Stellar models in Brane Worlds Francisco X. Linares,1,∗ Miguel A. Garc´ıa-Aspeitia,2,3,† and L. Arturo Uren˜a-L´opez4,5,‡ 1Departamento de F´ısica, Universidad Simo´n Bol´ıvar Apartado 89000, Caracas 1080A, Venezuela. 2Consejo Nacional de Ciencia y Tecnolog´ıa, Av, Insurgentes Sur 1582. Colonia Cr´edito Constructor, Del. Benito Jua´rez C.P. 03940, M´exico D.F. M´exico. 3Unidad Acad´emica de F´ısica, Universidad Auto´noma de Zacatecas, Calzada Solidaridad esquina con Paseo a la Bufa S/N C.P. 98060, Zacatecas, M´exico. 4Departamento de F´ısica, DCI, Campus Leo´n, Universidad de Guanajuato, C.P. 37150, Leo´n, Guanajuato, M´exico. 5Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, United Kingdom. (Dated: January 21, 2015) Weconsiderhereafullstudyofstellardynamicsfromthebrane-worldpointofviewinthecaseof constantdensityandofapolytropicfluid. Westartourstudycataloguingtheminimalrequirements 5 toobtainacompactobjectwithaSchwarszchildexterior,highlightingthelowandhighenergylimit, 1 the boundary conditions, and the appropriate behavior of Weyl contributions inside and outside of 0 the star. Under the previous requirements we show an extensive study of stellar behavior, starting 2 with stars of constant density and its extended cases with the presence of nonlocal contributions. n Finally, we focus our attention to more realistic stars with a polytropic equation of state, specially a in the case of white dwarfs, and study their static configurations numerically. One of the main J results is that the inclusion of the Weyl functions from braneworld models allow the existence of 0 more compact configurations than within General Relativity. 2 PACSnumbers: 04.50.-h,04.40.Dg ] c q I. INTRODUCTION could live in the bulk. This framework has been used for - r starswithaconstantdensityin[9],andalsoforpolytropic g Stellar astrophysics is one of the most characteristic matter with a given relationship between the quantities [ topics studied by General Relativity (GR), which has arising from the non-local Weyl terms in[10]. It has also 1 helped to describe the dynamic and evolution of stars been shown that the exterior solutions of these brane- v with unprecedented success[1]. In addition, the matter stars is not the Schwarzschild one[9, 11], and then the 9 inside a star may be in some cases in extreme conditions Weyl fluids in the exterior of the stars can have a non- 6 generating complicated high energy phenomena, princi- negligibleinfluenceintheinternalpressureandcompact- 8 4 pallyinwhitedwarfs,neutronstars,andothers,andthen ness of stellar objects. More recently, the conditions for 0 a complete description of the stellar properties requires stellar stability in brane-stars were revisited in[12] for a . the introduction of a particular equation of state (EoS) set of hypotheses called the minimal setup, which are 1 0 like in the case of polytropes[2], or even Bose-Einstein consistentwithaSchwarzschildexterior. Alsosee[13]for 5 Condensates (BEC)[3]. a study on the gravitational collapse of brane stars. 1 Another interesting possibility in recent times is to Withthepreviousbackground,thispaperisdedicated : consider alternative theories of gravity and to look for to the study of the stellar equations of motion that arise v i theirparticularsignaturesinstellarmodels, speciallyfor from the formalism of Brane-World theory, and the role X some of the extreme situations mentioned above. For of the Weyl functions in the regular behavior of a stellar r instance, the authors in[4] considered the corrections in- distribution. It is important to remark that our main a ducedbyaGalileonLagrangianinstarsofconstantden- objective is to consider models of stars as realistic as sity. Anotherexampleisgivenbytheso-calledmodelsof possible, and for this reason we will follow conventional Brane-Worlds (see[5, 6] for a good review) whose main wisdominthisregard: aSchwarzschildexterior,andreg- characteristicistheexistenceofbranes(fourdimensional ularityofallfunctionsinvolved. Basedonthesepremises, manifolds) embedded in a five dimensional bulk[7]. This we perform numerical studies of the so-called extended particular geometry allows a natural extension of Ein- GM solution with constant density, and of a polytropic stein’s equations[8], and introduces new degrees of free- fluid. dom through quadratic terms of the energy momentum The organization of the paper is as follows. In tensor, the non-local Weyl terms, and other fields that