Interior solutions of relativistic stars with anisotropic matter in scale-dependent gravity
EEPJ manuscript No. (will be inserted by the editor)
Interior solutions of relativistic stars with anisotropic matter inscale-dependent gravity
Grigoris Panotopoulos , ´Angel Rinc´on , and Il´ıdio Lopes Centro de Astrof´ısica e Gravita¸c˜ao-CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico-IST,Universidade de Lisboa-UL, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso, Avenida Brasil 2950, Casilla 4059, Valpara´ıso, Chile.Received: date / Revised version: date
Abstract.
We obtain well behaved interior solutions describing hydrostatic equilibrium of anisotropic rel-ativistic stars in scale-dependent gravity, where Newton’s constant is allowed to vary with the radialcoordinate throughout the star. Assuming i) a linear equation-of-state in the MIT bag model for quarkmatter, and ii) a certain profile for the energy density, we integrate numerically the generalized structureequations, and we compute the basic properties of the strange quark stars, such as mass, radius and com-pactness. Finally, we demonstrate that stability criteria as well as the energy conditions are fulfilled. Ourresults show that a decreasing Newton’s constant throughout the objects leads to slightly more massiveand more compact stars.
PACS.
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Einstein’s theory of General Relativity (GR) [1] is a rela-tivistic theory of gravitation, which not only is beautifulbut also very successful [2,3]. The classical tests and solarsystem tests [4], and recently the direct detection of grav-itational waves by the LIGO/VIRGO collaborations [5]have confirmed a series of remarkable predictions of GR.Despite its success, however, it has been known for along time that GR is a classical, non-renormalizable theoryof gravitation. Formulating a theory of gravity that incor-porates quantum mechanics in a consistent way is still oneof the major challenges in modern theoretical physics. Allcurrent approaches to the problem found in the literature(for a partial list see e.g. [6–14] and references therein),have one property in particular in common, i.e. the basicquantities that enter into the action describing the modelat hand, such as Newton’s constant, gauge couplings, thecosmological constant etc, become scale dependent (SD)quantities. This of course is not big news, as it is knownthat a generic feature in ordinary quantum field theory isthe scale dependence at the level of the effective action.As far as black hole physics is concerned, the impactof the SD scenario on properties of black holes, such asthermodynamics or quasinormal spectra, has been studiedover the last years, and it has been found that SD modifies a E-mail: [email protected] b E-mail: [email protected] c E-mail: [email protected] the horizon, thermodynamic properties and the quasinor-mal frequencies of classical black hole backgrounds [15–21]. Moreover, a scale dependent gravitational coupling isexpected to have significant cosmological and astrophysi-cal implications as well. In particular, since compact ob-jects are characterized by ultra dense matter and stronggravitational fields, a fully relativistic treatment is re-quired. Naturally, it would be interesting to investigatethe impact of the SD scenario on properties of relativisticstars.In the present work we propose to obtain for the firsttime interior solutions of relativistic stars with anisotropicmatter in the SD scenario, extending a previous work ofours where we studied isotropic compact objects [22]. Inparticular, here we shall focus on strange quark stars,which comprise a less conventional class of compact stars.Although as of today they remain hypothetical astronom-ical objects, strange quarks stars cannot conclusively beruled out yet. As a matter of fact, there are some claims inthe literature that there are currently some observed com-pact objects exhibiting peculiar features (such as smallradii for instance) that cannot be explained by the usualhadronic equations-of-state used in neutron star studies,see e.g. [23–25], and also Table 5 of [26] and referencestherein. The present study is also relevant for the pos-sible implications to understand the nature of compactstars. Recently, a few authors suggested that strange mat-ter could exist in the core of NS-hybrid stars [27–29], whileothers claim such stars are almost indistinguishable fromNS [30]. a r X i v : . [ g r- q c ] J a n Grigoris Panotopoulos et al.: Interior solutions of relativistic stars with anisotropic matter in scale-dependent gravity
Celestial bodies are not always made of isotropic mat-ter, since relativistic particle interactions in a very densenuclear matter medium could lead to the formation ofanisotropies [31]. The investigation of properties of anisotropicrelativistic stars has received a boost by the subsequentwork of [32]. Indeed, anisotropies can arise in many sce-narios of a dense matter medium, such as phase transi-tions [33], pion condensation [34], or in presence of type3 A super-fluid [35]. See also [36–39] as well as [40–42] formore recent works on the topic, and references therein. Inthe latter works relativistic models of anisotropic quarkstars were studied, and the energy conditions were shownto be fulfilled. In particular, in [40] an exact analyticalsolution was obtained, in [41] an attempt was made tofind a solution to Einstein’s field equations free of singu-larities, and in [42] the Homotopy Perturbation Methodwas employed, which is a tool that facilitates working withEinstein’s field equations.Currently there is a rich literature on