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DYNAMICAL INSTABILITY AND THE EXPANSION-FREE CONDITION L. Herrera∗ Departamento de F´ısica Teórica e Historia de la Ciencia, Universidad del Pa´ıs Vasco, Bilbao, Spain G. Le Denmat† Observatoire de Paris, Université Pierre et Marie Curie, LERMA(ERGA) CNRS - UMR 8112, 94200 Ivry, France. 2 N.O. Santos‡ 1 0 School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK. and 2 Observatoire de Paris, Université Pierre et Marie Curie, LERMA(ERGA) CNRS - UMR 8112, 94200 Ivry, France. n (Dated: January 24, 2012) a J We study the dynamical instability of a spherically symmetric anisotropic fluid which collapses 3 adiabaticallyundertheconditionofvanishingexpansionscalar. TheNewtonianandpostNewtonian 2 regimes are considered in detail. It is shown that within those two approximations the adiabatic index Γ1, measuring the fluid stiffness, does not play any role. Instead, the range of instability ] is determined by the anisotropy of the fluid pressures and the radial profile of the energy density, c independently of its stiffness, in a way which is fully consistent with results previously obtained q from thestudy on theTolman mass. - r g PACSnumbers: 04.40.-b,04.20.-q,04.40.Dg,04.40.Nr [ Keywords: Relativisticfluids,stability,localanisotropyofpressure. 2 v I. INTRODUCTION tribution, under the assumption of vanishing expansion 8 1 scalar. 5 The main motivation to undertake such an endeavour The problem of stability is of the utmost relevance in 1 is provided by the following argument: Highly energetic 0. the study of Newtonian and general relativistic models explosions in self-gravitatingfluid distributions are com- of self-gravitating objects. This becomes evident if we 1 mon events in relativistic astrophysics (see for example recallthat any static stellar model, in order to be of any 0 [9, 10] and references therein). Accordingly, a relevant use, has to be stable against fluctuations. Furthermore, 1 question related to this issue is : such a problem is closely related to the one of structure v formation, since different degrees of stability/instability i How the system evolves after the explosion? X willleadto differentpatterns ofevolutioninthe collapse • r of self-gravitating objects. It is therefore not surprising Now, in a recent series of paper it has been stablished a that a great deal of work has been devoted to this issue that the expansion-free condition is particularly suitable since the pioneering paper by Chandrasekhar [1]. for describing that kind of phenomena [11–13]. ExtensionsofChandrasekhar’sresultto non-adiabatic Indeed,sincetheexpansionscalardescribestherateof fluids [2, 3], anisotropic fluids [4] and shearing viscous changeofsmallvolumesofthefluid,itisintuitivelyclear fluids [5], have been carried out in the past. In all of that the evolution of an expansion–free spherically sym- these works the key variable is the adiabatic index Γ , metric fluid distribution is consistent with the existence 1 whose value defines the range of instability. Thus for a of a vacuum cavity within the distribution which would Newtonian perfect fluid, the system is unstable for Γ < be formedafteracentralexplosion. Thispointisfurther 1 4/3. Intheabovementionedreferencesitwasshownhow discussed in this paper. Therefore potential applications different physical aspects of the fluid affect the range of ofourresultsareexpectedforthoseastrophysicalscenar- instabilityofthesystem. Morerecently[6]thestabilityof ios where a cavity within the fluid distribution is likely anisotropic stars with quasi-local equation of state [7, 8] tobe present(seefor example[14]). Furthermore,asthe has been invetigated. fluid (under the expansion–free condition) reaches the centre entails a blowup of the shear scalar, whose conse- This work represents another forward step in that di- quences in the appearance of a naked singularity cannot rection. Our main goal here consists in studying the dy- be overlooked[15]. namical instability of a spherically symmetric fluid dis- The