existence of dark energy is a viable way.An interesting way out is to use a non–canonical self–interacting scalar field as suggested by Rastall gravitytheory [113] or analyzed cosmological models at the back-ground as well as perturbation level [114]. In this sense, Rastall’s proposal has proven to be a good candidate toexplain such problems. On the other hand the theory(10) is also a good alternative to explain dark componentspresence. Bertolami and P´aramos [110] studied and com-pared the known dark matter density profiles throughan appropriate power-law coupling f = ( R/R ) n (with n < III. FIELD EQUATIONS: MULTIPLE SOURCES
In this section we describe the field equations for mul-tiple matter sources. So, the standard field equations aregiven by R µν − Rg µν = ¯ T (tot) µν , (16)where ¯ T (tot) µν stands for¯ T (tot) µν = T (eff) µν + αθ µν . (17)The new sector θ µν always can be seen as correctionsto the theory and be consolidated as part of an effec-tive energy–momentum tensor. This extra source couldrepresent a scalar, vector or tensor fields and introducesanisotropies within the self–gravitating systems. It iscoupled to the matter sector through a dimensionlessconstant parameter ı.e, α . On the other hand, T (eff) µν represents the usual matter sector, that is isotropic,anisotropic, or charged distributions, among others. Inthe present case T (eff) µν is given by Eqs. (8)-(9). Aswe are interested in studying spherically symmetric andstatic fluid spheres, next we regard the most general lineelement in the standard Schwarzschild like coordinates { t, r, φ, θ } to be ds = e ν ( r ) dt − e η ( r ) dr − r ( dθ + sin θdφ ) . (18)The staticity of this space-time is ensured by consider-ing ν and η as functions of the radial coordinate r only.Putting together equations (8), (9), (16) and (18) onearrives at the following set of equationse − η (cid:18) η ′ r − r (cid:19) + 1 r = ρ (eff) + αθ tt , (19)e − η (cid:18) ν ′ r + 1 r (cid:19) − r = p (eff) − αθ rr , (20)e − η (cid:18) ν ′′ + ν ′ + 2 ν ′ − η ′ r − ν ′ η ′ (cid:19) = p (eff) − αθ ϕϕ . (21)where ρ (eff) = (3 λ − ρ + 3 λp λ − , (22) − p (eff) = − ( λ − p + λρ λ − . (23)The corresponding conservation law ∇ µ ¯ T (tot) µν = 0 asso-ciated with the system (19)-(21) reads − dp (eff) dr − α (cid:20) ν ′ (cid:0) θ tt − θ rr (cid:1) − dθ rr dr + 2 r ( θ ϕϕ − θ rr ) (cid:21) − ν ′ ρ (eff) + p (eff) ) = 0 . (24)It is found that the system of non–linear differential equa-tions (19)–(21) consists of seven unknown functions, themetric potentials { η, ν } , the thermodynamic observables { ρ (eff) , p (eff) } and the components of the extra source { θ tt , θ rr , θ ϕϕ } . In order to find these unknowns we adopt asystematic approach. Furthermore, for the system (19)–(21), the matter content (total energy–density, total ra-dial pressure and total tangential pressure) can be iden-tified as ¯ ρ (tot) ( r ) = ρ (eff) ( r ) + αθ tt ( r ) (25)¯ p (tot) r ( r ) = p (eff) ( r ) − αθ rr ( r ) (26)¯ p (tot) t ( r ) = p (eff) ( r ) − αθ ϕϕ ( r ) . (27)It is clear that an anisotropic behaviour arises into thesystem due to the presence of the θ –sector if θ rr = θ ϕϕ . So,in order to measure the anisotropic behaviour we definethe anisotropy factor as follows∆ = ¯ p (tot) t − ¯ p (tot) r = α (cid:0) θ rr − θ ϕϕ (cid:1) . (28)At this stage the system of Eqs. (19)– (21) could betreated as an anisotropic fluid, with five unknown func-tions, namely, the two metric functions ν and η , and thetotal functions in Eqs. (25)-(27). However, we are goingto implement a different approach, as explained below. IV. GRAVITATIONAL DECOUPLING: A MGDAPPROACH
As said before, gravitational decoupling by MGDscheme becomes a simple and powerful tool to ex-tent spherically and static isotropic fluid solutions toanisotropic domains [61]. To see how this approachworks let us start by turning off the coupling α , sowe are describing a perfect fluid solution given by { ξ, µ, ρ (eff) , p (eff) } , being ξ and µ the corresponding met-ric functions. The metric (18) now reads ds = e ξ ( r ) dt − dr µ ( r ) − r ( dθ + sin θdφ ) , (29) where µ ( r ) = 1 − m GR r is the standard GR expressioncontaining the mass function of the fluid configuration.Next, to see the effects of the θ –sector on the perfect fluiddistribution we turn on the coupling α . These effects canbe encoded in the geometric deformation undergone bythe perfect fluid geometry { ξ, µ } in Eq. (29), namely ξ → ν = ξ + αh (30) µ → e − η = µ + αf, (31)where h and f are the deformations introduced in thetemporal and radial metric components, respectively. Itis worth mentioning that the foregoing deformations arepurely radial functions, this feature ensures the sphericalsymmetry of the solution. The MGD scheme consists inset off either h or f . In this opportunity we set h = 0, itmeans that the temporal component remains unchangedand the anisotropy lies on the radial component [61]. So,we have µ ( r ) → e − η ( r ) = µ ( r ) + αf ( r ) . (32)Upon replacing Eq. (32) in the equations (19)-(21), thesystem splits into two sets of equations. The first onecorresponds to α = 0 that is, perfect fluid matter distri-bution − µ ′ r − µr + 1 r = ρ (eff) , (33) µ (cid:18) ν ′ r + 1 r (cid:19) − r = p (eff) , (34) µ (cid:18) ν ′′ + ν ′ + 2 ν ′ r (cid:19) + µ ′ (cid:18) ν ′ + 2 r (cid:19) = p (eff) . (35)From now on we shall call the above system of equationsthe Einstein-Rastall system. It can be solved for ρ and p by using Eqs.(22) and (23), in order to express thesequantities as functions of the metric potentials only [49].Explicitly we have − µ ′ r − µr + 1 r − λ (cid:20) − µ (cid:18) r + 3 ν ′ r (cid:19) + 4 r − µ ′ r (cid:21) = ρ, (36) µ (cid:18) ν ′ r + 1 r (cid:19) − r + λ (cid:20) r − µ (cid:18) r + 3 ν ′ r (cid:19) − µ ′ r (cid:21) = p, (37)14 (cid:20) µ (cid:18) ν ′′ + ν ′ + 2 ν ′ r (cid:19) + µ ′ (cid:18) ν ′ + 2 r (cid:19)(cid:21) + λ (cid:20) − µ (cid:18) r + 3 ν ′ r (cid:19) + 4 r − µ ′ r (cid:21) = p. (38)As was pointed out earlier, both ρ and p after some alge-braic manipulations, in their own expressions contain theRastall information as was expected. Besides by putting λ = 0 in Eqs. (36)-(38) one recovers the original GRfield equations for isotropic matter distributions. Fur-thermore, by adding (36) to (37) one regains the usualinertial mass density ρ + p which is given by ρ + p = µν ′ − µ ′ r . (39)Another interesting point to be noted here, is that theisotropic condition is exactly the same like in GR ı.e,4 (1 − µ ) + 2 r ( µ ′ − µν ′ ) + r (cid:0) µν ′′ + µν ′ + µ ′ ν ′ (cid:1) = 0 . (40)Equation (40) says that any solution describing a per-fect fluid matter distribution in GR is also a solution inthe arena of Rastall gravity theory. Obviously there isa subtlety, since both GR and Rastall theory share onlythe geometrical content but not the material one, is inthis sense that ”any” solution to Einstein theory of grav-ity can be seen as a solution in the gravitational Rastallframework. So, the other set of equations correspondingto the factor θ are given by, − f ′ r − fr = θ tt , (41) − f (cid:18) ν ′ r + 1 r (cid:19) = θ rr (42) − f (cid:18) ν ′′ + ν ′ + 2 ν ′ r (cid:19) − f ′ (cid:18) ν ′ + 2 r (cid:19) = θ ϕϕ . (43)The sets of equations (36)-(38) and (41)-(43) satisfythe following conservation equations, ν ′ ρ + p ) + dpdr − λ λ − ddr ( ρ − p ) = 0 , (44) − ν ′ θ tt − θ rr ) + dθ rr dr − r ( θ ϕϕ − θ rr ) = 0 (45)We note that the linear combination of conservationequations (44) and (45) via. coupling constant α pro-vides the conservation equation for the energy momen-tum tensor ¯ T µ (tot) ν = T µ (eff) ν + αθ µν , as follows − dpdr − α (cid:20) ν ′ θ tt − θ rr ) − dθ rr dr + 2 r ( θ ϕϕ − θ rr ) (cid:21) − ν ′ ρ + p )+ λ λ − ddr ( ρ − p ) = 0 . (46)The Eq. (46) is the same expression as Eq. (24) but inan explicit form. Furthermore, as can be seen there isan extra term (the last one) in (46), the so called Rastallforce (or simply the Rastall contribution). This addi-tional term could in principle be attractive or repulsivein nature, due to its behaviour depends on the sign ofthe Rastall parameter λ .At this point it is necessary to comment that fromnow on the total energy-momentum tensor T ( tot ) µν will be defined by the following components ρ ( tot ) ( r ) = ρ ( r ) + αθ tt ( r ) , (47) p ( tot ) r ( r ) = p ( r ) − αθ rr ( r ) , (48) p ( tot ) t ( r ) = p ( r ) − αθ ϕϕ ( r ) , (49)where ρ and p are given by Eqs. (36) and (37), respec-tively. This equations contain the additional geometricterms provided by the Rastall contribution. In this way,as we will see in the following sections, there will be a fullaffect of the Rastall sector in the decoupler function f ( r )and consequently in the θ -sector, as expected. Besides,the redefinition (47)-(49) does not change the anisotropyfactor ∆ definition given by Eq. (28). V. EXTERIOR SPACE-TIME: JUNCTIONCONDITIONS