relativistic stars,which indicates that it is an active and interesting field.For stars with a net electric charge see e.g. [43–51], forstars with anisotropic matter see e.g. [52–61], and for chargedansotropic objects see e.g. [62–69], and also compact starswith specific mass function [70].The plan of our work is the following: In the next sec-tion we briefly review the SD scenario. After that, in Sect.3 we present the generalized structure equations that de-scribe hydrostatic equilibrium of relativistic stars. Then,in the fourth section we introduce the equation-of-state,we obtain the interior solutions integrating the structureequations numerically, and we also show that the solutionsobtained here are realistic, well behaved solutions. Finally,we summarize our work and finish with some concludingremarks in the final section. We adopt metric signature,( − , + , + , +), and we work in units where the speed of lightin vacuum, c , and the usual Newton’s constant, G N , areset to unity. The aim of this section is to briefly introduce the formal-ism.The asymptotically safe gravity program is one of thevariety of approach of quantum gravity an this is, pre-cisely, the inspiration of our formalism. Also, close-relatedapproaches share similar foundations, for instance the well-known Renormalization group improvement method [71–74] (usually applied to black hole physics) or the runningvacuum approach [75–80] (usually implemented in cosmo-logical models). Following the same philosophy, recentlyscale-dependent gravity has provided us with non-trivialblack holes solutions as well as cosmological solutions, in-vestigating different conceptual aspects and offering novelresults (see, for instance [81–97] and references therein).Roughly speaking, scale-dependent gravity extends clas-sical GR solutions after treating the classical coupling asscale-dependent functions, which can be symbolically rep- resented as follows { A , B , ( · · · ) } → { A k , B k , ( · · · ) k } (1)Notice that the sub-index k is an arbitrary renormaliza-tion scale, which should be connected with one of the co-ordinates of the system. To account for the relevant inter-actions, we start by considering a effective action writtenas S [ g µν , k ] ≡ S EH + S M + S SD (2)where the terms above mentioned have he usual meaning,namely: i) the Einstein-Hilbert action S EH , ii) the mattercontribution S M , and finally iii) the scale-dependent term S SD . Moreover, notice that the contribution S M could ac-count for either isotropic or anisotropic matter. For ourconcrete case, the parameter allowed to vary is Newton’scoupling G k (or, equivalently, Einstein’s coupling κ k ≡ πG k ). There are two independent fields, i.e., i) the met-ric tensor, g µν ( x ), and ii) the scale field k ( x ). To obtain theeffective Einstein’s field equations, we take the variationof (2) with respect to g µν ( x ): R µν − Rg µν = κ k T effec µν (3)In scale-dependent gravity the effective energy-momentumtensor T effec µν is defined by T effec µν ≡ T µν − κ k ∆t µν (4) ∆t µν ≡ G k (cid:16) g µν (cid:3) − ∇ µ ∇ ν (cid:17) G − k (5)where the last tensor is obtained after an integration byparts.The conventional energy-momentum tensor, T µν , cor-responds to matter fields, whereas ∆t µν carries the infor-mation regarding the running of the gravitational coupling G k . In this sense, when the scale-dependent effect is ab-sent, the aforementioned tensor clearly vanishes.In this work, since we are interested in stars with anisotropicmatter content, we shall consider an energy-momentumtensor of the form T µν = diag (cid:0) − ρ, p r , p t , p t (cid:1) (6)with two different pressures, radial p r , and tangential, p t .Let us comment in passing that in principle one could in-clude the shear as well. It turns out, however, that shear ispresent only in cases where the metric components dependboth on the radial coordinate, r and the time, t . This isthe case for instance in gravitational collapse, see e.g. [98].Since here we are looking for static, spherically symmetricsolutions, shear does not contribute and therefore we shallignore it.Now, it is essential to improve our comprehension aboutthe running of Newton’s coupling, and how such a featureis affected by setting a certain renormalization scale k . Inthis respect, it is well known that General Relativity can rigoris Panotopoulos et al.: Interior solutions of relativistic stars with anisotropic matter in scale-dependent gravity 3 be treated as a low energy effective theory and, therefore,it may be viewed as a quantum field theory with an ultra-violet cut-off (see [99, 100] and references therein). Thatcut-off is parameterized by the renormalization scale k ,which allows us to surf between a classical and a quan-tum regime. In order to make progress, the external scale k is usually connected with the radial coordinate. In thescenario where G → G k , the corresponding gap equa-tions are not constant any more. The latter means thatnon-constant k = k ( x ) implies that the set of equationsof motion does not close consistently. Also, the energy-momentum tensor could be not conserved for a concretechoice of the functional dependence k = k ( x ). This pathol-ogy has been analyzed in detail in the context of renor-malization group improvement of black holes in asymp-totic safety scenarios. The source of the problem is that aconsistency equation is missing, and it can be computedvarying the corresponding action with respect to the field k ( · · · ), i.e., dd k S [ g µν , k ] = 0 (7)usually considered to be a variational scale setting proce-dure [74, 101]. The combination of Eq. (7) with the equa-tions