first known model satisfying the expansion-free condition is due to Skripkin [16] for the particular case of a non-dissipative isotropic fluid, with constant energy density. In that work,Skripkinaddressedthe very inter- ∗Electronicaddress: [email protected]. AlsoatU.C.V.,Caracas †Electronicaddress: [email protected] esting problem of the evolution of a spherically symmet- ‡Electronicaddress: [email protected] ric fluid distribution following a central explosion. As a 2 result of the conditions imposed by him a Minkowskian ition is no longer a reliable guide and Γ is expected to 1 cavity should surround the centre of the fluid distribu- play a role in the discussion. tion. The plan of the paper is as follows. In section II we Onlyrecentlythis modelhasbeenagainaddressed. In give the energy-momentum tensor, the field equations [11]ageneralstudyonshearingwithvanishingexpansion and the junction conditions. Section III is devoted to scalar Θ of spherical fluid evolution is presented, which explain the physical implications of the expansion-free includes pressure anisotropy and dissipation. While in evolution. The perturbation scheme is presented in sec- [12] it is shown that the Skripkin model is incompatible tion IV. In section V a brief discussion on Newtonian with Darmois’ junction conditions [17], and that inho- and post Newtonian approximations is presented. The mogeneous expansion-free dust models are deprived of dynamicalexpansion-freeequationis obtained in section physical interest since they imply negative energy den- VI. Finally there is a concluding section. sity distributions. Further analytical solutions describing expansion–free evolution may be found in [13]. Cav- ity evolution under kinematical conditions other than II. THE ENERGY-MOMENTUM TENSOR, THE expansion-free, have been considered in [18]. FIELD EQUATIONS AND JUNCTION From the above reasons, it follows that most, phys- CONDITIONS ically meaningful, expansion-free models should require anisotropy in the pressure and energy density inhomo- We consider a spherically symmetric distribution of geneity. Thus we shall consider in this paper locally collapsingfluid, boundedby a sphericalsurface Σ(e).The anisotropicfluids (furtherargumentstojustify suchkind fluid is assumed to be locally anisotropic. Choosing co- of fluid distributions may be found in [19–21] and refer- moving coordinatesinside Σ(e), the generalinterior met- ences therein). Also, even though it is already an estab- ric can be written lished fact, that gravitational collapse is a highly dissipative process (see [22, 23] and references therein), we ds2 = A2dt2+B2dr2+R2(dθ2+sin2θdφ2), (1) − − shall restrain here for simplicity to adiabatic evolution. The role played by different dissipative processes in the where A, B and R are functions of t and r and are as- instability of a fluid distribution have been discussed in sumed positive. We number the coordinates x0 = t, detail in [2–5]. x1 = r, x2 = θ and x3 = φ. Observe that A and B The fluid under consideration has two delimiting hy- are dimensionless, whereas R has the same dimension as persurfaces. The externalone separatingthe fluid distri- r. Two radii are determined for a collapsing spherical bution from a vacuum Schwarzschild spacetime and the fluid distribution by the metric (1). The first is deter- internal one, delimiting the cavity within which there is mined by R(t,r) representing the radius as measuredby Minkowski spacetime. Thus, we have to consider junc- its spherical surface, hence called its areal radius. The tion conditions on both hypersurfaces. second is obtained out its radial integration B(t,r)dr, hencecalledproper radius. Thesetworadii,inEinstein’s It should be mentioned that for cavities with sizes of R theory, in general need not to be equal unlike in Eu- theorderof20Mpcorsmaller,theassumptionofaspher- clideangeometry(fluidswithbothradiiequalarestudied ically symmetric spacetime outside the cavity is quite in [26]). reasonable, since the observed universe cannot be