A crucial aspect in the study of stellar distributions isthe junction conditions. These provide smooth matchingof the interior M − and the exterior M + geometries atthe surface Σ (defined by r = R ) of the stellar object, toinvestigate some significant characteristics of its evolu-tion. To study how the junction conditions work in thiscontext we will assume that the inner stellar geometry M − is given by the MGD metric, ds = e ν ( r ) dt − (cid:18) − m ( r ) r (cid:19) − dr − r ( dθ + sin θdφ ) , (50)where the interior mass function in this case is givenby4 π Z r ρ (tot) r dr ≡ m ( r ) = m GR ( r ) + m λ ( r ) − α r f ( r ) , (51)where we have defined m GR ( r ) as m GR ( r ) = r − µ ( r )] , (52)where from now on we shall call the total mass comingfrom the GR sector as m GR ( R ) = M . On the otherhand, the m λ ( r ) term is equal to m λ ( r ) = λ Z r r (cid:20) − µ (cid:18) r + 3 ν ′ r (cid:19) + 4 r − µ ′ r (cid:21) dr, (53)where at the boundary becomes M λ = m λ ( R ), theRastall mass hereinafter. So, when α = λ = 0 the fa-miliar gravitational mass definition is recovered. At thisstage it is of interest to contrast the so called binding en-ergy in this context with what is reported in GR. In fewwords the binding energy (B.E.) is the difference betweenthe total mass and the proper mass. Explicitly it readsB.E. = m ( R ) − m p ( R ) , (54)where the proper mass m p is given by m p ( R ) = 4 π Z R r ρ q − mr dr. (55)Since the factor q − mr , appearing in the proper mass m p is less than the unity. Then the proper mass is greaterthan the total mass, hence B.E. <
0. Respect to the GRcase, in the present one the proper mass will be little bitdifferent. The main difference is introduced in the factor m/r . As it is well known in the isotropic (uncharged)case this factor at the boundary Σ is bounded by theBuchdhal limit [115] ı.e, M R ≤ . On the other handin the anisotropic (uncharged) case, the above limit canbe overcome [116]. In this opportunity the ratio m/r isaltered by the Rastall and MGD contributions. So, wehave mr = m GR + m λ r − α f ( r ) . (56)Then, 1 − m GR + m λ ) r > αf ( r ) . (57)It is obvious that the above constraint impose some re-striction in order to avoid non-physical situations. Be-sides, (57) also imposes some restrictions on the constant α , since α is not restricted to be a strictly positive quan-tity. So, the proper mass in this case may be greater orless than that the GR case. Therefore the B.E. will bechange according the MGD contribution.Next, the internal manifold (50) should be joined ina smoothly way with outer space-time. This exteriormanifold in principle could contain some contributionscoming from the θ -sector. So, this means that the ex-terior space-time surrounding the compact structure isno longer a vacuum space-time. The most general outermanifold is described by ds = e ν + ( r ) dt − e η + ( r ) dr − r ( dθ + sin θdφ ) . (58)To match the internal configuration (50) with the exteriorone (58) we employ the well known Israel–Darmois (IDhereinafter) junction conditions [117, 118] (for a recentand more clear discussion of how these conditions worksee [119–121]). These conditions are the most generalones. The ID matching conditions involve the first andsecond fundamental forms. The first fundamental formexpress the continuity of the metric potentials across theboundary Σ. More specifically, the metric potentialsdescribe the intrinsic geometry of the manifolds. So,the continuity of the first fundamental form across theboundary of the compact structure, reads (cid:2) ds (cid:3) Σ = 0 , (59)concisely e ν − ( r ) | r = R − = e ν + ( r ) | r = R + , (60) and 1 − MR − = e − η + ( r ) | r = R + , (61)being M = m ( R ) the total gravitational mass containedby the fluid sphere. The second fundamental form isrelated with the continuity of the extrinsic curvature K µν induced by M − and M + on Σ. The continuity of K rr component across Σ yields to h p (tot) r ( r ) i Σ = [ p ( r ) − αθ rr ( r )] Σ = 0 . (62)At this stage some comments are in order. First, p (tot) r has a little different fashion in comparison with the ex-pression (26) since ρ and p were separated implying thatthe Rastall terms are no longer contained in p (eff) through λ as shown Eq. (9). Now the terms coming from theRastall sector are encoded in separate expressions for ρ and p given by Eqs. (36)–(37). From this point of viewit is clear how Rastall contribution comes into the fieldequations. Moreover, from now on we shall denote theRastall input as follows F λ ( r ) = λ (cid:20) − µ (cid:18) r + 3 ν ′ r (cid:19) + 4 r − µ ′ r (cid:21) . (63)It should be noted that the form of F λ depends on thechoice of T µν which in our case is given by Eq. (7) de-scribing a perfect fluid matter distribution. Hence, p ( r )in Eq. (62) is given by Eq. (37). Second, in this way theRastall sector will come into the θ –sector through thedecoupler function f ( r ) (as we will see later) in order tosee the effects on it. So Eq. (62) reads[ p ( r ) − αθ rr ( r )] r = R − = [ − F λ ( r ) − αθ rr ( r )] r = R + . (64)Equation (64) tells us that the outer space-time receivescontributions from the θ –sector, as well as from theRastall non–minimal coupling matter assumption. Inthis respect, in the study of compact structures withinthe framework of modified gravity theories such as f ( R ),the exterior space–time receives contributions from theinclusion of higher order derivative terms coming fromthe Ricci scalar. In principle, these contributions can al-ter or introduce some modifications on the usual junctionconditions. Moreover, the outer manifold could be differ-ent from the usual ones ı.e Schwarzschild vacuum solu-tion, Reissner–Nordstr¨om, for example. At this stage andbased on the previous discussion, a couple of commentsare pertinent in order to clarify how to proceed in mod-ified type gravity theories. In this direction, Capozzielloet.al [13] have argued that in the f ( R ) domain the mass–radius profile undergoes modifications due to the pres-ence of high order curvature terms such as R , R , etc.Besides, in [122] was discussed the well–known ID match-ing conditions in the framework of f ( R ) gravity in con-sidering both isotropic and anisotropic matter distribu-tions. They conclude that ID matching conditions arenot satisfied at all in the f ( R ) gravity arena. However, inthe present situation, one could dropped out the Rastallcontribution F + λ from the external space–time. To do so,one needs to consider an outer geometry free from ma-terial content ı.e, a vanishing energy-momentum tensor T + µν = 0. Then from Eqs. (4) and (5) one arrives to G µν = 0 . (65)The above expression implies (as said before) that bothEinstein and Rastall gravity theories share exactly thesame vacuum solution ı.e, the exterior Schwarzschild so-lution. If one wants to consider contributions comingfrom the Rastall sector, the outer space–time is no longervacuum, since it is filled by an effective cosmological con-stant describing a (anti) de Sitter space–time [54]. So,Eq.(64) becomes[ p ( r ) − αθ rr ( r )] r = R − = [ − αθ rr ( r )] r = R + . (66)It remains to be analyzed how the θ –sector affects the ex-terior geometry. In this case the external solution comesfrom solving the field equations R µν − R g µν = αθ µν , (67)in conjunction with (58). Hence, the resulting outwardgeometry is described by ds = (cid:18) − M Sch r (cid:19) dt − (cid:18) − M Sch r + αg ( r ) (cid:19) − dr − r d Ω , (68)where g ( r ) is the geometric deformation of the exteriorSch-warzschild space–time associated to the source θ µν ,and M Sch denotes the Schwarzschild mass. By using Eq.(42) in (66) we obtain p ( R ) + αf ( R ) (cid:18) R + ν ′ ( R ) R (cid:19) = αg ( R ) (cid:20) R +2 M Sch R (cid:0) − M Sch R (cid:1) (cid:21) , (69)where R − = R + = R at the surface. It should be notedthat if the geometric deformation function g ( r ) of theouter manifold is taken to be null, then one recovers theoriginal Schwarzschild exterior solution. In consequenceEq. (69) leads to the condition p (tot) r ( R ) = p ( R ) + αf ( R ) (cid:18) R + ν ′ ( R ) R (cid:19) = 0 . (70)Equation (70) is an important result, since the compactobject will be in equilibrium in a true exterior space–time without material content (vacuum) only if the totalradial pressure at the surface vanishes. The condition(70) determines the size of the object ı.e the radius R ,which means that the material content is confined withinthe region 0 ≤ r ≤ R . Furthermore the continuity ofthe remaining components of the extrinsic curvature K θθ and K φφ yield to m ( R ) = M. (71) VI. STELLAR INTERIOR: TOLMAN IV MODEL
In this section we solve the set of equations (41)–(43) by imposing some suitable constraints on the θ µν components in order to obtain the deformation function f ( r ) and then compute the full energy–momentum tensor T (tot) µν . Among all the possibilities, to tackle the systemof equations (41)–(43) we follow the same procedure asgiven in [61]. The imposition of some extra information isnecessary in order to close the system of equations (41)–(43). Furthermore, to obtain the deformation function f ( r ) also is necessary provide a seed solution satisfyingequations (36)–(38). To illustrate how gravitational de-coupling by means of MGD works in the Rastall gravityscenario, we apply it to the well known Tolman IV solu-tion. This space–time was already studied in the contextof MDG in [61] and in the framework of Rastall the-ory [49]. So, this allows us to compare the resulting de-formed solution immersed in an anisotropic scenario withprevious results already obtained and therefore establishwhether the study of the compact structures whitin thearena of Rastall gravity + gravitational decoupling byMGD approach is plausible, when the matter distribu-tion contains local anisotropies. Before to proceed wepresent the well–known Tolman IV space–time within theRastall framework, which is described by the followingmetric potentials e ν ( r ) = B (cid:18) r A (cid:19) , (72) µ ( r ) = (cid:16) − r C (cid:17) (cid:16) r A (cid:17)(cid:0) r A (cid:1) , (73)and characterized by the following thermodynamic ob-servables (in the Rastall context) ρ = 1 C ( A + 2 r ) (cid:26) λ (cid:20) r (cid:18) C − A (cid:19) − (cid:18) A +4 r (cid:19)(cid:21) + 3 A + A (cid:18) C + 7 r (cid:19) + 2 r (cid:18) C + 3 r (cid:19)(cid:27) , (74) p = 1 C ( A + 2 r ) (cid:26) λ (cid:20) r (cid:18) A − C (cid:19) + 3 (cid:18) A +4 r (cid:19)(cid:21) + (cid:18) A + 2 r (cid:19)(cid:18) C − A − r (cid:19)(cid:27) . (75)At this point a couple of comments are in order. First, itis worth mentioning that taking the limit λ → A. θ -effects: Mimicking the pressure for anisotropy The so called mimic constraints are some restrictionsimposed at the level of the field equations (36)–(38) and(41)–(43) after introduce the decoupler mechanism (32).In principle, these choices lead to well–behaved solutions,that is, free of undesired physical and mathematical be-haviors such as singularities, non–decreasing thermody-namic functions, violation of causality condition, amongothers. However, other options can be considered, forexample a direct and adequate representation for the ge-ometric deformation function f ( r ) [73, 74, 98] which sat-isfies the basic requirements of physical and mathemati-cal admissibility, or relate only the θ –sector componentsthrough a barotropic, polytropic or linear equation ofstate. In this opportunity, an acceptable interior solutionis deduced when forcing the associated radial pressure θ rr to mimic the isotropic pressure p ( r ). Explicitly it reads θ rr ( r ) = p ( r ) . (76)This constraint implies that the stress–energy tensor forthe seed solution coincides with the anisotropy in the ra-dial direction. Consequently Eq. (37) and Eq. (42) areequals. Thus, this yields to an algebraic general expres-sion for the deformation function f ( r ) = − µ ( r ) + (cid:20) r − F λ ( r ) (cid:21) (cid:20) ν ′ r + 1 r (cid:21) − , (77)where F λ ( r ) and µ ( r ) are given by Eq. (63) and Eq. (73),respectively and ν ′ can be obtained from Eq. (72). Then,the general minimally deformed radial metric potential isexpressed as e − η = (1 − α ) µ ( r )+ α (cid:20) r − F λ ( r ) (cid:21) (cid:20) ν ′ r + 1 r (cid:21) − . (78)The resulting expression of f ( r ) after inserting the cor-responding elements in Eq. (77), is given by f ( r ) = − r C ( A + 2 r ) ( A + 3 r ) (cid:20)(cid:18) A + r (cid:19)(cid:18) A λ +22 A r λ − C r λ + 24 r λ − A r + 2 C r − r − A + A C (cid:19)(cid:21) . (79) In general, the deformed Tolman IV solution by virtue of(77) can be expressed as ds = B (cid:18) r A (cid:19) dt − (cid:20) (1 − α ) (cid:16) − r C (cid:17) (cid:16) r A (cid:17)(cid:0) r A (cid:1) − α (cid:20) r − F λ ( r ) (cid:21) (cid:20) ν ′ r + 1 r (cid:21) − (cid:21) dr − r d Ω . (80)Next, following the discussion in section V, the constantparameters, namely A , B and C defining the interiorsolution can be obtained from e ν ( r ) | r = R − = [ µ ( r ) + αf ( r )] | r = R − = 1 − M Sch R + , (81)where the Schwarzschild mass M Sch coincides at theboundary Σ with total mass M contained by the sphere.Furthermore, from Eqs. (62) and (76) we have(1 − α ) p ( r ) | r = R − = 0 . (82)This last expression (82), imposes a natural constrainton the free parameter α given by α < , (83)in order to preserve p (tot) t > p (tot) r at all points inside thecollapsed structure, which in addition ensures ∆ > p (tot) t ( r ) = p (tot) r ( r ) + αr C ( A + 2 r ) ( A + 3 r ) (cid:20) A λ +4 A C λ + 24 A r λ + 12 A r λ − C r λ +12 A C r + 12 C r + 3 A C (cid:21) , (84)remembering that p (tot) r = (1 − α ) p . As it is observed p (tot) t imposes a lower bound on α ı.e, α >
0. Thus, thepositiveness of the total tangential pressure throughoutthe compact object is ensured. Therefore we have0 < α < , (85)in order to get a well behaved stellar interior. Anotherinteresting point to be noted here, is that the condition(82) leads to C = 12 R + A − R λ (cid:20)(cid:18) R λ − R − A (cid:19)(cid:18) A λ +22 A R λ + 24 R λ − A − A R − R (cid:19)(cid:21) / , (86)which shows that C is α independent. Moreover, a de-tailed computation from (81) shows that the remainingparameters namely A and B also are α independent whenone chooses the constraint (76) (these expressions are to1long to be displayed here, for this reason we only give theappropriate comments). So, expressions (81) and (86) arethe sufficient and necessary conditions to obtain the fullset of constant parameters A , B and C describing theinterior solution. On the other hand, the remaining ther-modynamic observable ρ (tot) can be obtained as follows ρ (tot) ( r ) = ρ ( r ) + αθ tt ( r ) , (87)where ρ ( r ) is given by Eq. (36) and θ tt ( r ) has the follow-ing expression θ tt ( r ) = 1 C ( A + 2 r ) ( A + 3 r ) (cid:20) A a + 146 A ar − A C ar + 522 A ar − A C ar + 1014 A ar − A C ar + 1020 A ar − C ar + 432 ar − A +3 A C − A r + 16 A C r − A r + 29 A C r − A r + 24 A C r − A r + 12 C r − r (cid:21) . (88)The anisotropy factor ∆ is given by the following expres-sion∆( r ) = αr C ( A + 2 r ) ( A + 3 r ) (cid:20) A λ + 4 A C λ +24 A r λ + 12 A r λ − C r λ +12 A C r + 12 C r + 3 A C (cid:21) . (89)It should be noted that at the center of the compact con-figuration ∆(0) = 0. This is so because at the center ofthe star p (tot) t (0) = p (tot) r (0). Besides, p (tot) t > p (tot) r and p (tot) t > > θ components andthe decoupler function f ( r ) can be assumed. In the nextsection, a different constrain is considered yielding to adifferent anisotropic solution. B. θ -effects: Mimicking the density for anisotropy Another way to close the system (41)–(43) and obtaina physically and mathematically admissible solution, is toconsider that the isotropic density given by (36) mimics its ”simile” of the anisotropic sector given by (41). Thenwe have θ tt ( r ) = ρ ( r ) . (90)So, by equating Eqs. (36) and (41) we arrive to a generalexpression for the decoupler function f ( r ) given by f ( r ) = µ ( r ) − r Z F λ ( r ) r dr + Dr , (91)being D an integration constant. To avoid divergent be-havior in the stellar interior we set D = 0. Thus (91)becomes f ( r ) = µ ( r ) − r Z F λ ( r ) r dr. (92)Thus the deformed radial metric potential e − η is givenby e − η = (1 + α ) µ ( r ) + α (cid:18) r Z F λ ( r ) r dr − (cid:19) . (93)Therefore the general deformed Tolman IV solution iswritten as ds = B (cid:18) r A (cid:19) dt − (cid:20) (1 + α ) µ ( r )+ α (cid:18) r Z F λ ( r ) r dr − (cid:19) (cid:21) dr − r d Ω . (94)As before, the parameters A , B and C are obtained fromthe junction conditions. However, the imposition of con-straint (90) slightly changes the information obtainedfrom condition (62). Now from (62) one gets an expres-sion for C in terms of α parameter. The consequencesof this α dependency will be a matter of the followingsection.The rest of the principal variables are obtained afterinserting the following decoupler function f ( r ) f ( r ) = 18 rC ( A + 2 r ) (cid:20) √ λA (cid:18) A + 2 A C + 2 A r +4 C r (cid:19) arctan r √ A ! − A rλ − A C rλ +8 A r λ − C r λ + 32 r λ − A r − C r − r (cid:21) (95)into equations (42)–(43). As it is observed from Eq. (95),there is a global factor 1 /r . This factor arises in thefinal expression of the decoupler function f ( r ) after solveEq. (90). In principle, this factor introduces a singularbehavior at r = 0. However, this is not a true singularityin the present case, because the arctan( r √ /A ) functionis smooth and continuous for all r , hence one can expandthis function in a Taylor series around r = 0 up to firstorder in r to eliminated the singular behaviour and keep2 FIG. 1:
Mimic Constraint p ( r ) = θ rr . To obtain the trend of the principal thermodynamic observables we have consideredthroughout the study the following mass–radius ratio M /R = 0 .
2. Moreover, the red curve (dashed) representing Rastall +MGD corresponds to α = 0 . λ = − .
09, for the blue (dashed–dotted) one corresponding to pure Rastall gravity, α and λ are 0 . − .
09, respectively. Next, the green (short–dashed) line corresponding to GR + MGD takes α = 0 . λ = 0 . α = λ = 0 .
0. Upper row:
Left panel illustrates the monotonicbehaviour from the center to the boundary of the total radial pressure at all points inside the structure. As it is observed thisquantity vanishes at surface. The
Right panel : shows the trend of the total tangential pressure inside the compact object.Lower row:
Left panel exhibits the behaviour of the total energy–density. Finally, the
Right panel displays a comparisonbetween the total radial and total tangential pressure. It should be noted that the presence of anisotropies causes the pressuresvalues to drift apart. the dimensionality of all terms. So, the concrete formonce the expansion is performed, is given by f ( r ) = (cid:0) A λ + 2 C λ + 8 λr − A − C − r (cid:1) r C ( A + 2 r ) . (96)After that, the thermodynamic observables can be com-puted as follows p (tot) r = p ( r ) − αθ rr (97) p (tot) t = p ( r ) − αθ ϕϕ , (98)and by virtue of (90) ρ (tot) = (1 + α ) ρ ( r ) , (99)where ρ ( r ) and p ( r ) are given by Eqs. (36) and (37),respectively. The final expressions are too long to be displayed here, for this reason we have omitted them.However, the expression corresponding to the anisotropyfactor ∆ is quite small and has the following form∆( r ) = α (cid:0) θ rr − θ ϕϕ (cid:1) = α (cid:0) A λ − C λ + 2 C (cid:1) r C ( A + r ) . (100)As it is observed, the above expression has the usual be-haviour ı.e, null at r = 0 and positive defined everywhereinside the compact configuration iff α > ˜˜FIG. 2: Mimic Constraint ρ ( r ) = θ tt . To obtain the trend of the principal thermodynamic observables we have consideredthroughout the study the following mass-radius ratio M /R = 0 .