Sec. II, we describe the equations of stellar dynamics withbranes,emphasizingthehighandlowenergylimits, boundary conditions, and the role played by the Weyl functions in providing consistent and regular solutions. ∗ [email protected] Subsequently in Sec. III, we study the case of constant † aspeitia@fisica.uaz.edu.mx density and the extended GM solution. Also, in Sec. IV ‡ lurena@fisica.ugto.mx we study polytropic brane-stars. Finally, in Sec. V we 2 give some conclusions and remarks. where a prime indicates derivative with respect to r. We havealsodefinedV =6U/κ4 ,N =4P/κ4 ,andA(r)= (4) (4) [1−2G M(r)/r]−1,whereasp andρ areexplicitly II. STELLAR DYNAMICS WITH BRANES N eff eff given by: Let us start by writing the equations of motion for (cid:16) ρ(cid:17) ρ2 V N p =p 1+ + + + , (6a) an embedded brane in a five dimensional bulk using the eff λ 2λ 3λ λ Randall-Sundrum II model[7]. We first assume that the (cid:16) ρ (cid:17) V ρ =ρ 1+ + . (6b) Einsteinequationsarethegravitationalequationsofmo- eff 2λ λ tion of the 5-dimensional Universe, Now, we are in position of analyze the following impor- G +Λ g =κ2 T . (1) tant points. AB (5) AB (5) AB Following an appropriate computation, the modified 4dim Einstein’s equation can be written as[6, 8] A. Numerical analysis G +ξ +Λ g =κ2 T +κ4 Π +κ2 F , (2) µν µν (4) µν (4) µν (5) µν (5) µν In order to have a numerical solution of the equations where κ(4) and κ(5) are respectively the four and five- of motion, we choose the following dimensionless vari- dimensional coupling constants, which are related one to ables: each other in the form: κ2 = 8πG = κ4 λ/6, λ is (4) N (5) (cid:112) defined as the brane tension, and GN is Newton’s con- x= GNM/R(r/R), ρ¯=ρ/(cid:104)ρeff(cid:105), p¯=p/(cid:104)ρeff(cid:105)(,7a) stant. For purposes of simplicity, we will not consider λ¯ =λ/(cid:104)ρ (cid:105), V¯ =V/(cid:104)ρ (cid:105)2, N¯ =N/(cid:104)ρ (cid:105)2, (7b) eff eff eff bulk matter, which translates into F = 0, and dis- µν card the presence of the four-dimensional cosmological for which Eqs. (5) read constant, Λ = 0, as we do not expect it to have any (4) M¯(cid:48) =x2ρ¯ , (8a) important effect at astrophysical scales (for a recent dis- eff cussion about it see[14]). Additionally, we will neglect 3 (cid:18)x3p¯ +M¯ (cid:19) p¯(cid:48) =− eff (p¯+ρ¯), (8b) any nonlocal energy flux, which is allowed by the static x2 1−6M¯/x spherically symmetric solutions we will study below[6]. 6 (cid:18)x3p¯ +M¯ (cid:19) In the case of a perfect fluid, the energy-momentum V¯(cid:48)+3N¯(cid:48) =− eff (2V¯+3N¯) x2 1−6M¯/x T tensor,thequadraticenergy-momentumtensorΠ , µν µν and the Weyl ξµν can be written as: −9N¯ −3(ρ¯+p¯)ρ¯(cid:48), (8c) x T =ρu u +ph , (3a) µν µ ν µν 1 where now a prime denotes derivatives with respect to Πµν = 12ρ[ρuµuν +(ρ+2p)hµν], (3b) x, the mean effective density is just given by (cid:104)ρeff(cid:105) = (cid:20) (cid:21) 3M/4πR3, and also 6 U −P ξ =− Uu u +Pr r + h ,(3c) µν κ2 λ µ ν µ ν 3 µν (cid:16) ρ¯ (cid:17) V¯ (4) ρ¯ =ρ¯ 1+ + , (9a) eff 2λ¯ λ¯ where p = p(r) and ρ = ρ(r) are respectively, the pres- (cid:16) ρ¯(cid:17) ρ¯2 V¯ N¯ sureanddensityofthestellarmatterofinterest,U isthe p¯ =p¯ 1+ + + + . (9b) nonlocal energy density, P is the nonlocal anisotropic eff λ¯ 2λ¯ 3λ¯ λ¯ stress scalar, u is the fluid four-velocity, that also satis- α Note that the ratio ρ/λ is invariant under the change fies the condition gµνuµuν = −1, and hµν = gµν +uµuν of variables, ρ¯/λ¯ = ρ/λ, and then we will omit the bar is orthogonal to u . Under the assumption of spherical µ whenever the ratio appears in the equations of motion. symmetry, the metric can be written as: ds2 =−B(r)dt2+A(r)dr2+r2(dθ2+sin2θdϕ2). (4) B. High and low energy limits The equations of motion for stellar structure then are[9, 12]: There are two very clear limiting expressions of the M(cid:48) =4πr2ρ , (5a) equations of motion in terms of normalized brane ratio eff λ¯, as the latter represents the energy ratio of the brane 1B(cid:48) p(cid:48) =− (p+ρ), (5b) tension with respect to the mean energy density of the 2 B compact star of interest, see Eq. (7b). It is usually as- B(cid:48) 9 V(cid:48)+3N(cid:48) =− (2V +3N)− N −3(ρ+p)ρ(cid:48),(5c) sumed that the brane corrections are measured in terms B r of the absolute value of the brane tension λ, but in our B(cid:48) 2G (cid:18)4πr3p +M(cid:19) studywefindthebraneratioλ¯ =λ/(cid:104)ρ (cid:105)tohaveamore = N eff , (5d) eff B r2 1−2G M/r meaningful