con- Thematterenergy-momentumT− insideΣ(e) hasthe sidered homogeneous on scales less than 150-300 Mpc. αβ form Howeverforlargercavitiesitshouldbemoreappropriate to consider their embedding in an expanding Lemaˆıtre- T− =(µ+P )V V +P g +(P P )χ χ , (2) Friedmann-Robertson-Walkerspacetime (for the specific αβ ⊥ α β ⊥ αβ r − ⊥ α β case of voidmodeling in expanding universessee [24, 25] where µ is the energy density, P the radial pressure, and references therein). r P the tangential pressure, Vα the four-velocity of the Nextweshallpresentourperturbativeschemewhichis ⊥ fluidandχ aunitfour-vectoralongtheradialdirection. very similar to that employed in [2–5], but now with the α These quantities satisfy additional expansion-free condition. The study of the resulting equations shows that at Newtonian and post VαV = 1 , χαχ =1 , χαV =0. (3) Newtonianapproximations,the rangeofinstabilityisin- α − α α dependent on Γ . This result is intuitively clear if we 1 Thefour-accelerationa andtheexpansionΘofthefluid recallthat, onthe onehandthe expansion-freecondition α are given by implies, roughly speaking, that (at least close to New- tonian regime), the fluid evolves without being “com- a =V Vβ, Θ=Vα , (4) α α;β ;α pressed”, and on the other that Γ somehow measures 1 the (in)compressibility of the fluid. Therefore it is to be and its shear σ by αβ expected that not very far from Newtonian regime Γ 1 does not appear in the discussion under expansion-free 1 condition. Beyond post Newtonian approximation,intu- σαβ =V(α;β)+a(αVβ)− 3Θ(gαβ +VαVβ). (5) 3 Since we assumed the metric (1) comoving then The mass function m(t,r) introduced by Misner and Sharp [27] (see also [28]) reads Vα =A−1δα, χα =B−1δα. (6) 0 1 From(4)with(6)wehavethe nonzerocomponentfor R3 R R˙ 2 R′ 2 m= R 23 = +1 , (18) the four-accelerationand its scalar, 23 2 2  A! −(cid:18)B(cid:19)  A′ A′ 2 a1 = , aαaα = , (7) andthe nontrivialcomponents ofthe Bianchiidentities, A AB (cid:18) (cid:19) T−αβ =0, from (12) yield ;β and for the expansion 1 B˙ R˙ T−αβV = 1 µ˙ +(µ+P )B˙ +2(µ+P )R˙ =0, Θ= A B +2R , (8) ;β α −A" r B ⊥ R# ! (19) where the prime stands for r differentiation and the dot or using (8) stands for differentiation with respect to t. With (6) we obtain for the shear (5) its non zero components R˙ µ˙ +(µ+P )AΘ+2(P P ) =0, (20) r ⊥ r 2 1 − R σ = B2σ, σ =σ sin−2θ = R2σ, (9) 11 3 22 33 −3 and and its scalar 1 A′ R′ T−αβχ = P′+(µ+P ) +2(P P ) =0. σαβσαβ = 23σ2, (10) ;β α B (cid:20) r r A r− ⊥ R(cid:21) (21) where Introducing the proper time derivative DT given by 1 B˙ R˙ 1 ∂ σ = . (11) DT = , (22) A B − R! A∂t we can define the velocity U of the collapsing fluid (for Einstein’sfieldequationsfortheinteriorspacetime(1) another definition of velocity see section III) as the vari- to Σ(e) are given by ationofthearealradiuswithrespecttopropertime,i.e., G− =κT−, (12) αβ αβ U =D R. (23) T anditsnonzerocomponentswith(1),(2)and(6)become Using field equations and (18) we may write κT0−0 =κµA2 = 2BB˙ + RR˙!RR˙ m′ = κ2µR′R2. (24) A 2 R′′ R′ 2 B′R′ B 2 Outside Σ(e) we assume we have the Schwarzschild 2 + 2 ,(13) −(cid:18)B(cid:19) " R (cid:18)R(cid:19) − B R −(cid:18)R(cid:19) # spacetime, i.e. κT0−1 =0=−2 RR˙′ − BB˙ RR′ − RR˙ AA′!,(14) ds2 =−(cid:18)1− 2Mρ (cid:19)dv2−2dρdv+ρ2(dθ2+sin2θdφ2), (25) κT− =κP B2 where M denotes the total mass, and v is the retarded 11 r B 2 R¨ A˙ R˙ R˙ time. = 2 2 The matching of the adiabatic fluid sphere to −(cid:18)A(cid:19) " R − A − R!R# Schwarzschildspacetime, onthe surfacer =rΣ(e) = con- A′ R′ R′ B 2 stant (or ρ = ρ(v)Σ(e) in the coordinates of (25)) , re- + 2 + ,(15) quires the continuity of the first and second differential A R R − R (cid:18) (cid:19) (cid:18) (cid:19) forms (Darmois conditions), implying κT− =κT−sin−2θ =κP R2 22 33 ⊥ R 2 B¨ R¨ A˙ B˙ R˙ B˙ R˙ AdtΣ=(e) dv 1 2M , (26) = + + + − ρ −(cid:18)A(cid:19) "B R − A B R! BR# (cid:18) (cid:19) R 2 A′′ R′′ A′ B′ R′ B′R′ Σ(e) + + .