2. Moreover, the red curve (dashed) representing Rastall+ MGD corresponds to α = 0 . λ = − .
09, for the blue (dashed–dotted) one corresponding to pure Rastall gravity α and λ are 0 . − .
09 respectively. Next, the green (short–dashed) line corresponding to GR + MGD takes α = 0 . λ = 0 .
0. Finally, the black curve (solid) representing GR theory α = λ = 0 .
0. Upper row:
Left panel illustrates the monotonicbehaviour from the center to the boundary of the total radial pressure at all points inside the structure. As it is observedthis quantity vanishes at surface. The
Right panel : shows the trend of the total tangential pressure everywhere inside thecompact object. Lower row:
Left panel exhibits the behaviour of the total energy–density. Finally, the
Right panel displaysa comparison between the total radial and total tangential pressure. It should be noted that the presence of anisotropies causesthe pressures values to drift apart.
VII. THE MASS FUNCTION ANDCOMPACTNESS FACTOR
In this section we analyze the incidences induced bygravitational decoupling via MGD on the mass function m ( r ) and compactness factor u ≡ m ( R ) R . In order toprovide a pedagogic explanation, we will start by dis-cussing the implications of MGD in the context of Ein-stein’s gravity theory and then we will move to Rastall’sapproach.As was pointed out in Sec. IV, the MGD process meansto deform one of the metric potentials, namely ν or e − η .This mechanism allows to separate the seed space–timewith its matter distribution from the new sector θ µν ,geometrically described by the decoupler function f ( r ).However, it should be noted that the only way to splitthe system of equations (33)–(35) is through the map expressed by Eq. (32). This is so because, the t − t component of the Einstein field equations only dependson the g rr metric potential and its derivative. So, if theMGD is realized on the temporal g tt component of themetric tensor characterized by ν , there is not way to sep-arate ρ from θ tt . Hence, the anisotropic behaviour entersto the system via the radial component of the metrictensor. This entails an important consequence on someof the macro physical observables defining the compactstructure, specifically its mass and the associated mass–radius ratio. As it is well–known the gravitational massfunction can be obtained by direct integration of the t − t component of the Einstein equations, yielding to m GR ( r ) = r h − e − λ ( r ) i , (101)4which under MGD becomes m ( r ) = m GR ( r ) + m MGD ( r ) = r − µ ( r ) − αf ( r )] , (102)where we have defined the MGD mass function as follows m MGD ( r ) ≡ − α rf ( r )2 . (103)As it is observed, the original GR mass function m GR ( r )is altered by a quantity (103). In principle, this extrapiece induced by MGD grasp could increase or reduce themass of the compact structure. Obviously this dependson the sign of α and f ( r ). Given that these process canoccur by different situations, it is very important to keepin mind that there is a mandatory condition to satisfy: astrictly positive and increasing mass function everywhereinside the structure. So, in order to clarify this situationwe summarize all the possibilities as follows • Case 1:
Positive α and positive and increasingdecoupler function f ( r ) . In this case the massof the object is reduced. Nevertheless, this extrapiece must grow less than the original mass function m GR ( r ). • Case 2:
Positive α and negative and decreasing f ( r ) function. In this opportunity the mass willincrease. • Case 3:
Negative α and positive and increasingdecoupler function f ( r ). This case is equal to thecase 2. • Case 4:
Negative α and negative and decreasingdeformation function f ( r ). This case is equal tothe case 1.It is obvious that the above general analysis, is validfor all r belonging to the interval [0 , R ]. Now, the sign ofthe constant α depends on several factors. The most im-portant is associated with the anisotropy factor, which ingeneral if the seed solution is described by a perfect fluidmatter distribution is defined by (28). So, if θ rr > θ ϕϕ then α must be positive, otherwise α is negative. Onthe other hand, the behaviour of the decoupler function f ( r ) is subject to the closure of the θ –sector. As waspointed out earlier, to close the problem at least fromthe mathematical point of view it is necessary to sup-plement with additional information. For example in[73, 74, 98], the decoupler function f ( r ) was specifiedwithout assuming any relation between the seed and θ µν sector or by imposing some constraints on the θ µν com-ponents. Thus, the behaviour of f ( r ) in the mentionedcases, is only determined by the behaviour of the µ ( r )metric potential ı.e, positive defined and strictly increas-ing function with increasing r within the star. In thepresent study the situation is quite different because thedeformation function f ( r ) is obtained by imposing theso–called mimic constraint, Eqs. (76) and (90). In this respect, the first of these restrictions involves an inter-esting situation. When the θ rr component is mimickingthe isotropic pressure ( p ( r )), then the decoupler function f ( r ) has the general form p ( r ) = θ rr ( r ) ⇒ f ( r ) = 11 + rν ′ ( r ) − µ ( r ) . (104)From the r − r component of the Einstein field equations p ( r ) = − r + µ ( r ) (cid:18) ν ′ ( r ) r + 1 r (cid:19) , (105)as the pressure in the radial direction must be vanish atthe boundary Σ ≡ r = R , from (105) one gets Rν ′ ( R ) = 1 µ ( R ) − . (106)Next, evaluating f ( r ) at r = R from expression (104)and replacing (106) one arrives to f ( R ) = 0 . (107)This implies that the total mass m GR ( R ) + m MGD ( R )contained by the sphere and observed by a distant ob-server coincides with the original mass m GR ( R ). Thisis a general result independent of the theory. Now, inconsidering the present case, the Rastall mass is also notmodified by MGD. To see this, we obtain the total massfunction m ( r ) from Eqs. (51)–(53) as m ( r ) = r − µ ( r )] [1 − λ ] − α r f ( r )+ 3 λ Z r [ µ ( r ) ν ′ ( r ) − µ ′ ( r )] dr. (108)It is evident that when α = λ = 0 the GR mass functionis recovered, what is more if λ = 1 / Rν ′ ( R ) = 1 − λ + λrµ ′ ( R ) + 4 λµ ( R ) − µ ( R ) µ ( R ) (1 − λ ) , (109)subject to λ = 1 /
3. Then, replacing the above result inEq. (77) evaluated at the boundary, after some algebraone gets again f ( R ) = 0 . (110)So, in this case ( p r = θ rr ) the mass of the compact con-figuration is given by the GR gravitational mass plusRastall contribution. The mass function m ( r ) associated5to the model under study is given by m ( r ) = r − (cid:16) − r C (cid:17) (cid:16) r A (cid:17)(cid:0) r A (cid:1) (cid:20) − λ (cid:21) + αr C ( A + 2 r ) ( A + 3 r ) (cid:20)(cid:18) A + r (cid:19) × (cid:18) A λ + 22 A r λ − C r λ + 24 r λ − A r + 2 C r − r − A + A C (cid:19)(cid:21) + λ (cid:0) A + 2 C (cid:1) r C ( A + 2 r ) . (111)Therefore, if the mass remains unchanged when theisotropic pressure is mimicked the same occurs withthe compactness factor u , since it depends on the to-tal mass M . To support the previous discussion Fig.3 shows the behaviour of the MGD mass and the de-coupler function inside the star (upper panels). As canbe seen both m MGD ( r ) and f ( r ) are vanishing at theboundary of the configuration implying that there is notcontribution coming for the MGD sector to the totalmass of the object. In fact. this is the peculiarity ofthe constraint (76), the mass inside the star is redis-tributed around the core while towards the boundaryis attenuated. One can confirm this by contrasting theRastall+MGD and pure Rastall density profiles given inFig. 1 (left lower panel), where it is observed that thedensity is greater in the Rastall+MGD scenario than inthe pure Rastall case at some point before reaching thesurface, where the Rastall+MGD curve (red) has a dis-continuity ı.e, is dominated by the pure Rastall scenario.The other test that shows that the mass is only redis-tributed within the structure is reflected in Fig. 3 (lowerpanels) which clearly illustrates that both the mass func-tion m ( r ) and the compactness parameter as a functionof the radial coordinate u ( r ) remain unchanged in bothdomains: Rastall+MGD and pure Rastall. Besides, wehave displayed in the same plots the GR and GR+MGDcases to prove that this is a general result, independent ofthe underlying theory. To finalize the discussion regard-ing the mimic constraint (76), it should be noted thatthe mass function curve in the Rastall+MGD and pureRastall setting coincide at r = 0 and r = R while from r/R = 0 . r/R = 0 . θ tt mimics the seed density ρ ( r ) (90). In this case,the mass function m ( r ) associated with this constraint is FIG. 3:
Mimic Constraint p ( r ) = θ rr . The top left panelshows the MGD mass function (103) versus radial coordi-nate r/R while the right panel represents the deformationfunction f ( r ) against the radial coordinate r/R correspond-ing to Rastall+MGD (dashed curve) and GR+MGD (dottedcurve). We observe from figures (top one) that the valueof the MGD mass function is same at the boundary r = R in both Rastall and GR theories, because the deformationfunction vanishes at the boundary r = R . bottom left fig-ure illustrates mass function (111) inside the stellar struc-ture for the Rastall+MGD (dashed), Rastall (dot–dashed),GR+MGD (dotted), and GR (solid) scenario. The bottomright figure shows the compactness u ≡ m ( r ) r versus radialcoordinate r/R with the same description the curve as in leftpanel. As we see, clearly the mass function m ( r ) and compact-ness u coincide at the boundary r = R for Rastall+MGD andpure Rastall case. This situation also occurs for GR+MGDand GR scenarios. This happens due to the no contributionof MGD mass at the boundary i.e. m MGD ( R ) = 0. given as m ( r ) = (1 + α ) r − (cid:16) − r C (cid:17) (cid:16) r A (cid:17)(cid:0) r A (cid:1) (cid:20) − λ (cid:21) + λ (cid:2) A + 2 C (cid:3) r C [ A + 2 r ] ! . (112)As Fig. 4 demonstrates, the usage of this restrictionleads to an increase in the mass of the compact object (seeleft panel in the lower row). This is so because the totalmass M = m ( R ) is proportional to the total seed mass M + M λ by a factor (1 + α ). Of course, this happenssince α > f ( r ) < FIG. 4:
Mimic Constraint ρ ( r ) = θ tt . The top left panelshows the MGD mass function (103) versus the radial co-ordinate r/R , while the right panel represents the deforma-tion function f ( r ) versus r/R corresponding to Rastall+MGD(dashed curve) and GR+MGD (dotted curve). Here the sit-uation is different than the previous mimic constraint. TheMGD mass is increasing monotonically throughout within thestellar object and it has greater value in Rastall case, whichis happening due to the trend of deformation function f ( r )within the object. bottom left figure illustrates mass func-tion (112) inside the stellar structure for the Rastall+MGD(dashed), Rastall (dot-dashed), GR+MGD (dotted), and GR(solid) scenarios. The bottom right figure shows the com-pactness u ≡ m ( r ) r against the dimensionless radial coordi-nate r/R with the same description of the curve as in leftpanel. It is clear from both bottom figures that the mass m ( r )and compactness u in Rastall+MGD case is larger than thepure Rastall, GR+MGD and pure GR cases within the stellarstructure. However, m ( r ) and u ( r ) have approximately thesame value in pure Rastall and GR+MGD within the object. u = M /R = 0 .