character for compact objects in general. N 3 Under this line of reasoning, we first present the low C. Boundary conditions energy limit of the equations of motion, represented by the operation λ¯ →∞, under which Eqs. (8) become the The change of variables (7) is very appropriate to ex- usual Tolman-Oppenheimer-Volkoff (TOV) equations of plorethesolutionsoftheTOVequations(8),asallphys- GR[1, 2]: ical quantities involved are normalized in terms of two important observables in stellar astrophysics, which are M¯(cid:48) =x2ρ¯, (10a) the mass M and the radius R of the star. Furthermore, 3 (cid:18) x3p¯+M¯ (cid:19) these two parameters appear in the single combination p¯(cid:48) =− (p¯+ρ¯), (10b) x2 1−6M¯/x GNM/Rthatrepresentsthecompactnessofthestar. For instance, the interior range of the new radial variable is (cid:112) where the effective pressure and density are directly rep- x=[0, GNM/R], which means that the surface of the (cid:112) resented by their normalized physical values. starislocatedatx(R)≡X = G M/R. Also,thenew N We have called it the low energy limit because we mass function changes to: are assuming that the mean density of the star is much lower than the brane tension λ, so that any brane cor- 1(cid:18)G M(cid:19)3/2 M(r) rectionsintheequationsofmotionarehighlysuppressed M¯(x)= N , (13) 3 R M by the brane energy scale. We cannot say here whether the brane tension is at a very energy scale, or it is just that the star density is not high enough. In strict sense, and then the total mass is M¯(X) = (1/3)(GNM/R)3/2. Eq. (8c) can still be considered for the integration of the In other words, the compactness of the star will deter- Weyl functions, but their values will not make any dif- mine the mass and size of the numerical solutions. ference in the final integration of the physical variables. The equations of motion will be integrated from the There is also the high energy limit of the equations center up to the surface of the star defined by the con- of motion represented by λ¯ → 0, for which the effective dition p(X) = 0; the latter only refers to the physical density and pressure read pressure, and we will take it as a reasonable physical as- sumptioneventhoughitisnotnecessarilyrequiredinthe ρ¯2 V¯ case of brane stars. Finally, at the center of the star we ρ¯eff (cid:39) 2λ¯ + λ¯ , (11a) will also assume that M¯ → 0 as x → 0, so that there is ρ V¯ N¯ not a discontinuity of the different quantities in the cen- p¯ (cid:39) (2p¯+ρ¯)+ + . (11b) ter of the star, and the central value of the pressure (or eff 2λ 3λ¯ λ¯ anyotherrelatedquantity)willbesetasafreeparameter that will characterize the numerical solutions. We can see that there is an overall factor of λ¯ in the Eventhoughwewillnotconsiderexteriorsolutions,we above expressions (11), which will also appear as such in mustanywaytakeintoaccounttheinformationprovided Eqs. (8) in the high energy limit. The brane ratio can bytheIsrael-Darmois(ID)matchingcondition,whichfor then be absorbed in the equations of motion by means the case under study can be written as[9]: of the following change of variables: M¯ → M¯λ¯1/2, and x → xλ¯1/2, and then we finally find the equations of (3/2)ρ¯2(X)+V¯−(X)+3N¯−(X)=V¯+(X)+3N¯+(X), motion for the high energy limit: (14) M¯(cid:48) = x2 (cid:0)ρ¯2+2V¯(cid:1) , (12a) wriohre)revatlhueessuopfetrhsceridpitffe−re(n+t)qdueannottiteisesthaet itnhteersiuorrfa(ecxetoe-f 2 the star, and we also assumed that ρ¯(x>X)=0. 3 (cid:20)x3(p¯ρ¯+ρ¯2/2+V¯/3+N¯)+M¯ (cid:21) p¯(cid:48) =− × A desirable property we want in our solutions is a x2 1−6M¯/x Schwarzschild exterior,whichcanbeeasilyaccomplished (p¯+ρ¯), (12b) undertheboundaryconditionsV¯+(X)=0=N¯+(X),as 6 (cid:18)x3(p¯ρ¯+ρ¯2/2+V¯/3+N¯)+M¯ (cid:19) for them the simplest solution that arises from Eq. (8c) V¯(cid:48)+3N¯(cid:48) =−x2 1−6M¯/x × is the trivial one: V¯(x≥X)=0=N¯(x≥X). Thus, for the purposes of this paper, we will refer hereafter to the 9 (2V¯+3N¯)− N¯ −3(ρ¯+p¯)ρ¯(cid:48). (12c) restricted ID matching condition given by: x (3/2)ρ¯2(X)+V¯−(X)+3N¯−(X)=0. (15) In contrast to the TOV equations of GR in (10), we shallcallthiscasethehighenergylimitbecausethemean density is much larger than the brane tension, even if For completeness, we just note that the exterior solu- we cannot say whether this is because the brane tension tionsofthemetricfunctionsaregivenbythewellknown attains a very small energy value, or it is just that the expressions B(r)=A−1(r)=1−2G