of motion ensures the conservation of the energy-momentum tensor, although an unavoidable problem ap-pears in this approach, i.e., we should know the corre-sponding β -functions of the theory. Given that they arenot unique, we circumvent the above mentioned compu-tation and, instead of that, we supplement our problemwith a auxiliary condition. The energy conditions are fourrestrictions usually demanded in General Relativity, be-ing the Null Energy Condition (NEC hereafter) the lessrestrictive of them. We take advantage of this, and we pro-mote the classical coupling to radial-dependent couplingsto solve the functions involved.Thus, this philosophy of assuring the consistency of theequations by imposing a null energy condition will also beapplied for the first time in the following study on interior(anisotropic) solutions of relativistic stars. In this section we briefly review relativistic anisotropicstars in General Relativity and, after that, we will gener-alize the structure equations in the scale-dependent sce-nario. Clearly, this work is a natural continuation of ourprevious work where isotropic relativistic star in the scale-dependent scenario were studied [22].The starting point is Einstein’s field equations withouta cosmological constant G µν = 8 πT µν (8)where G µν is Einstein’s tensor, and T µν is the matterstress-energy tensor, which for anisotropic matter takesthe form [36, 38] T µν = diag( − ρ, p r , p t , p t ) (9) where ρ is the energy density, p r is the radial pressure and p t is the transverse pressure.Considering a non-rotating, static and spherically sym-metric relativistic star in Schwarzschild coordinates, ( t, r, θ, φ ),the most general metric tensor has the form: ds = − e ν d t + 11 − m ( r ) /r d r + r dΩ (10)where we introduce for convenience the mass function m ( r ), and dΩ is the line element of the unit two-dimensionalsphere. One obtains the Tolman-Oppenheimer-Volkoff equa-tions for a relativistic star with anisotropic matter [36,38] m (cid:48) ( r ) = 4 πr ρ ( r ) (11) ν (cid:48) ( r ) = 2 m ( r ) + 4 πr p r ( r ) r (1 − m ( r ) /r ) (12) p (cid:48) r ( r ) = − (cid:0) p r ( r ) + ρ ( r ) (cid:1) m ( r ) + 4 πr p r ( r ) r (1 − m ( r ) /r ) + 2 ∆ ( r ) r (13)where we define the anisotropic factor ∆ ≡ p t − p r , andthe prime denotes differentiation with respect to the radialcoordinate r . The special case in which p r = p t (i.e., when ∆ = 0) one recovers the usual Tolman-Oppenheimer-Volkoffequations for isotropic stars [102, 103].The exterior solutions is given by the well-known Schwarzschildgeometry [104] ds = − f ( r )d t + f ( r ) − d r + r dΩ (14)where f ( r ) = 1 − M/r , with M being the mass of theobject. Matching the solutions at the surface of the star,the following conditions must be satisfied m ( r ) (cid:12)(cid:12)(cid:12) r = R = M (15) p r ( r ) (cid:12)(cid:12)(cid:12) r = R = 0 (16) e ν ( r ) (cid:12)(cid:12)(cid:12) r = R = 1 − MR (17)The second condition allows us to compute the radius ofthe star, the first one allows us to compute the mass ofthe object, while the last condition allows us to determinethe initial condition for ν ( r ). Finally, depending on thephysics of the matter content the appropriate equation-of-state should be also incorporated, see next section.Next, we shall now generalize the standard structureequations (valid in General Relativity) in the scale-dependentscenario which accounts for quantum effects. As we al-ready mentioned before, Newton’s constant is promotedto a function of the radial coordinate, G ( r ), and thereforethe effective field equations now take the form R µν − Rg µν = 8 πG ( r ) T effec µν (18)where the effective stress-energy tensor has two contri-butions, namely one from the ordinary matter, T µν , andanother due to the G-varying part, ∆t µν T effec µν = T µν − πG ∆t µν (19) Grigoris Panotopoulos et al.: Interior solutions of relativistic stars with anisotropic matter in scale-dependent gravity where the G-varying part was introduced in 5 (see e.g.[105] and references therein for additional details).Similarly to the classical case, the structure equationsvalid in the scale-dependent scenario are found to be( G ( r ) m ) (cid:48) = 4 πG ( r ) r ρ eff (20) ν (cid:48) ( r ) = 2 G ( r ) m + 4 πr p eff r r (1 − G ( r ) m/r ) (21)and we spare the details for the last equation, since it istoo long to be shown here. Notice also that when G (cid:48) ( r ) =0 = G (cid:48)(cid:48) ( r ) (classical case), the previous set of equations isreduced to the usual TOV equations.Finally, there is an additional differential equation ofsecond order for G ( r ), which is the following [105]2 G ( r ) (cid:48)(cid:48) G ( r ) (cid:48) − G ( r ) (cid:48) G ( r ) = (cid:20) ln (cid:18) e ν ( r ) − m ( r ) /r (cid:19)(cid:21) (cid:48) (22)and which must be supplemented by two initial conditionsat the center of the star, G ( r ) (cid:12)(cid:12)(cid:12) r =0 = G c (23) G ( r ) (cid:48) (cid:12)(cid:12)(cid:12) r =0 = G . (24) From the formulation of the problem it it clear that thereare four equations, namely the three Einstein’s field equa-tions plus the additional one for Newton’s constant, andsix unknown quantities, namely two metric potentials, ther-varying gravitational coupling, and the energy densityand the pressures of anisotropic matter. Therefore one isallowed to start by assuming two conditions. As usual instudies of relativistic stars with anisotropic matter, weshall assume a given density profile with a reasonable be-havior as well as a certain equation-of-state (EoS). There-fore, before we proceed to integrate the structure equa-tions, we must specify the matter source first.