2, it is clear that the MGD contributionalways arises as a mass generator, taking into accountthat this strongly depends on the sign of α and f ( r ). A. Bounds induced on α and λ by observationaldata In considering that the previous results depend on thechoice made on α and λ , in order to provide a more realpicture and reliable model we bound the mentioned pa-rameters by using real observational data. In this op-portunity we have selected the well–known millisecondpulsar PSR J1614–2230. The mass of this neutron staris 1 . ± . M ⊙ and corresponds to the highest pre-cisely measured neutron star mass determined to date.It is worth mentioning that to determine the radius R of these structures, it strongly depends upon two mainingredients: i) the equation of state (EoS) driven the in- teraction inside the star, and ii) the method employed toobserve the object, for example X–rays, Shapiro delay,etc. In this regard, the EoS could describe the inter-action of nucleons, nucleons coupled to exotic hadronicmatter such as hyperons or kaon condensates and strangematter (formed by u, d and s quarks), to name a few. Onthe other hand, the method could or not provide infor-mation about the size of the configuration ı.e, its radius R . For example, the measures based on X–rays provideinformation on the mass–radius ratio of the object, thusthe radius R can be predicted. However, the Shapirodelay provides no information about the neutron star ra-dius. In this case to fix the radius R and explore thelimit of the theory, we shall follow the information givenin [123]. In that work, the radius R of the millisecondpulsar PSR J1614–2230, can be inferred by looking atthe M–R curve. In this concern, Demorest et al.[123]used the Shapiro delay approach together with differentEoS, thus determining which interaction leads to obtain-ing the reported mass value for PSR J1614–2230. Whatis more, when nucleon interaction is considered the ra-dius range is 11 −
15 [km], while for the strange matter,it is 10 −
11 [km] (for further details see Fig. 3 in [123]).Before starting to discuss about the boundedness of theparameters α and λ , we would like to highlight some it isimportant to mention that we shall concentrate the anal-ysis only in the mimic constraint given by Eq. (90). Aswas discussed previously, the restriction (76) only causesa mass redistribution inside the compact structure. Thenthe total mass M and its associated compactness pa-rameter u are not altered. Besides, α is automaticallybounded by above and below as shown Eq. (85). To un-derstand how these parameters are disturbed under themimic constraint (90). For this purpose, we start revis-iting the GR+MGD scenario and then the pure Rastallcase. Finally, the Rastall+MGD framework will be dis-cussed.Since any affection on the total mass M is reflectedin the compactness factor u , it is better to deal with u instead of M . This is so because u tell us how thecompact the object is. So, by virtue of Eq. (99) one gets M = (1 + α ) M , (113)remembering that the subscript 0 stands for pure GRscenario. Then, u = (1 + α ) u , (114)where the condition u/u > α >
0. On theother hand, for isotropic fluid spheres the Buchdahl limitin the context of GR says that u ≤ /
9. The extremecase u = 4 / α > u BH = 1 /
2, that is,the black hole limit. Of course, a compact object whosemass–radius ratio corresponds to 4 / u = u (1 − λ ) + 3 λ R Z R r ( µν ′ − µ ′ ) dr. (115)Then, to assure u > u one requires λ " R Z R r ( µν ′ − µ ′ ) dr − u > . (116)It is evident that to satisfy the above restriction, the signof λ depends on the sign of the bracket, what is morethe result of the integral term depends on the choice ofthe metric potentials. Now if MGD is incorporated andtaking into account (99), equation (115) becomes u = [1 + α ] " u (1 − λ ) + 3 λ R Z R r ( µν ′ − µ ′ ) dr . (117)As can be seen α and λ are quite involved. So in general,it is not an easy task to determine the magnitude andsign of these parameters, in order to have u > u . Butfor this specific situation, where ν and µ are given by(72)–(73), the total mass M reads M ≃ [1 + α ] " (cid:0) A + C + R (cid:1) − λ (cid:0) A + 6 C + 8 R (cid:1) C ( A + 2 R ) R , (118)and the expression (117) becomes u ≃ [1 + α ] " u (1 − λ ) + λ (cid:0) A + 2 C (cid:1) R C ( A + 2 R ) . (119)As was pointed out before, α and λ are too much in-volved to be bounded separately. So, to explore the limitof the theory ı.e, Rastall+MGD, we shall proceed in thesame way as proposed by Linares et al. [124]. In thatarticle, the authors studied the impact of Weyl contri-butions inside and outside of the star in the frameworkof the brane world. The maximum compactness factorwas obtained by analyzing the total mass M behaviouragainst the radius R and by considering different ordersof magnitude for the brane world parameter. So, follow-ing the same spirit, we take different orders of magnitudefor λ to elucidate the limit of the theory under this partic-ular solution. At this point some comments are in order.First, to assure M > λ < (cid:0) A + C + R (cid:1) A + 6 C + 8 R . (120)As can be seen, the maximum order of magnitude of theright member in (120) is 10 − . This bound is indepen-dent of the values that A , C and R take. Second, al-though α >
0, it cannot be arbitrarily large, since the
FIG. 5: The M–R curve for different values of λ mentioned intable I and α = 0 .
1. See text for more details.TABLE I: The total mass M and compactness factor u fordifferent values of λ , R = 11 . α = 0 . M = 1 . M ⊙ and u = 0 . λ M/M ⊙ u − .
099 0 . − .
160 0 . − − .
174 0 . − − .
236 0 . − − .
919 0 . extreme limit for the compactness factor corresponds tothe black hole one ı.e, u BH = 1 /
2. Then from (114), itis evident that the exceeding order of magnitude morethan 10 − can create an unrealistic situation for some u . Thus, plausible lower and upper bounds for α are0 < α ≤ − . (121)So, as it is appreciated in Fig. 5 the maximum to-tal mass M and compactness factor u (purple curve)correspond to the lowest value of λ (see table I). Re-spect to the compactness factor, its value is bounded bythe extreme GR isotropic case, that is, 4 /
9. However, u = 0 . M = 2 .
919 is also above the apprised experimentaldata. Hence, there is a clear tendency to increase thetotal mass and mass–radius ratio values when λ changesboth its magnitude and sign. VIII. ANALYSIS AND DISCUSSION
In this section we analyze the physical consequences ofthe results obtained in VI A and VI B. So, the pertinent8comments for the resulting interior solution obtained inVI A are :1. Regarding the junction conditions. The first funda-mental form carries an interesting conclusion aboutthe extra piece introduced by the decoupler func-tion in the radial metric potential e − η . In perform-ing the matching conditions with the vacuum ex-terior space–time described by Schwarzschild solu-tion (see Eq. (81)), the geometric sector describingthe deformed part is totally vanished. This meansthat the total mass contained by the sphere seenby a distant observer matches the original mass ofthe object ı.e, m ( R ) = M + M λ . An importantpoint to be noted is that for r < R the mass func-tion m ( r ) carries the MGD contribution. This isbecause the energy–density has and extra contri-bution coming from the θ –sector. Therefore, theseed energy–density ρ is altered by the presence ofthis additional contribution. Nonetheless, the massremains the same with or without θ –sector. Fromthe physical point of view, as it is shown by theleft panel of Fig 1 (lower row) in the framework ofRastall+MGD (red curve) the total energy–densitydominates other scenarios. The object is denser atthe center in the Rastall+MGD background, buttowards the surface, the total energy–density isdominated by other scenarios. This behavior sug-gests that the same mass is redistributed aroundthe core of the star and diminishing towards thesurface of the object.2. From the second fundamental form, explicitly givenby Eq. (82) one gets and expression for the con-stant C independent of the free parameter α . thisimplies that C takes the same values in consideringpure Rastall gravity and Rastall+MGD approach.Moreover, as p (tot) r = (1 − α ) p ( r ), the resultingcentral pressure is below the central pressure inRastall gravity as illustrates Fig. 1 in the upperrow (left and right panels). This α –independenceis a feature of the mimic constraint (76). In addi-tion as α is restricted to belong to (0 ,
1) in order toobtain an admissible interior solution, by imposing(76) the resulting object’s core will be denser whenadding MGD approach to other theories and thecentral pressure will be less than the seed centralpressure.3. Local anisotropies arising in the system due to thepresence of the extra source θ µν , introduce a pos-itive anisotropy factor ∆( r ) at all points insidethe compact structure. This is a very importantissue in the study of compact configurations be-cause a positive anisotropy factor introduces a re-pulsive force that counteracts the gravitational gra-dient. Therefore preventing the compact objectfrom collapsing below its Schwarzschild radius. Inaddition stability and balance mechanisms are en-hanced [125, 126]. Besides, as was pointed out by Gokhroo and Mehra [127] a positive anisotropy fac-tor allows to build more compact objects. In thepresent study, this feature is depicted in the rightpanel (lower row) of Fig. 1. It is observed thatboth the total radial and total tangential pressurescoincide at the center and then drift apart towardsthe boundary of the object.4. Another relevant point is the macro information ofthe compact structure ı.e, the mass and radius, ob-tained from astrophysical observations. Both quan-tities are related by means of the compactness fac-tor u , which is related with the surface gravitationalred–shift z s . Explicitly z s = 1 √ − u − . (122)In considering isotropic fluid spheres u has and up-per bound known as Buchdahl’s limit [115] givenby u ≤ /