M/r. In addition, N starhassuchalargedensitythatthelattersurpassesthe it can be shown from Eq. (5d) that the interior solution energy scale of the brane tension. of the lapse function, in terms of the normalized vari- 4 ables (7), is given by: There is though a non-divergent interior solution of N if we drop the condition of constant density, which is: 1−2BG(Nx)M/R =exp(cid:34)−(cid:90)xXdxx62 (cid:18)x13p¯−ef6fM+¯/Mx¯ (cid:19)(cid:35) , N(x<X)= B(x1)x3 (cid:90)0xBx3(ρ¯+p¯)ρ¯(cid:48)dx. (20) (16) ButtheIDmatchingcondition(15)nowindicatesthatat andthenwewillnotsolveitexplicitlyinanyofthecases the surface of the star we must have ρ¯(X)(cid:54)=0, or either presented below. give up the Schwarzschild exterior. In consequence, the Thenumericalrecipedescribedabovewillbeappliedto only interior solution of the nonlocal anisotropic stress differentcasesandconfigurationsinSecs.IIIandIV.The under the conditions of a Schwarzschild exterior, and results that will be obtained will have a universal char- non-constant density with ρ¯(X) = 0, which are the con- acter, astheywillnotdependupontheparticularvalues ditions we expect to have in realistic stars, is the trivial of the mass and radius of a given star, but they will rep- one: N(x)≡0 (see also [12]). resentgeneralclassesofstarsaccordingtotheircommon Thereareotherpossibilitiesthathavebeenexploredin compactness G M/R. This will allow us to reach wide N the specialized literature, like for instance a relationship general conclusions about the physical properties of the between the Weyl functions in the form N =σV, where different configurations by means of numerical methods. σ isaconstantparameter[10]. Clearly,thesolutions(17) and(18)arespecialcasesforwhichσ =0. Inthegeneral case, Eq. (8c) can be written as: D. Weyl functions B(cid:48) 9 (1+3σ)V¯(cid:48) =− (2+3σ)V¯− σV¯−3(ρ¯+p¯)ρ¯(cid:48), (21) B x It must be noticed that the interior solutions cannot as long as σ (cid:54)= −1/3. If the density is constant, then evadethepresenceoftheWeyltermseveniftheexterior there is a solution which is similar to Eq. (19): solution is Schwarzschild. For example, let us put by handthatN¯(x)≡0. Ifthedensityisconstantρ¯(X)(cid:54)=0, (cid:104) (cid:105)1/(1+3σ) the ID matching condition (15) implies that V¯−(X) = V¯ =C (p+ρ)2(2+3σ)x−9σ , (22) −(3/2)ρ¯2(X), and then the full solution must be[9]: where C is an integration constant that could be deter- mined with the help of the ID matching condition (15). V(x<X)=−(3/2)ρ2(1+p/ρ)4. (17) However, this solution is not appropriate for the interior of the star because, as it happened too for Eq. (19), it The full consequences of this nonlocal energy density V diverges in the center of the star (x=0). are explored in Sec. IIIB below. We can also consider the case of non-constant density, EventheconditionofaSchwarzschildexteriortogether in which case we find a similar solution to Eq. (20): with ρ¯(X)=0 do not directly imply that the Weyl func- (cid:104) (cid:105)1/(1+3σ) tions must vanish in the interior, as it can be shown[12] N(x<X)= B(x)−2(2+3σ)x−9σ × that in such a case Eq. (8c) must have the following so- lution: (cid:90) x(cid:104)B2(2+3σ)x9σ(cid:105)1/(1+3σ)(ρ¯+p¯)ρ¯(cid:48)dx(2.3) 3 (cid:90) X 0 V(x<X)= B2(ρ¯+p¯)ρ¯(cid:48)dx, (18) Even though this solution is well behaved in the interior B2(x) x of the star, it needs a non-trivial boundary condition at thesurfaceasdictatedbytheIDmatchingcondition(15), which accomplishes the boundary condition V−(X) = 0 and for that we require either to have ρ¯(X) (cid:54)= 0, or to and is also regular at x = 0. This is particularly impor- give up the Schwarzschild exterior. tant for all cases in which the density is not constant, as We see that the imposition of a Schwarzschild exterior we shall see below for the polytropes in Sec. IV. has strong consequences for the interior solutions of the In the opposite case when V¯(x) ≡ 0 and the density Weylfunctions,mostlybecauseitisdifficultingeneralto is constant, the ID matching condition (15) implies that find for them a well behaved interior solution. The most at the surface of the star: N¯−(X) = −(1/2)ρ¯2(X), and problematiccaseisthatoftheanisotropicstressfunction then Eq. (8c) integrates into N, andforthisreasonwewillnottakeitintoaccountas part of the brane gravitational corrections, but assume 1(cid:18)X(cid:19)3 thatthelatterareonlygivenbythequadraticcorrections N(x<X)=− (p+ρ)2. (19) 2 x of the density ρ¯2 and the nonlocal energy