Matter inside the stars is modelled as a relativistic gas ofde-confined quarks described by the MIT bag model [106,107], where there is a simple analytic function, relatingthe energy density to the pressure of the fluid, that is p r = k ( ρ − ρ s ) (25)where k is a dimensionless numerical factor, while ρ s isthe surface energy density. The MIT bag model is charac-terized by 3 parameters, namely i) the QCD coupling con-stant, α c , ii) the mass of the strange quark, m s , and iii)the bag constant, B . The numerical values of k and ρ s de-pend on the choice of m s , α c , B . In this work we shall con-sider the extreme model SQSB40, where m s = 100 MeV, α c = 0 . B = 40 MeV fm − . In this model k = 0 . ρ s = 3 . × g cm − [108].What is more, given that the number of unknown quan-tities exceeds the number of equations, we may assume aparticular density profile ρ ( r ) as was done for instancein [36,38]. In our case, we have selected the following den-sity profile: ρ = b ar (1 + ar ) (26)which is a monotonically decreasing function of the radialcoordinate r , the central value of which is ρ c ≡ ρ (0) = 3 b .The two free parameters a, b have dimensions [ L ] − , andthey will be taken to be a = ˜ a (45 km ) (27) b = ˜ b (45 km ) (28)where now ˜ a, ˜ b are dimensionless numbers. Next, since the energy density and the radial pressure areknown, we obtain a closed system for m ( r ) , ν ( r ) , G ( r ) us-ing the tt and the rr field equations combined with theequation for G ( r ). Once these are determined, the lastfield equation allows us to compute the transverse pres-sure p t and the anisotropic factor ∆ .Since in this work we assume a vanishing cosmologicalconstant, the exterior solutions is still given by the well-known Schwarzschild geometry, and therefore the match-ing conditions remains the same as in GR. In the SD sce-nario there is an additional condition, which requires thatNewton’s constant must take precisely the classical valueat the surface of the star. Therefore, in the SD scenario,the matching conditions are the following: m ( r ) (cid:12)(cid:12)(cid:12) r = R = M (29) p r ( r ) (cid:12)(cid:12)(cid:12) r = R = 0 (30) e ν ( r ) (cid:12)(cid:12)(cid:12) r = R = 1 − MR (31) G ( r ) (cid:12)(cid:12)(cid:12) r = R = 1 (32)To integrate the structure equations for m ( r ) , G ( r ) , ν ( r )we impose the initial conditions at the center of the star m ( r = 0) = 0 (33) ν ( r = 0) = ν c (34) G ( r = 0) = G c (35) G (cid:48) ( r = 0) = ± . km (36)where we consider two distinct cases, namely that G ( r )can be either a decreasing or an increasing function of rigoris Panotopoulos et al.: Interior solutions of relativistic stars with anisotropic matter in scale-dependent gravity 5 r , and we fix the absolute value of G (cid:48) ( r = 0). The cen-tral values G c , ν c , in principle unknown, are determineddemanding that the matching conditions for G ( R ) , ν ( r ),i.e. G ( r = R ) = 1 , e ν ( r = R ) = 1 − MR (37)are satisfied. It should be emphasized here that if G c and ν c are picked up at random, the above matching conditionsare not satisfied. Instead, they are satisfied only for thespecific initial conditions shown in tables 1 and 2. Our main numerical results are summarized in the tablesand in the figures below. In particular, first we present indetail five representative solutions for G (cid:48) ( r = 0) <
0, andfive more for G (cid:48) ( r = 0) >
0. The numerical values of ˜ a, ˜ b are shown as well. In Tables 1 and 2 we show the initialconditions for G ( r ) and ν ( r ) for a negative and a posi-tive G (cid:48) ( r = 0), respectively, while in Tables 3 and 4 weshow the properties (i.e. mass, radius and compactness) ofstrange quark stars for positive and negative G (cid:48) ( r = 0), re-spectively. Our results show that for a given density profile(given pair a, b and given radius), a decreasing Newton’sconstant (Table 4) implies a more massive star and con-sequently a higher compactness factor in comparison withan increasing Newton’s constant (Table 3). The mass-to-radius profiles as well as the factor of compactness versusthe mass, and the surface red-shift, z s , versus the radiusof the objects are shown in the three panels of Fig. 6. Thesurface red-shift, an important quantity to astronomers,is given by [109–111] z s = − (cid:18) − MR (cid:19) − / (38)Fig. 1 shows the metric potentials e ν ( r ) and e λ ( r ) as afunction of the dimensionless radial coordinate r/R . Fig. 2shows the (normalized) energy density as well as the radialand transverse pressure versus r/R for the five plus fivecases considered here, while Fig. 3 shows the anisotropicfactor ∆/ρ s vs r/R . Fig. 4 shows the scale-dependent grav-itational coupling as a function of radial coordinate. Allsolutions, irrespectively of the sign of G (cid:48) ( r ), are found tobe well behaved, realistic solutions,, which tend to G ( r = R ) = 1 at the surface of the star, which was imposed rightfrom the start. The interior solutions obtained here must be able to de-scribe realistic astrophysical configurations. In this sub-section we check if stability criteria as well as the energyconditions are fulfilled or not. First, regarding stability,we impose the condition