9. By taking the equality, the maxi-mum allowed gravitational surface red–shift for anisotropic spherical matter distribution is z s = 2.Nevertheless, Ivanov studies [128] suggested thatwhen anisotropies are included in the stellar inte-rior, the surface gravitational red–shift increases itsmaximum value in comparison with its isotropiccounterpart. Obviously, z s can not be arbitrar-ily large and its maximum value depends on themechanism to introduce anisotropies into the sys-tem. In this respect as we discussed above, themimic constraint (76) does not alter the total mass(keeping the same radius) of the configuration, onlyis redistributed within the object. So, in this case,the observed z s does not change despite the systemcontains local anisotropies [61, 77]. The reasons be-hind this behavior is due to the MGD contributionencode in the decoupler function f ( r ) is vanishingat the boundary r = R of the structure. Thereforethe original total mass of the object M = M + M λ remains unaltered. This is corroborated in the up-per panels in Fig. 3, as can be seen the MGD masshas not the usual strictly increasing behaviour withincreasing radius, it has two identical minimum val-ues ( m MGD = 0) at r = 0 and at r = R . Thisis supported by the upper right panel where, it isevident that f ( R ) = 0. Besides, the mass func-tion and compactness parameter illustrated in thelower panels of Fig. 3 certify that for Rastall andRastall+MGD, the mass function and the mass–radius ratio are exactly the same (to reinforce thispoint we have added the GR and GR+MGD casesto show that this result is independent of the the-ory).Now we proceed with the appropriated comments forthe results obtained in VI B. In this respect, the mimicconstraint (90) gives more interesting results than themimic constraint (76).91. By imposing the mimic constraint (90), the result-ing junction conditions provide new insights in con-sidering the total mass of the compact object. Thistime the extra piece f ( r ) contributes to the match-ing conditions. This means that the coupling con-stant α has an active role. So, the observed massis no longer the same. This is so because m ( r ) = 4 π Z ρ (tot) r dr = 4 π Z (1 + α ) ρr dr, (123)then m ( r ) = (1 + α ) ( m GR + m λ ) ( r ) . (124)Therefore, by virtue of (90) the mass function m ( r )is proportional to the mass of the seed solutiongiven by m GR + M λ . However, the maximum valuethat the mass and energy–density of the compactstructure can take, is strongly constrained by thevalues taken by the dimensionless constant α . Thisis because parameter C now depends on α and λ .This is evident from the condition of null pressureon the surface of the object, which is different fromthe case previously considered where the constant C only depended on the Rastall parameter λ . Thebehavior of the constant C , in this case, determinesthe energy–density behavior at the center of the ob-ject and consequently the value of its mass. So, toobtain a denser object, we need to go in the direc-tion of increasing energy–density and mass. Thisis possible by assigning small values to parame-ter α , which implies that the constant C decreasesin module. Nevertheless, the value of parameter α cannot be arbitrarily small since the effects ofanisotropy on the stellar interior would be negligi-ble. In addition, negative values of α would intro-duce instabilities in the system since the total tan-gential pressure would be less than the total radialpressure, which represents a physically inadmissiblesituation.2. With respect to the central pressure, it increases ifthe magnitude of α decreases. If α is very small(close to zero) the anisotropy from θ –sector willbe negligible. If α is negative then the anisotropyfactor ∆ will be too, introducing into the systema force attractive in nature. In conclusion, α isbound to be positive defined. However, it should benoted that the restrictions imposed by choice (90)on the parameter α depend on the chosen seed so-lution. For example, for the Heintzman IIa [75] andDurgapal-Fuloria [77] isotropic models, studied inthe framework of GR+MGD, the mimic constraint(90) only allows negative values for α , which en-sures a physically acceptable solution.3. Finally, it is important to highlight that in thepresent case the observational differences betweenisotropic and anisotropic distributions are evident. Due to the total mass varies then compactness fac-tor u changes. This fact alters the surface gravi-tational red–shift z s value. It follows immediatelyfrom the definition of z s z s ( α ) = 1 p − u ( α ) − , (125)where the α dependency is explicit, if α increasesthen the mass grows, in consequence u increases.Hence, the factor 1 / p − u ( α ) increases imply-ing that z s grows its value as it is expected whenthe compact object becomes denser. To verify theimpact of MGD on the main macro physical ob-servables by using (90) we have performed the sameanalysis as before. As Fig. 4 illustrates, the m MGD mass (upper left panel) behaves as usual ı.e, in-creasing function with increasing radial coordinate,reaching its maximum value at the surface. On theother hand, the deformation function f ( r ) (upperright panel) has a decreasing behaviour with in con-junction with a positive α provides in principle atotal mass increased by a certain amount (it shouldbe noted that the total mass also depends on λ ).The lower panels sketched the mass function andmass–radius relation, where the MGD effects onthese quantities are evident.To conclude this section 7.1, it is important to highlightthat we have explored the limits of the theory under theassumption of this particular model, by bounding thefree parameters α and λ with the help of real observa-tional data. For the limits we mean the maximum andminimum order of magnitude of α and λ yielding to a rea-sonable results from the astrophysical point of view ı.e,plausible values for the total mass M and mass–radiusratio u . The analysis was performed by considering therestriction (90) only. This consideration is based on ear-lier discussions about how mimetic constraints modifythe main features of the system. As illustrated by ta-ble I and Fig. 5, in the scenario of Rastall+MGD it ispossible to get more massive and compact objects thanin the GR domain. Notwithstanding, the extreme value u = 4 / / α and λ are subject to themetric functions and thermodynamic variables. In thecritical case u = 4 / ⇒ u > /
9, then the black holevalue u BH = 1 / u . In the present case, it is evident thatwhen λ decreases M and u increase. However, it is clearthat λ is not a variable parameter, but it is necessaryto take at least different orders of magnitude to establishwhat are the maximum values of M and u allowed by thetheory. On the other hand, all this analysis allows to elu-cidate how the Buchdahl limit is affected when one movesfrom GR framework to Rastall scenario (with and with-out MGD). As the expressions (115) and (117) shows the0new contributions to this important quantity are non–trivial. Besides, these contributions strongly depend onthe selected model, since it also determines the signatureof the parameters α and λ which are too much involvedin the mass–radius ratio. It is worth mentioning thatany order of magnitude outside the range considered for λ and α in this opportunity, is ruled out. This is becausethe space parameter { A, B, C } becomes imaginary. IX. CONCLUDING REMARKS
We extended gravitational decoupling via minimal ge-ometric deformation approach into the Rastall gravityscenario. To illustrate how this methodology works inthe background of Rastall gravity, the well known Tol-man IV space–time describing a spherically symmetricand static perfect fluid sphere was analyzed. This modelwas already studied in the light of general relativity +minimal geometric deformation scheme [61] and in thearena of pure Rastall theory [49]. In both cases, the re-sulting model respects the general requirements in orderto describe a well–behaved solution.Since Rastall theory of gravity contains an extra termwhich deviates the attention from general relativity be-havior, in this work we have investigated the effects ofthis extra term and the possibility to obtaining com-pact structures which could serve to describe neutronor quark stars. Due to the presence of this additionalterm, the minimal coupling matter breaks down and inconsequence, Bianchi’s identities are violated (the con-servation law of the energy–momentum tensor). Thisissue could in principle modified the junction conditionmechanism as happened in f(R) gravity, for example. Inthis respect, we have discussed extensively how Rastallcontribution remains inside the compact configuration,allowing the implementation of the most general match-ing conditions ı.e, the Israel-Darmois junction conditions[117, 118]. Moreover, as was pointed out by Rastall [1],his proposal and Einstein theory share the same vacuumsolution, the outer Schwarzschild space–time.To translate the Tolman IV solution to an anisotropicdomain in the Rastall framework, we have followed thesame approach given in [61]. This approach consists inimposing some suitable conditions relating the thermo-dynamic seed observables with the corresponding com-ponents of the new sector ı.e, the θ –sector. With thisextra information at hand, the problem is closed becausethe decoupler function f ( r ) and the full θ –sector is deter-mined. The methodology followed in this work in orderto tackle the system of equations (41)–(42) is known asthe mimic constraints approach. Among all the possibil-ities the most common ones worked in the literature are:i) p ( r ) = θ rr , ii) ρ = θ tt , that is the r − r component of the θ –sector mimics the seed pressure p ( r ) and the t − t onemimics the seed energy–density ρ ( r ). However, it shouldbe noted that an adequate decoupler function f ( r ) canbe imposed in order to close the problem (for more de- tails see [73, 74, 98]). The advantage of both proposalsare evident. Regarding the first one, it allows to obtainthe decoupler function f ( r ) in an easy way. This is so be-cause, one obtains after equating the corresponding fieldequation for p ( r ) and θ rr is an algebraic equation (see Eq.