density V. Needless to say, this solution diverges at the center of the star and cannot be considered as a useful interior III. THE CASE OF CONSTANT DENSITY solution. That is, in the case of constant density there is not a regular interior solution with the only presence of One of the simplest possibilities of star modes is that the nonlocal anisotropic stress N. ofconstantdensityρ,whichcanbesolvedunderdifferent 5 gravitational schemes. In this section we will work out the effective density and pressure: suchacasewithinthebraneworldschemeandexplainthe (cid:16) ρ (cid:17) 3 ρ additionalphysicalandboundaryconditionsthatmaybe ρ¯ =ρ¯ 1+ − ρ¯ (1+p¯/ρ¯)4, (25a) eff 2λ 2 λ needed in order to reach well posed numerical solutions. (cid:16) ρ(cid:17) ρ¯ρ ρ¯ρ p¯ =p¯ 1+ + − (1+p¯/ρ¯)4. (25b) eff λ 2λ 2λ Here we have taken into account that the nonlocal en- A. The case of the Germani-Maartens solution of ergy density is given by Eq. (17). As it can be seen in brane stars Eqs. (25), there are negative contributions in both the effective density and pressure originated from the pres- To start with we consider here the GM interior solu- ence of the Weyl nonlocal energy, and the solutions now tion, which was thoroughly studied in[9], and that does depend upon three separate parameters: the constant not take into account corrections induced by the Weyl density ρ, the brane ratio ρ/λ, and the compactness of terms: V¯ = 0 = N¯. The modified TOV equations the star G M/R. N are given again by Eqs. (5a) and (5b) with the follow- As we expect to have p(x)>0, then the effective den- ing identifications: ρeff = ρ(1 + ρ/2λ), and peff = sity must be an increasing function, ρeff(x) ≤ ρeff(X), p(1+ρ/λ)+ρ2/2λ. which may even attain negative values at the interior Because the density is constant, we find, in terms of points where the pressure is largest. Moreover, we also the variables in (7), that (cid:104)ρ (cid:105) = ρ(1+ρ/2λ) and then infer from this information that (cid:104)ρ (cid:105) < ρ (X) = eff eff eff ρ¯ = 1. Likewise, we find that ρ¯= (1+ρ/2λ)−1, and ρ(1 − ρ/2λ), and then we expect that in general ρ¯ > eff then its value is directly determined by the ratio ρ/λ. (1−ρ/2λ)−1. IncontrasttotheGMcaseabove,ρ¯cannot Notice that ρ¯≤1, and that we recover ρ¯=1 in the GR begivenafixedvaluebeforehandandbecomesavariable limit ρ/λ → 0. The boundary conditions depend upon that must be adjusted appropriately so that the numer- the compactness of the star G M/R, as in the case of ical solutions accomplish all boundary conditions. This N GR, but also upon the ratio ρ/λ, as expected in brane time, however, the GR limit ρ¯ = 1 is a lower bound as models. Theexactsolutionofthepressurefunctionis[9]: ρ/λ→0,whichisanearlyindicationthattheknownGR upper bound on the star compactness could in principle p¯ (cid:112)1−6M¯/X−(cid:112)1−6M¯x2/X3 be surpassed by the new solutions. ρ¯ = (cid:112)1−6M¯x2/X3−3ζ−1(cid:112)1−6M¯/X, (24) The equations of motion are more easily solved if we take the following change of variables: where ζ ≡ (1+2ρ/λ)/(1+ρ/λ). The brane ratio ρ/λ x→xρ¯−1/2, M¯ →M¯ρ¯−1/2, w ≡p/ρ, (26) lowers the maximum value of the compactness of the star, and the numerical solutions satisfies the analytic where w is the EoS, and then the Eqs. (8) become: bound found from the exact GM solution: G M/R = N (cid:20) ρ 3ρ (cid:21) (1/2)(1−ζ2/9). The GR limit is obtained when ρ/λ → M¯(cid:48) =x2 1+ − (1+w)4 , (27a) 0: G M/R ≤ 4/9, whereas in the opposite direction 2λ 2λ N ρ/λ→∞ we obtain: G M/R≤5/18. 3 (cid:18)x3w +M¯ (cid:19) N w(cid:48) =− eff (1+w), (27b) According to the discussion in Sec. IID, the GM solu- x2 1−6M¯/x tfoiornthcaantnroeatsboenmitaitschuesduatlolyaaSscsuhwmaerdzstchhaitldoetxhteerrieoxrt,earinodr weff = p¯eρ¯ff =w(cid:16)1+ λρ(cid:17)+ 2ρλ(cid:2)1−(1+w)4(cid:3) . solutionswiththepresenceoftheWeylfunctionmustbe the correct ones for brane stars. However, it has been The only free parameter that appears explicitly in recently shown[12], under very general conditions, that Eqs. (27) is the brane ratio ρ/λ. Moreover, the outer the GM solution plays also the role of being the limiting boundary conditions, see for instance Eq. (13), must case of realistic stars when brane corrections are consid- be adjusted to the values X = (GNM/R)1/2ρ¯1/2 and ered, and then gives an upper bound in the compactness M¯(X)=(GNM/R)3/2ρ¯1/2/3. of stars with both brane corrections and a Schwarzschild Examples of the numerical solutions allowed