Γ > / Γ is defined by Γ ≡ c s (cid:20) ρp r (cid:21) (39) with c s being the sound speed defined by c s ≡ dp r dρ (40)For the linear EoS considered here, the speed of sound isa constant, c s = k .Fig. 5 shows that Γ > / G (cid:48) ( r = 0)(right panel).Next, regarding energy conditions, we require that [40–42, 115, 116] WEC: ρ ≥ , ρ + p r,t ≥ , (41)NEC: ρ + p r,t ≥ , (42)DEC: ρ ≥ | p r,t | , (43)SEC: ρ + p r,t ≥ , ρ + p r + 2 p t ≥ . (44)According to the interior solution shown in Fig. 2, weobserve that i) all three quantities ρ, p r , p t are positivethroughout the star, and ii) the energy density always re-mains larger that both p r , p t . Clearly all energy conditionsare fulfilled. Hence, we conclude that the interior solu-tions found here are well behaved solutions within scale-dependent gravity, capable of describing realistic astro-physical configurations.As a final remark it should be stated here that inthe present article we took a modest step towards the in-vestigation of spherically symmetric, anisotropic strangequark stars in the scale-dependent scenario assuming theMIT bag model equation-of-state. Regarding future work,one may consider i) more sophisticated equations-of-statefor quark matter [117–119], ii) rotating stars, or iii) othertypes of compact objects, such as neutron stars or whitedwarfs. It would be interesting to study those issues inscale-dependent gravity in forthcoming articles. Summarizing our work, in the present article we haveobtained well behaved interior solutions for relativisticstars with anisotropic matter in the scale-dependence sce-nario. In particular, we have investigated the propertiesof strange stars adopting the extreme SQSB40 MIT bagmodel, and assuming for quark matter a linear EoS. Firstwe derived the generalized structure equations describ-ing the hydrostatic equilibrium of the stars for a non-vanishing anisotropic factor. Those new equations gener-alize the usual TOV equations valid in GR, which arerecovered when Newton’s constant is taken to be a con-stant, G (cid:48) ( r ) = 0 = G (cid:48)(cid:48) ( r ). Next, assuming a certain pro-file for the energy density, we numerically integrated thestructure equations for the system m ( r ) , ν ( r ) , G ( r ), andwe computed the radius, the mass as well as the fac-tor of compactness of the stars for a varying Newton’sconstant, either increasing or decreasing, throughout theobjects. Moreover, we have shown that the energy condi-tions are fulfilled, and that the Bondi’s stability condition, Grigoris Panotopoulos et al.: Interior solutions of relativistic stars with anisotropic matter in scale-dependent gravity
Table 1.
Initial conditions for five interior solutions for G (cid:48) ( r = 0) = − . /km . No of solution G c ν c Table 2.
Initial conditions for five interior solutions for G (cid:48) ( r = 0) = 0 . /km . No of solution G c ν c Table 3.
Properties of five interior solutions assuming a positive G (cid:48) ( r = 0). No of solution R [ km ] M [ M (cid:12) ] C = M/R ˜ a ˜ b Γ > /
3, is satisfied as well. In both cases, G (cid:48) ( r = 0) > G (cid:48) ( r = 0) <
0, we obtained well behaved solutionsdescribing realistic astrophysical configurations, althougha decreasing Newton’s constant throughout the objectsleads to slightly more massive and more compact stars.
Acknowlegements
We wish to thank the anonymous reviewer for valuablecomments and suggestions. The authors G. P. and I. L.thank the Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT),Portugal, for the financial support to the Center for As-trophysics and Gravitation-CENTRA, Instituto SuperiorT´ecnico, Universidade de Lisboa, through the Project No. UIDB/00099/2020and No. PTDC/FIS-AST/28920/2017. The author A. R.acknowledges DI-VRIEA for financial support through ProyectoPostdoctorado 2019 VRIEA-PUCV.
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Properties of five interior solutions assuming a negative G (cid:48) ( r = 0). No of solution R [ km ] M [ M (cid:12) ] C = M/R ˜ a ˜ b r / R M e t r i c P o t e n t i a l s r / R M e t r i c P o t e n t i a l s Fig. 1.