(77)). The second choice does not lead to an algebraicequation, but to a first order differential equation (Eq.(92)). At this point, it is worth mentioning that in thecase of general relativity + minimal geometric deforma-tion, the Rastall contribution F λ is not there. So, obtainthe decoupler function f ( r ) is easier than our case, dueto the Rastall piece F λ strongly depends on the metricpotentials µ ( r ) and ν ( r ) (63). So, when the t − t com-ponent of the θ –sector mimics the seed energy–density ρ ,this additional term could introduce some mathematicalcomplications.The emergence of Rastall term after impose the mimicconstraints, is due to after splitting the system of equa-tions (19)–(21) by introducing the minimal geometric de-formation (32), the resulting seed sector (33)–(35) wassolved in order to express p ( r ) and ρ ( r ) in a separate way.The resulting expressions for ρ ( r ) and p ( r ) are (36) and(37) respectively that contain the usual Einstein termsand the Rastall contribution. This additional term iscoupled to the field equations via a dimensionless con-stant λ , the so–called Rastall’s parameter [1]. Clearly, inthe limit, λ → p ( r ) and ρ ( r ) introduces theRastall contribution into the θ –sector through the defor-mation function f ( r ). Therefore, the incidence of Rastallcontributions are evident. Since, not only Rastall’s pa-rameter λ is affecting the dynamic of the solution, butalso the extra geometrical terms.Mimic constraint methodology does not introduce newinformation into the problem. Because, these constraintsare imposed at the level of the field equations, relatingthem after separate the system of equations (33)–(35) bymeans of minimal geometric deformation approach. Theconsistency of these choices is reflected in the obtainedsolutions. Where in both cases, the evolution of thermo-dynamic parameters reveals an appropriate behavior asdictated by the basic requirements associated with thestudy of compact structures. Furthermore, the mimicconstraint grasp plays an important role in some obser-vational parameters such as the surface gravitational red-shift z s . As it is well–known the surface gravitationalred–shift relates the macro observables features of anycompact configuration ı.e, the mass and radius. In thisrespect, Ivanov studies [128] suggest that z s changes inmagnitude when anisotropies are present in the materialcontent. Moreover, B¨ohmer and Harko [116] discussedthe effects on the compactness factor in the anisotropicmatter distribution case. Notwithstanding, in the presentstudy, the mimic constraint p ( r ) = θ rr does not modifythe total mass of the compact object, it only redistributesthe mass inside the stellar interior. Consequently, thecompactness factor u and surface gravitational red-shift z s remain unchanged, which makes it difficult to dis-1tinguish between an object whose material content isisotropic from an anisotropic one. In distinction withthe case ρ ( r ) = θ tt where the total mass of the objectis modified, therefore the observational implications aredifferent.To show how the anisotropic effects introduced by the θ –sector work in the Rastall framework, we have revisitedthe behavior of the main salient features in the arena ofgeneral relativity, general relativity + gravitational de-coupling minimally deformed and pure Rastall theory.In this concern we have fixed the space parameter tobe { u, α, λ, } = { GR { . , , } ; GR + MGD { . , . , } ;RT { . , , − . } ; RT + MGD { . , . , − . }} (RTmeans Rastall theory). From fig. 1 (these plots corre-spond to p ( r ) = θ rr solution) it is clear that the RT+MGDradial and tangential pressures dominate the correspond-ing ones in the picture of GR and GR+MGD. Nonethe-less, RT dominates all frames. Particularly, in com-paring RT with RT+MGD, the final radial pressure inRT+MGD represents only a portion of the pressure ofthe RT, indeed p ( r ) (RT+MGD) = (1 − α ) p ( r ) (RT) . On theother hand, the final energy–density in the RT+MGDdominates all scenarios. So, by using p ( r ) = θ rr the fi-nal configuration is denser than GR, GR+MGD and RT.However, the increase in energy–density does not reflecta change in the total mass of the object (as discussedearlier). In Fig. 2 (these plots correspond to ρ ( r ) = θ tt solution), the salient radial and tangential pressure inthe RT+MGD picture dominate GR, GR+MGD and RT,what is more the salient energy–density also dominatesover GR and GR+MGD and RT. Finally, both solutionspresent a positive anisotropy factor ∆. In fact, this char-acteristic avoids the system to undergo unstable behav-ior. To back up this analysis and in order to providesome physical meaning to the mimic constraint approach,we have done a detailed study of the impact on the to-tal mass contained by the fluid sphere and its associatedmass–radius ratio. This study is displayed in Sec. VIIand supported by Figs. 3 and 4.Moreover, by fixing the real observational data M =1 . M ⊙ and R = 11 . M and compactness fac-tor allowable for the theory under this particular model.To do so, we have bounded α and λ by using (90) and thesalient physical variables associated to this constraint (wehave not analyzed the situation when the restriction (76)is imposed, because as was discussed this constraint doesnot modify the mass), determining that the maximumvalues for M and u are 2.919 and 0.367612, respectively.In table I are displayed different values for M and u cor-responding to different orders of magnitude for λ and α = 10 − . Any other order of magnitude for λ and α isdiscarded, since the parameter space that describes thegeometry of the considered model ı.e, { A, B, C } becomesimaginary. In this concern, it should be highlighted thatthis is the first time that Buchdahl limit is exploredwithin the framework of Rastall gravity theory (with and without MGD). As can be seen the non–trivial contribu-tions coming from Rastall side (and also from MGD sec-tor), provides a numerical data which is outside the scopeof what is usually reported in the study of compact struc-tures. However, the value 4 / / u equalto this quantity, then the resulting numerical data for u will be greater than 4 /
9. Nevertheless, in that case theupper bound becomes the black hole one ı.e, u BH = 1 / p ( r ) = θ rr (76) and ρ ( r ) = θ tt (90) can beelucidated. In considering the first mimic constraint, itis clear that anisotropy enters the system by perturbingthe seed pressure. This in principle suggests, that someof the macro physical observables of the system such asthe total mass M and related quantities such as the com-pactness factor u and the surface gravitational red–shift z s are not altered. On the other hand, the second mimicconstraint introduces anisotropy by disturbing the den-sity of the seed solution. Clearly, the aforementionedobservables and their related quantities are directly af-fected. In conclusion, if the anisotropy enters the systemthrough a change in the original pressure of the system,from the physical point of view, certain quantities of theoriginal system are preserved, while if it enters due todensity disturbances, these quantities are modified.As a final remark, we want to highlight two things.First, it is possible to obtain well behaved stellar inte-riors in the framework of Rastall gravity by using grav-itational decoupling via minimal geometric deformationapproach. The two families of solutions found in thiswork satisfy and share all the physical and mathemat-ical properties required in the study of compact con-figurations, which serve to understand the behavior ofreal astrophysical objects such as neutron stars, for ex-ample. Second, it was found that Rastall theory is apromising scenario to study the existence of compactstructures described by an anisotropic matter distribu-tion, which results can be contrasted with the well–posedgeneral relativity theory. Moreover, as was discussed inSec. II among all the features that Rastall gravity the-ory shares with other non–conservative modified gravitytheories [107–110], it should be noted that in the cosmo-logical scenario [113, 114] stands as a viable and promis-ing proposal which, together with the study carried outin this work on stellar interiors, can potentially answersome of the unknowns that are open today. X. ACKNOWLEDGEMENTS