by exterior. Eqs. (27) are shown in Fig. 1 for the brane ratios ρ/λ= 10−1,10−6, where it is confirmed that there are numeri- calsolutionswellbeyondtheGRlimitofG M/R≤4/9. N We only considered cases in which the star has an over- B. The extended GM solution all positive mass, for which it must also have a positive density at its surface. The latter can be translated into We will now review the interior brane solution with the condition ρ¯ (X) > 0, and then from Eq. (25a) we eff constant density, a Schwarzschild exterior, and a non- find the constraint ρ/λ<1. null Weyl term V. This cases was also briefly considered There are two main reasons for the surpass of the GR in[9], but lacks an analytical solution. The equations of limit. The first one is that the extra free parameter ρ¯is motion are again (8) with the following expressions for onlyboundedfrombelow,andthenitisatourdisposalto 6 findnumericalsolutionsthatcansurpasstheGRlimitfor n as γ ≡ (n+1)/n. For example, white dwarfs can be any given value of the brane ratio ρ/λ. Correspondingly, modeledbythepolytropicindexn=3,andneutronstars the second reason is that the effective density ρ¯ is bypolytropeswithanindexintherangen=0.5−1[15]. eff an increasing function that can become as negative as The equations of motion (8) can be simplified if we necessary in the interior of the star. Actually, as far as follow the usual recipe for polytropes and make the fol- the numerical experiments are concerned, the only true lowing change of variable for the density: ρ¯=θn, where limit that could be found for the numerical solutions is n is the polytropic index defined above. For the reasons the Schwarzschild one G M/R<1/2. explained in Sec. IID, we set N¯ = 0. Eqs. (8) are then N written in the form: 0.15 (cid:20) (cid:18) θn(cid:19) V¯(cid:21) 0.1 ρ/λ=10-1 M¯(cid:48) =x2 θn 1+ 2λ¯ + λ¯ , (28a) 0.05 3 (cid:18)x3p¯ +M¯ (cid:19)(cid:0)1+K¯θ(cid:1) θ(cid:48) =− eff , (28b) 0 x2 1−6M¯/x K¯(n+1) -0.05 12(cid:18)x3p¯ +M¯ (cid:19) V¯(cid:48) =− eff V¯ M -0.1 x2 1−6M¯/x -0.15 9 (cid:18)x3p¯ +M¯ (cid:19)n(1+K¯θ)2 + eff θ2n−1,(28c) -0.2 G M/R =0.30 x2 1−6M¯/x K¯(n+1) N G M/R =0.35 -0.25 GNN M/R =0.40 and the effective pressure (9b) now reads G M/R =0.45 -0.3 GN M/R =0.48 -0.35 GNN M/R =0.49 p¯ =K¯θn+1(cid:18)1+ θn(cid:19)+ θ2n + V¯ . (29) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 eff λ¯ 2λ¯ 3λ¯ x It must be stressed out that in our case the density pa- 0.15 0.1 ρ/λ=10-6 rameter θ gives an indication of the values of the density ρ with respect to the mean value of the effective density 0.05 (cid:104)ρ (cid:105), given by the dimensionless density ρ¯, in contrast eff 0 tothestandardcaseinwhichthevalueofreferenceisthe density at the center of the star ρ(0). -0.05 It can also be shown that, again like in the standard M -0.1 case of polytropes, the polytropic coefficient K¯ is a re- -0.15 dundant constant and can be hidden in the equations of -0.2 G M/R =0.30 motion. If we further consider the following change of N G M/R =0.35 variables: -0.25 GN M/R =0.40 N -0.3 GGN MM//RR ==00..4458 x→xK¯n/2, θ →θK¯−1, M¯ →M¯K¯n/2,(30a) N G M/R =0.49 -0.35 N p¯ →p¯ K−n, λ¯ →λ¯K¯−n, V¯ →V¯K¯−2n,(30b) eff eff 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x then Eqs. (28) simply read FIG.1. TheprofileoftheintegratedmassM(x)correspond- M¯(cid:48) =x2(cid:20)θn(cid:18)1+ θn(cid:19)+ V¯(cid:21) , (31a) ingtotheextendedGMsolutionwithratiosρ/λ=10−1(Top) 2λ¯ λ¯ andρ/λ=10−6(Bottom). WecanseethattheextendedGM 3 (cid:18)x3p¯ +M¯ (cid:19) (1+θ) solution allows the existence of stars with a compactness be- θ(cid:48) =− eff , (31b) yond the GR limit but below the extreme Schwarzshild limit x2 1−6M¯/x (n+1) GmNasMs f/uRnc<tio1n/2c.anNoatciqcueirtehantegoanteivreeavsaolnuefsorinthtahteiisnttehraitorthoef V¯(cid:48) =−x122 (cid:18)x13p¯−ef6fM+¯/Mx¯ (cid:19)V¯ the star for the most compact cases. See the text for more details. + 9 (cid:18)x3p¯eff +M¯ (cid:19)n(1+θ)2θ2n−1, (31c) x2 1−6M¯/x (n+1) where IV. POLYTROPIC BRANE STARS (cid:18) θn(cid:19) θ2n V¯ p¯ =θn+1 1+ + + . (32) eff λ¯ 2λ¯ 3λ¯ In this section we study brane stars with a polytropic fluid and an EoS in the form p(r)=Kργ(r). Here, K is Our main interest are the numerical solutions of stars thepolytropicconstant,andγisthepolytropicexponent, with a finite size, as determined by the boundary con- which can be written in terms of the polytropic index dition p(X) = 0, which in the case of the polytropes 7 translates into ρ(X) = 0, and from this into θ(X) = 0. 