Metric potentials e ν and e λ vs dimensionless radial coordinate r/R for the five plus five interior solutions obtainedhere, see Tables I and II for the initial conditions, and Tables III and IV for the properties of the stars. They exhibit theusual behavior of interior solutions of relativistic stars, i.e. they are increasing functions of r , and e ν always remains below e λ . Since m ( r = 0) = 0, the metric potential e λ always starts from unity. LEFT:
Solutions 1-5 corresponding to the case G (cid:48) ( r = 0) = − . /km (Table I). Shown are: i) Solution 1 (solid red line), ii) Solution 2 (short dashed blue line), iii) Solution3 (dotted brown line), iv) Solution 4 (dot-dashed magenta line), v) Solution 5 (long dashed orange line). RIGHT:
Same as leftpanel, but for solutions 1-5 corresponding to the case G (cid:48) ( r = 0) = +0 . /km (Table II).25. A. Aziz, S. Ray, F. Rahaman, M. Khlopov andB. K. Guha, Int. J. Mod. Phys. D (2019) no.13, 1941006[arXiv:1906.00063 [gr-qc]].26. F. Weber, Prog. Part. Nucl. Phys. (2005) 193.27. S. Benic, D. Blaschke, D. E. Alvarez-Castillo, T. Fis-cher and S. Typel, Astron. Astrophys. (2015), A40[arXiv:1411.2856 [astro-ph.HE]].28. Yazdizadeh, T., Bordbar, G. H., & Eslam Panah, B. 2019,arXiv:1902.0488729. Eslam Panah, B., Yazdizadeh, T., & Bordbar, G. H. 2019,European Physical Journal C, 79, 815.30. P. Jaikumar, S. Reddy and A. W. Steiner, Phys. Rev. Lett. (2006), 041101 [arXiv:nucl-th/0507055 [nucl-th]].31. R. Ruderman, Ann. Rev. Astron. Astrophys., (1972)427.32. R. L. Bowers and E. P. T. Liang, Astrophys. J., (1974)657.33. A. I. Sokolov, JETP (1980) 1137.34. R. F. Sawyer, Phys. Rev. Lett., (1972) 382.35. R. Kippenhahn and A. Weigert, Stellar structure and evo-lution , Springer, Berlin, 1990.36. R. Sharma and S. Maharaj, Mon. Not. Roy. Astron. Soc. (2007), 1265-1268 [arXiv:gr-qc/0702046 [gr-qc]].37. L. Gabbanelli, ´A. Rinc´on and C. Rubio, Eur. Phys. J. C (2018) no.5, 370 [arXiv:1802.08000 [gr-qc]].38. I. Lopes, G. Panotopoulos and ´A. Rinc´on, Eur. Phys. J.Plus (2019) no.9, 454 [arXiv:1907.03549 [gr-qc]]. 39. F. Tello-Ortiz, M. Malaver, ´A. Rinc´on and Y. Gomez-Leyton, Eur. Phys. J. C (2020) no.5, 371[arXiv:2005.11038 [gr-qc]].40. M. K. Mak and T. Harko, Chin. J. J. Astron. Astrophys. , 248 (2002).41. D. Deb, S. R. Chowdhury, S. Ray, F. Rahaman andB. K. Guha, Annals Phys. , 239 (2017).42. D. Deb, S. Roy Chowdhury, S. Ray and F. Rahaman, Gen.Rel. Grav. , no. 9, 112 (2018).43. F. de Felice, F., Yu, Y., and Fang, J. 1995, MNRAS, 277,L17.44. F. de Felice, S. m. Liu and Y. q. Yu, Class. Quant. Grav. (1999), 2669-2680 [arXiv:gr-qc/9905099 [gr-qc]].45. Zhang, J. L., Chau, W. Y., and Deng, T. Y. 1982, APSS,88, 81.46. S. Ray, A. L. Espindola, M. Malheiro, J. P. S. Lemosand V. T. Zanchin, Phys. Rev. D (2003), 084004[arXiv:astro-ph/0307262 [astro-ph]].47. Siffert, B. B., de Mello Neto, J. R. T., & Calv˜ao, M. O.2007, Brazilian Journal of Physics, 37, 609.48. J. D. V. Arba˜nil, J. P. S. Lemos and V. T. Zanchin, Phys.Rev. D (2013), 084023 [arXiv:1309.4470 [gr-qc]].49. J. D. V. Arba˜nil, J. P. S. Lemos and V. T. Zanchin, Phys.Rev. D (2014) no.10, 104054 [arXiv:1404.7177 [gr-qc]].50. R. P. Negreiros, F. Weber, M. Malheiro and V. Usov, Phys.Rev. D (2009), 083006 [arXiv:0907.5537 [astro-ph.SR]].51. J. D. V. Arba˜nil and M. Malheiro, Phys. Rev. D (2015),084009 [arXiv:1509.07692 [astro-ph.SR]]. Grigoris Panotopoulos et al.: Interior solutions of relativistic stars with anisotropic matter in scale-dependent gravity r / R X i r / R X i Fig. 2.
Dimensionless energy density and pressures (both radial and tangential) X i ≡ ( ρ, p r , p t ) /ρ s vs dimensionless radialcoordinate r/R for the 5th solution obtained here. LEFT:
Solution for the case G (cid:48) ( r = 0) = − . /km (5th solution inTable I). RIGHT:
Solution for the case G (cid:48) ( r = 0) = +0 . /km (5th solution in Table II ). Shown are: i) dimensionless radialpressure p r /ρ s (solid blue line), ii) dimensionless transverse pressure p t /ρ s (short dashed orange line), iii) dimensionless density ρ/ρ s (long dashed green line). The energy density starts from its central value, ρ c , and it monotonically decreases until it reachesits surface value, ρ s . The pressures start from the same value at the centre of the stars (which implies that the anisotropy factorvanishes there, see next figure), and they monotonically decrease until p r vanishes at the surface, whereas p t does not have tovanish. - r / R Δ ( r ) / ρ s - r / R Δ ( r ) / ρ s Fig. 3.