S. K. Maurya and F. Tello-Ortiz acknowledge thatthis work is carried out under TRC project, grantNo.-BFP/RGP /CBS/19/099, of the Sultanate ofOman. S. K. Maurya is thankful for continuoussupport and encouragement from the administrationof University of Nizwa. F. Tello-Ortiz is partially2supported by grant Fondecyt No. 1161192, Chile.F. Tello-Ortiz thanks the financial support by theCONICYT PFCHA/DOCTORADO-NACIONAL/2019- 21190856 and projects ANT-1856 and SEM 18-02 at theUniversidad de Antofagasta, Chile. F. Tello-Ortiz thanksto Luciano Gabbanelli for fruitful discussions. [1] P. Rastall, Phys. Rev. D, 6, 3357 (1972).[2] H. A Buchdhal, Mon. Not. Roy. Astron. Soc. 150, 1(1970).[3] A. Starobinsky, Phys. Lett. B 91, 99 (1980).[4] S. D. Odintsov and V. K. Oikonomou Phys. Rev. D 99,064049 (2019).[5] S. D. Odintsov and V. K. Oikonomou Class. QuantumGrav. 36, 065008 (2019).[6] S. Capozziello, R. DAgostino and O. Luongo, Gen. Rel.Grav. 51, 2 (2019).[7] S. V. Chervon, A. V. Nikolaev, T. I. Mayorova, S. D.Odintsov and V. K. Oikonomou, Nuc. Phys. B 936, 597(2018).[8] S. Capozziello, S. Nojiria and S. D. Odintsov Phys. Lett.B 781, 99 (2018).[9] S. Capozziello, C. A. Mantica and L. G. Molinari,arXiv:1810.03204 (2018).[10] C. S. Santos, J. Santos, S. Capozziello and J. S. Alcaniz,Gen. Rel. Grav. 49, 50 (2017).[11] A. V Astashenok et al, Class. Quantum Grav. 34,205008 (2017).[12] V. B. Jovanovi, S. Capozziello, P. Jovanovi and D.Borka, Phys. D. Univ. 14, 73 (2016).[13] S. Capozziello, M. De Laurentis, R. Farinelli and S. D.Odintsov, Phys. Rev. D 93, 023501 (2016).[14] A. V. Astashenok, S. Capozziello and S. D. Odintsov,Phys. Lett. B 742, 160 (2015).[15] A.V. Astashenok, S. Capozziello and S.D. Odintsov, As-trophys. Space. Sci. 355, 333 (2015).[16] A. V. Astashenok, S. Capozziello and S. D. Odintsov,J. Cosmol. Astropart. Phys. 1312, 040 (2013).[17] S. Capozziello, N. Frusciante and D. Vernieri, Gen. Rel.Grav. 44, 1881 (2012).[18] S. Capozziello, M. De Laurentis, S. D. Odintsov and A.Stabile, Phys. Rev. D 83, 064004 (2011).[19] S. Capozziello, M. De Laurentis and A. Stabile Class.Quant. Grav. 27, 165008 (2010).[20] S. Nojiri, S. D. Odintsov, D. Sez-Gmez Phys. Lett. B681, 74 (2009).[21] S. Capozziello, E. De Filippis and V. Salzano, Mont.Not. R. Astron. Soc. 394, 947 (2009).[22] S. Capozziello, A. Stabile and A. Troisi, Class. Quant.Grav. 25, 085004 (2008).[23] T. Harko, F. S. N. Lobo, S. Nojri and S. D. Odintsov,Phys. Rev. D 84, 024020 (2011).[24] J. Wu , G. Li, T. Harko, and S. D. Liang, Eur. Phys. J.C 78, 430 (2018).[25] E. Barrientos , F. S. N. Lobo, S. Mendoza, G. J. Olmo,and D. Rubiera-Garcia, Phys. Rev. D 97, 104041 (2018).[26] D. Deb , B. K. Guha, F. Rahaman, and S. Ray, Phys.Rev. D 97, 084026 (2018).[27] P. H. R. S. Moraes and P. K. Sahoo, Eur. Phys. J. C77, 480 (2017).[28] A. Das, S. Ghosh, B. K. Guha, S. Das, F. Rahaman andS. Ray, Phys. Rev. D 95, 124011 (2017).[29] P. H. R. S. Moraes, J. D. V. Arba˜nil and M. Malheiro, J. Cosmol. Astropart. Phys., 06, 005 (2016).[30] K. Koyama, Rep. Prog. Phys. 79, 046902 (2016).[31] Z. Yousaf, K. Bamba and M. Z. Bhatti, Phys. Rev. D93, 064059 (2016).[32] Z. Yousaf, K. Bamba and M. Z. Bhatti, Phys. Rev. D93, 124048 (2016).[33] R. Zaregonbadi and M. Farhoudi, Gen. Relativ. Grav.48, 142 (2016).[34] E. H.Baffou, M. J. S. Houndjo, M. E. Rodrigues, A.V. Kpadonou and J. Tossa, Phys. Rev. D 92, 084043(2015).[35] A. Alhamzawi and R. Alhamzawi, Int. J. Mod. Phys. D25, 1650020 (2015).[36] H. Shabani and M. Farhoudi, Phys. Rev. D 90, 044031(2014).[37] T. Harko, Phys. Rev. D 90, 044067 (2014).[38] S. Chakraborty, Gen. Relativ. Gravit. 45, 2039 (2013).[39] K. Bamba, S. Capozziello, S. Nojiri and S. D. Odintsov,Astrophys. Space Sci. 342, 155 (2012).[40] S. Capozziello, M. De Laurentis, Phys. Rept. 509, 167(2011).[41] S. Capozziello and V. Faraoni, Beyond Einstein Gravity(Springer, New York) (2010).[42] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451(2010).[43] E. H. Baffou, M. J. S. Houndjo, M. E. Rodrigues, A. V.Kpadonou and J. Tossa, Chin. J. Phys. 55, 467 (2007).[44] S. Nojiri and S. D. Odintsov, Int. J. Geom. Meth. Mod.Phys. 4, 115 (2007).[45] S. K. Maurya, Abdelghani Errehymy, Debabrata Deb,Francisco Tello-Ortiz, Mohammed Daoud, Phys. Rev.D 100, 044014 (2019); arXiv:1907.10149.[46] S. K. Maurya, Ayan Banerjee, Francisco Tello-Ortiz,Physics of the Dark Universe 27, 100438 (2020);arXiv:1907.05209.[47] S. K. Maurya, Francisco Tello-Ortiz, Annals of Physics414, 168070 (2020); arXiv:1906.11756.[48] Debabrata Deb, Sergei V. Ketov, S. K. Maurya, MaximKhlopov, P. H. R. S. Moraes, Saibal Ray, Mon. Not. R.Astr. Soc. 485, 5652 (2019).[49] S. Hansraj, A. Banerjee and P. Channuie, Ann. Phys.400, 320 (2019).[50] C. Abbas and M. R. Shahzad, Eur. Phys. J. A 54, 211(2018).[51] C. Abbas and M. R. Shahzad, Astrophys. Space Sci.364, 50 (2019).[52] G. Abbas and M. R. Shahzad, Astrophys. Space Sci.363, 251 (2018).[53] Y. Heydarzade and F. Darabi, Phys. Lett. B 771, 365(2017).[54] Y. Heydarzade, H. Moradpour and F. Darabi, Can. J.Phys. 95, 1253 (2017).[55] K. Bamba, A. Jawad, S. Rafique and H. Morad-pour,Eur. Phys. J. C 78, 986 (2018).[56] I. Lobo, P. Moradpour, J. P. M. Graca and I. G. Salako,Int. J. Mod. Phys. D 27, 1850069 (2018). [57] M. S. Ma and R. Zhao, Eur. Phys. J. C 77, 629 (2017).[58] R. Kumar and S. G. Ghosh, Eur. Phys. J. C 78, 750(2018).[59] Z. Xu, X. Hou, X. Gong and J. Wangr, Eur. Phys. J. C78, 513 (2018).[60] J. Ovalle, Phys. Rev. D 95, 104019 (2017).[61] J. Ovalle, R. Casadio, R. da Rocha and A. Sotomayor,Eur. Phys. J. C 78, 122 (2018).[62] J. Ovalle, Laszl´o A. Gergely and R. Casadio, Class.Quantum Grav. 32, 045015 (2015).[63] J. Ovalle, Int. J. Mod. Phys. Conf. Ser. 41, 1660132(2016).[64] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370(1999).[65] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690(1999).[66] J. Ovalle, Mod.Phys.Lett. A 23, 3247 (2008).[67] J. Ovalle and F. Linares, Phys.Rev. D 88, 104026 (2013).[68] J. Ovalle, F. Linares, A. Pasqua, A. Sotomayor,Class.Quantum Grav., 30, 175019 (2013).[69] R. Casadio, J. Ovalle, R. da Rocha, Class. QuantumGrav. 30, 175019 (2014).[70] R. Casadio, J. Ovalle and R. da Rocha, Europhys. Lett.110, 40003 (2015).[71] R. Casadio, J. Ovalle and R. da Rocha, Class. QuantumGrav. 32, 215020 (2015).[72] J. Ovalle, R. Casadio and A. Sotomayor, Adv. HighEnergy Phys. 2017, 9 (2017).[73] S. K. Maurya and F. Tello-Ortiz, Eur. Phys. J. C 79, 85(2019).[74] E. Morales and F. Tello-Ortiz, Eur. Phys. J. C 78, 841(2018).[75] M. Estrada and F. Tello-Ortiz, Eur. Phys. J. Plus 133,453 (2018).[76] E. Morales and F. Tello-Ortiz, Eur. Phys. J. C 78, 618(2018).[77] L. Gabbanelli, A. Rinc´on and C. Rubio, Eur. Phys. J.C 78, 370 (2018).[78] C. Las Heras and P. Le´on, Fortsch. Phys. 66, 1800036(2018).[79] A. R. Graterol, Eur. Phys. J. Plus 133, 244 (2018).[80] J. Ovalle and A. Sotomayor, Eur. Phys. J. Plus 133, 428(2018).[81] L. Gabbanelli, J. Ovalle, A. Sotomayor, Z. Stuchlik andR. Casadio, Eur.Phys.J. C 79, 486 (2019).[82] S. Hensh, Z. Stuchl´ık Eur.Phys.J. C 79, 834 (2019).[83] E. Contreras, A. Rinc´on and P. Bargue˜no, Eur. Phys.J. C, 79, 216 (2019).[84] K. N. Singh, S. K. Maurya, M. K. Jasim, F. Rahaman,Eur. Phys. J. C 79, 851 (2019).[85] F. Tello-Ortiz, S. K. Maurya, Y. Gomez-Leyton, Eur.Phys. J. C 80, 324 (2020).[86] J. Ovalle, R. Casadio, R. da Rocha , A. Sotomayor andZ. Stuchlik, Eur. Phys. J. C 78, 960 (2018).[87] E. Contreras and P. Bargue˜no, Eur. Phys. J. C 78, 558(2018).[88] E. Contreras and P. Bargue˜no, Eur. Phys. J. C, 78, 985(2018).[89] E. Contreras, Eur. Phys. J. C, 78, 678 (2018).[90] E. Contreras, Class. Quant. Grav. 36, 095004 (2019).[91] G. Panotopoulos and A. Rinc´on, Eur. Phys. J. C 78,851 (2018). [92] J. Ovalle, R. Casadio, R. Da Rocha, A. Sotomayor andZ. Stuchlik, EPL 124, 20004 (2018).[93] C. Las Heras and P. Le´on Eur. Phys. J. C 79, 990 (2019).[94] J. Ovalle, C. Posada and Z. Stuchl´ık, Class. QuantumGrav. 36, 205010 (2019).[95] J. Ovalle, Phys.Lett. B, 788, 213 (2019).[96] E. Contreras and P. Bargue˜no, Class. Quantum Grav.36, 215009 (2019).[97] S. K. Maurya, Eur. Phys. J. C 79, 958 (2019)[98] M. Estrada, R. Prado, Eur.Phys.J.Plus 134, 168 (2019).[99] M. Estrada, Eur. Phys. J. C 79, 918 (2019).[100] S. K. Maurya and F. Tello-Ortiz, Phys. Dark Univ. 27,100442 (2020).[101] F. X. L. Cede˜no and E. Contreras, Phys. Dark Univ.28, 100543 (2020).[102] M. Visser, Phys. Lett. B, 782 83 (2018).[103] A. De Felice and S. Tsujikawa, Living Rev. Rel. 13, 3(2010).[104] F. Darabi, H. Moradpour, I. Licata, Y. Heydarzade andC. Corda, Eur. Phys. J. C 78, 25 (2018).[105] W. A. G. De Moraes and A. F. Santos, Gen. Rel. Grav. , 167 (2019).[106] A. F. Santos and S. C. Ulhoa, Mod. Phys. Lett. A ,1550039 (2015).[107] O. Bertolami, C. Bohmer, T. Harko and F. Lobo, Phys.Rev. D 75, 104016 (2007).[108] O. Bertolami and J. p´aramos, Phys. Rev. D 77, 084018(2008).[109] O. Bertolami and A. Martins, Phy. Rev. D 85, 024012(2012).[110] O. Bertolami and J. P´aramos, JCAP 1003, 009 (2010).[111] A. S. Al-Rawaf and M. O. Taha, Phys. Lett. B366, 69(1996).[112] H. Seeliger, Astro. Nach. 137, 129 (1895).[113] J. C. Fabris, O. F. Piattella, D. C. Rodrigues, C. E. M.Batista and M. H. Daouda, Int. J. Mod. Phys. Conf.Ser. 18, 67 (2012).[114] W. Khyllep and J. Dutta, Phys. Lett. B 797, 134796(2019).[115] H. A. Buchdahl, Phys. Rev. D 116, 1027 (1959).[116] C. G. B¨ohmer and T. Harko, Class. Quantum Gravit.23, 6479 (2006).[117] W. Israel, Nuovo Cim. B 44, 1 (1966).[118] G. Darmois, M´emorial des Sciences Mathematiques(Gauthier-Villars, Paris, 1927), Fasc. 25 (1927).[119] P. Musgrave and K. Lake, Class.Quant.Grav. 13, 1885(1996).[120] K. Lake, Phys. Rev. D 67, 104015 (2003) .[121] K. Lake, Gen. Relativ. Gravit. 49, 134 (2017).[122] J. M. M. Senovilla, Phys. Rev. D 88, 064015 (2013).[123] P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E.Roberts and J. W. T Hessels, Nat. , 1081 (2010).[124] F. X. Linares, M. A. Garcia–Aspeitia and L. A. Ure˜na–Lopez,
Phys. Rev. D92