0.2 G M/R=0.170 Inordertoavoidanysingularitiesintheequationsofmo- λ=102 GN M/R=0.180 N tionatthesurfaceofthestar,inparticularforEq.(31c), GN M/R=0.190 G M/R=0.200 we must constraint the values of the polytropic index 0.15 GN M/R=0.210 N G M/R=0.220 in the range n ≥ 1/2. Needless to say, such a con- N G M/R=0.230 N straintdoesnotexisteitherinthecaseofnon-relativistic 0.1 (Newtonian) or relativistic (GR) polytropes. It must M be noticed as well that the boundary conditions should also be adjusted so that X = (G M/R)1/2K¯−n/2 and 0.05 N M¯(X)=(1/3)(G M/R)3/2K¯−n/2. N We now include brane corrections with the contribu- 0 tion of one of the Weyl terms, with the boundary condi- tion V¯(X) = 0, so that the ID matching condition (14) allowsaSchwarzschildexteriorforthepolytropeanddic- -0.05 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 tates that the interior solution for the nonlocal energy x density is given by Eq. (18). As discussed in Sec. IIB, the brane terms must contribute to the effective density FIG. 2. Numerical solutions of the interior profile of the and pressure inside the star, which means that we can- massM ofpolytropicbranestars,seeEqs.(31),withλ=102. not have in this case a counterpart of the GM solution, We can observe for G M/R≥0.180 that the mass becomes N unless the Schwarzschild condition were waived. negative, like in the constant density case shown in Fig. 1. As in the cases studied in Sec. III, we will integrate This is due to the contribution of the Weyl function V¯, see inwards the equations of motion under the same bound- text for more details. ary conditions presented in Sec. IIC, with the central value of θ(0) being a free parameter that will help us to classify the numerical solutions. The most compact star the effective density at the center becomes negative for will be given by the maximum in the plot of the com- GNM/R ≥ 0.175. It is clear that the geometric term V¯ pactness as a function of the central density: G M/R contributes notoriously for large values of the compact- N vs θ(0). The value of the compactness will be read off ness in the high energy limit. from the outermost points of the numerical solution as Thus, at least for the range of compactness we nu- G M/R = 3M¯(X)/X, whereas the polytropic coeffi- merically explored, the contribution of the Weyl tensor N cient can be calculated from: K¯ =[3M¯(X)]1/3/X. affects the internal configurations of the stars in such This time we have to give explicit values to the brane a way that there is no maximum for the compactness tension λ¯ and the polytropic index n. For the latter, we that can be reached, except for the Schwarzschild bound considerinthefollowingsectionsthecaseofwhitedwarfs GNM/R < 0.5. This can be seen in Fig. 4, where we with n = 3, whereas the brane tension will remain free note that for λ = 106,105 the curves reach a maximum to label the different brane star solutions. value just as in the case of polytropic stars in GR. How- ever, as λ decreases it is possible to find stellar configu- rations with a larger compactness beyond the standard GR bound. A. Numerical solutions Solutions for the high energy limit with λ = 102 al- lowed by Eqs. (31) are shown in Figs. 2 and 3. In par- V. CONCLUSIONS AND REMARKS ticular, the interior mass profile M(x) is shown in Fig. 2 for a range of compactness: G M/R=0.170−0.230, in In this paper we studied the equilibrium configura- N whichthemainfeaturewecanobserveisachangeinsign tionsofstarswithgravitationalcorrectionsinbraneworld close to the center of the star for G M/R≥0.180. This models, and provided numerical solutions when neces- N behavior is due to the contribution of the Weyl function sary. For that we considered the high and low energy V¯ in Eq. (31a). Note that the same behavior occurs in limits of the equations of motion to show the threshold the constant density case in Section IIIB, so this type between GR and braneworlds, and explored the appro- of effect from the Weyl function is present also in more priateboundaryconditionstoobtaingeneralconclusions realistic stars. about the physical properties of the different stellar con- Ontheotherhand,numericalsolutionswithλ=102in figurations. Fig. 3 show that for low compactness the effective den- OuranalysistookintoaccountthecorrespondingWeyl sity has the expected decreasing behavior as we move functions which provide non-local terms in the pressure outwards from