Normalized anisotropic factor ∆/ρ s vs dimensionless radial coordinate r/R for the five plus five interior solutionsobtained here. It starts from zero at the centre of the stars, but it does not have to vanish at the surface. LEFT:
Solutions 1-5corresponding to the case G (cid:48) ( r = 0) = − . /km (Table I). Shown are: i) Solution 1 (solid red line), ii) Solution 2 (shortdashed blue line), iii) Solution 3 (dotted brown line), iv) Solution 4 (dot-dashed magenta line), v) Solution 5 (long dashedorange line). RIGHT:
Same as left panel, but for solutions 1-5 corresponding to the case G (cid:48) ( r = 0) = +0 . /km (Table II).52. R. Sharma and S. D. Maharaj, Mon. Not. Roy. Astron.Soc. (2007), 1265-1268 [arXiv:gr-qc/0702046 [gr-qc]].53. Mak, M. K. and Harko, T. 2002, ChJAA, 2, 248.54. Deb, D., Chowdhury, S. R., Ray, S., et al. 2017, Annals ofPhysics, 387, 239.55. D. Deb, S. Roy Chowdhury, S. Ray and F. Rahaman, Gen.Rel. Grav. (2018) no.9, 112 [arXiv:1509.00401 [gr-qc]].56. P. Bhar, M. Govender and R. Sharma, Eur. Phys. J. C (2017) no.2, 109 [arXiv:1607.06664 [gr-qc]].57. L. Gabbanelli, ´A. Rinc´on and C. Rubio, Eur. Phys. J. C (2018) no.5, 370 [arXiv:1802.08000 [gr-qc]].58. S. K. Maurya, S. T.T., Y. K. Gupta and F. Rahaman, Eur.Phys. J. A (2016) no.7, 191 [arXiv:1512.01667 [gr-qc]].59. D. Deb, S. R. Chowdhury, S. Ray, F. Rahamanand B. K. Guha, Annals Phys. (2017), 239-252[arXiv:1606.00713 [gr-qc]]. 60. S. R. Chowdhury, D. Deb, S. Ray, F. Rahaman andB. K. Guha, Eur. Phys. J. C (2019) no.7, 547[arXiv:1902.01689 [gr-qc]].61. S. R. Chowdhury, D. Deb, F. Rahaman, S. Ray andB. K. Guha, Int. J. Mod. Phys. D (2020) no.01, 2050001[arXiv:1903.03514 [gr-qc]].62. S. Thirukkanesh and S. D. Maharaj, Class. Quant. Grav. (2008), 235001 [arXiv:0810.3809 [gr-qc]].63. V. Varela, F. Rahaman, S. Ray, K. Chakrabortyand M. Kalam, Phys. Rev. D (2010), 044052[arXiv:1004.2165 [gr-qc]].64. Maurya, S. K., Ayan Banerjee, and Phongpichit Channuie.Chinese Physics C 42.5 (2018): 055101.65. D. Deb, M. Khlopov, F. Rahaman, S. Ray and B. K. Guha,Eur. Phys. J. C (2018) no.6, 465 [arXiv:1802.01332 [gr-qc]].rigoris Panotopoulos et al.: Interior solutions of relativistic stars with anisotropic matter in scale-dependent gravity 9 r / R G ( r ) r / R G ( r ) Fig. 4.
Running (r-varying) Newton’s constant vs dimensionless radial coordinate r/R for the five plus five interior solutionsobtained here.
LEFT:
Solutions 1-5 corresponding to the case G (cid:48) ( r = 0) = − . /km (Table I). Shown are: i) Solution 1(solid red line), ii) Solution 2 (short dashed blue line), iii) Solution 3 (dotted brown line), iv) Solution 4 (dot-dashed magentaline), v) Solution 5 (long dashed orange line). RIGHT:
Same as left panel, but for solutions 1-5 corresponding to the case G (cid:48) ( r = 0) = +0 . /km (Table II). r / R Γ ( r ) r / R Γ ( r ) Fig. 5.