the center of the star. However, as the and density, and which can have noticeable effects in di- compactness increases the maximum value of the den- versefeaturesofastar. Thisstudyallowsustorelinquish sity is displaced from the center, and then the density the nonlocal anisotropic stress under the conditions of a profile is not just a decreasing function. Also note that Schwarzschild exterior and non-constant density, which 8 0.00012 0.3 λ=102 GGNN MM//RR==00..004450 λλ==110056 0.0001 GGNN MM//RR==00..005505 0.25 λλ==110043 GN M/R=0.060 λ=102 G M/R=0.070 N 8e-05 0.2 R ρeff 6e-05 M/N 0.15 G 4e-05 0.1 2e-05 0.05 0 0 0 50 100 150 200 250 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 x θ(0) 0.006 G M/R=0.165 0.005 λ=102 GGNN MM//RR==00..117705 FceInGt.ra4l.valuTehθe(0c)ofmorpathctenpesoslyGtrNopMic/Rconafisguarafutinocntsioonbtoafintehde N GN M/R=0.180 from Eqs. (31). It can be seen that the compactness of the G M/R=0.185 0.004 GN M/R=0.190 polytropic star with n = 3 is not bounded as λ → 0. For N λ = 106,105 the curves coincides with that of polytropes in 0.003 GR. eff 0.002 ρ 0.001 ered the classical compactness reported in the literature for the case of GR. All these results are in agreement 0 withthestudyin[12],whereitwasshownthatoneofthe -0.001 mainassumptionsfortheexistenceofanupperboundin the compactness of a star was that the effective energy -0.002 density in the interior should be a decreasing function. 0 5 10 15 20 25 30 x The results presented were based on a clear method- ology that make the equations of motion of braneworlds FIG.3. (Top)Theeffectiveenergydensityρ¯ asafunction more tractable in numerical terms that in other analy- eff of the radial coordinate x, which results from the numerical ses in the literature. As noted, the presence of brane solution of Eqs. (31) for low compactness and λ = 102. It correctionsmodifiesinanotoriouswaythecompactness, canbeseenthat,asthecompactnessincreases,themaximum mass,andotherphysicalcharacteristicsinstellardynam- of the effective density is displaced from the center of the ics, even under the assumption of a Schwarzschild exte- star. (Bottom) The interior profiles of the effective density rior. Extended studies along the lines suggested in this ρ¯ for high compactness. The effect of the Weyl term V¯ is eff paper can be used to constrain the value of the brane sufficientlylargetochangethesignoftheeffectivedensityat tension using observational data provided by stellar dy- the center; actually, ρ¯ (x=0)<0 for G M/R≥0.175. eff N namics, and with that to find evidence for the presence of extra dimensions. As a final note, we cannot say if all theconfigurationsfoundwouldbegravitationallystable, are conditions rightly expected for a real star. but it is very likely that those with a negative effective Asinitialtest,werevisitedthecaseofconstantdensity, densityintheinteriormaynotbeabletopreventthecol- corresponding to the GM solution, but later studied the lapse into configurations well within the general bound so-calledextended GMsolution,forwhichexiststarswith found in[12]. This is work in progress that will be re- a compactness beyond the standard GR bound. This is ported elsewhere. due mainly to the existence of the non-local terms which provoketheappearanceofnegativevaluesoftheeffective density and mass in the interior of the star. Finally, we considered the case of a white dwarf star ACKNOWLEDGMENTS modeled with a polytropic EoS and index n=3. In sim- ilarity with the extended GM case, our results proved MAG-A acknowledges support from C´atedra- the existence of dwarf stars with a compactness beyond CONACYT and SNI, also thanks the Departamento theGRlimit,becauseofthepresenceofnon-localterms. de F´ısica-UG for its kind hospitality. This work was Also, the compactness of dwarf stars is not bounded as partially supported by PROMEP, DAIP (534/2015), the brane tension tends to zero, which correspond to the PIFI, and by CONACyT M´exico under grants 232893 highenergylimit,whileinthelowenergylimitwerecov- (sabbatical), 167335 and 179881. We also thank the 9 support of the Fundaci´on Marcos Moshinsky, the Insti- tuto Avanzado de Cosmolog´ıa (IAC), and the Beyond Standard Theory Group (BeST) collaborations. [1] S. Chandrasekhar, The mathematical theory of black [8] T. Shiromizu, K. Maeda, and M. Sasaki, Phys. Rev. D holes (Oxford University Press, 1985); R. C. Tolman, 62, 024012 (2000). 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