Adiabatic index Γ versus dimensionless radial coordinate r/R for the five plus five interior solutions obtained here. Thehorizontal dashed line corresponds to the Newtonian limit 4 / LEFT:
Solutions 1-5 corresponding to the case G (cid:48) ( r = 0) = − . /km (Table I). Shown are: i) Solution 1 (solid red line), ii) Solution 2 (short dashed blue line), iii) Solution 3 (dottedbrown line), iv) Solution 4 (dot-dashed magenta line), v) Solution 5 (long dashed orange line). RIGHT:
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Factor of compactness, C , versus mass, M (insolar masses). Solid red line corresponds to the case G (cid:48) ( r =0) = − . /km , while dashed cyan line corresponds to thecase G (cid:48) ( r = 0) = +0 . /km . MIDDLE:
Mass-to-radio pro-files for the two cases regarding the sign of G (cid:48) ( r = 0). Thecolor code is the same as in the top panel. BOTTOM:
Sur-face red-shift, z s , (see text) versus radius R (in km). The colorcode is the same as in the top panel.85. E. Contreras, ´A. Rinc´on, B. Koch and P. Bargue˜no,Int. J. Mod. Phys. D , no. 03, 1850032 (2017)[arXiv:1711.08400 [gr-qc]].86. ´A. Rinc´on and G. Panotopoulos, Phys. Rev. D , no. 2,024027 (2018) [arXiv:1801.03248 [hep-th]]. 87. E. Contreras, ´A. Rin´on, B. Koch and P. Bargue˜no, Eur.Phys. J. C , no. 3, 246 (2018) [arXiv:1803.03255 [gr-qc]].88. ´A. Rinc´on and B. Koch, Eur. Phys. J. C , no. 12, 1022(2018) [arXiv:1806.03024 [hep-th]].89. ´A. Rinc´on, E. Contreras, P. Bargue˜no, B. Koch andG. Panotopoulos, Eur. Phys. J. C , no. 8, 641 (2018)[arXiv:1807.08047 [hep-th]].90. E. Contreras, ´A. Rinc´on and J. M. Ram´ırez-Velasquez,Eur. Phys. J. C , no. 1, 53 (2019) [arXiv:1810.07356[gr-qc]].91. ´A. Rinc´on, E. Contreras, P. Bargue˜no and B. Koch, Eur.Phys. J. Plus , no. 11, 557 (2019) [arXiv:1901.03650[gr-qc]].92. ´A. Rinc´on and J. R. Villanueva, arXiv:1902.03704 [gr-qc].93. E. Contreras, ´A. Rinc´on and P. Bargue˜no,arXiv:1902.05941 [gr-qc].94. M. Fathi, ´A. Rinc´on and J. R. Villanueva,arXiv:1903.09037 [gr-qc].95. G. Panotopoulos and ´A. Rinc´on, Eur. Phys. J. Plus ,no. 6, 300 (2019) [arXiv:1904.10847 [gr-qc]].96. E. Contreras, J. M. Ramirez-Velasquez, ´A. Rinc´on, G. Pan-otopoulos and P. Bargue˜no, Eur. Phys. J. C , no. 9, 802(2019) [arXiv:1905.11443 [gr-qc]].97. F. Canales, B. Koch, C. Laporte and ´A. Rinc´on, JCAP , no. 01, 021 (2020) [arXiv:1812.10526 [gr-qc]].98. N. F. Naidu, M. Govender and K. S. Govinder, Int. J.Mod. Phys. D (2006), 1053-1066 [arXiv:gr-qc/0509088[gr-qc]].99. D. Dou and R. Percacci, Class. Quant. Grav. (1998),3449-3468100. J. F. Donoghue, Phys. Rev. D , 3874 (1994)101. B. Koch and I. Ramirez, Class. Quant. Grav. , 055008(2011) [arXiv:1010.2799 [gr-qc]].102. R. C. Tolman, Phys. Rev. , 364 (1939).103. J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. (1939) 374.104. K. Schwarzschild, Sitzungsber. Preuss. Akad. Wiss.Berlin (Math. Phys. ) (1916) 189 [physics/9905030].105. ´A. Rinc´on and B. Koch, J. Phys. Conf. Ser. (2018)no.1, 012015 [arXiv:1705.02729 [hep-th]].106. A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn andV. F. Weisskopf, Phys. Rev. D (1974) 3471.107. A. Chodos, R. L. Jaffe, K. Johnson and C. B. Thorn,Phys. Rev. D (1974) 2599.108. D. Gondek-Rosinska and F. Limousin, [arXiv:0801.4829[gr-qc]].109. O. Zubairi, A. Romero and F. Weber, J. Phys. Conf. Ser. (2015) no.1, 012003.110. L. Gabbanelli, ´A. Rinc´on and C. Rubio, Eur. Phys. J. C (2018) no.5, 370 [arXiv:1802.08000 [gr-qc]].111. G. Panotopoulos and ´A. Rinc´on, Eur. Phys. J. Plus (2019) no.9, 472 [arXiv:1907.03545 [gr-qc]].112. H. Bondi, Mon. Not. R. Astron. Soc. , 39 (1964).113. S. Chandrasekhar, Astrophys. J., , 417 (1964).114. C. C. Moustakidis, Gen. Rel. Grav. (2017) no.5, 68.115. P. Bhar, M. Govender and R. Sharma, Eur. Phys. J. C , no. 2, 109 (2017) [arXiv:1607.06664 [gr-qc]].116. G. Panotopoulos and ´A. Rinc´on, Eur. Phys. J. Plus ,no. 9, 472 (2019) [arXiv:1907.03545 [gr-qc]].117. E. S. Fraga, A. Kurkela and A. Vuorinen, Astrophys. J. (2014) no.2, L25 [arXiv:1311.5154 [nucl-th]].rigoris Panotopoulos et al.: Interior solutions of relativistic stars with anisotropic matter in scale-dependent gravity 11118. V. D. Toneev, E. G. Nikonov, B. Friman, W. Noren-berg and K. Redlich, Eur. Phys. J. C (2003) 399 [hep-ph/0308088].119. Y. B. Ivanov, A. S. Khvorostukhin, E. E. Kolomeitsev,V. V. Skokov, V. D. Toneev and D. N. Voskresensky, Phys.Rev. C72