(16) R = ρ(v), (27) B A R − A B − R − B R (cid:18) (cid:19) (cid:20) (cid:18) (cid:19) (cid:21) The component (14) can be rewritten with (8) and (11) as m(t,r)Σ=(e) M, (28) 1 R′ (Θ σ)′ σ =0. (17) and 3 − − R 4 R˙′ B˙ R′ R˙ A′ Σ(e) B R¨ A˙ R˙ R˙ A A′ R′ R′ B 2 2 = 2 2 + 2 + , (29) R − B R − R A! −A " R − A − R!R# B "(cid:18) A R(cid:19) R −(cid:18)R(cid:19) # Σ(e) whereas D (δl), has also the meaning of “velocity”, be- where = meansthatbothsidesoftheequationareeval- T uated on Σ(e). Comparing (29) with (13) and (14) one ing the relative velocity between neighboring layers of matter, and can be in general different from U. obtains WeshallseenowthattheconditionΘ=0isassociated Σ(e) to the existence ofacavitysurroundingthe centre ofthe P = 0. (30) r fluid distribution. Indeed, if Θ = 0 then it follows from Thus the matching of(1)and(25) onΣ(e) produces (28) (35) and (36) that and (30). U As we mentioned in the introduction, the expansion- σ = 3 , (37) − R free models present an internal vacuum cavity (reasons for the formation of this cavity are discussed in the fol- and feeding back (37) into (17) we get lowing section.). If we call Σ(i) the boundary surface U′ R′ between the cavity and the fluid, then the matching of = 2 , (38) U − R the Minkowski spacetime within the cavity to the fluid distribution, implies whose integration with respect to r yields ζ(t) Σ(i) U = , (39) m(t,r) = 0, (31) R2 where ζ is an integration function of t, implying P Σ=(i) 0. (32) 3ζ(t) r σ = . (40) − R3 Inthecasewhenthefluidfillsallthesphere,including III. ON THE PHYSICAL IMPLICATIONS OF thecentre(R(t,0)=0),wehavetoimposetheregularity THE EXPANSION-FREE EVOLUTION condition ζ = 0, implying U = 0. Therefore if we want theexpansion–freeconditionapplied to all fluid elements In section II we introduced the variable U which, as to be compatible with a time dependent situation (U = mentionedbefore,measuresthe variationofthe arealra- 6 0) we must assume that either dius R per unit proper time. Another possible definition of “velocity” may be introduced, as the variation of the the fluid has no symmetry centre, or • infinitesimal proper radial distance between two neighboring points (δl) per unit of proper time, i.e. D (δl). T the centre is surrounded by a compact spherical Then, it can be shown that (see [11] for details) • section of another spacetime, suitably matched to the rest of the fluid. D (δl) 1 T = (2σ+Θ), (33) δl 3 Here we discard the first possibility since we are particularly interested in describing localized objects without or, by using (8) and (11), theunusualtopologyofasphericalfluidwithoutacentre. Also,withinthecontextofthesecondalternativewehave D (δl) B˙ T = . (34) chosen an inner vacuum Minkowski spherical vacuole. δl AB Thus the kinematical condition Θ = 0 is consistent Then with (8), (11), (23) and (34) we can write with an evolving spherically symmetric fluid if there is a vacuum cavity surrounding its centre. D (δl) D R D (δl) U We can also arrive at this latter conclusion by the fol- T T T σ = = , (35) δl − R δl − R lowing qualitative argument. Since the expansion scalar describestherateofchangeofsmallvolumesofthefluid, and itisintuitively clearthatinthe caseofanoverallexpan- D (δl) 2D R D (δl) 2U sion(contraction),the increase(decrease)in volume due T T T Θ= + = + , (36) totheincreasing(decreasing)areaoftheexternalbound- δl R δl R ary surface must be compensated with the increase (de- Thus the “circumferential”(or “areal”)velocityU, is re- crease) of the area of the internal boundary surface (de- latedtothechangeofarealradiusRofalayerofmatter, limiting the cavity) in order to keep Θ vanishing. 5 IV. THE PERTURBATIVE SCHEME where0<ǫ 1andusingthefreedomallowedbythera- ≪ dial coordinate we choose the Schwarzschild coordinates We shall now describe in some detail the perturbative with R0(r) = r. Considering (41-46) we have from (13- scheme which will provide us with the main equation 16) for the static configuration required for our (in)stability analysis. We assume that initially the fluid is in static equilib- rium, which means that the fluid is described by quanti- tiesonlywithradialcoordinatedependence. Thesequan- tities are denoted by a subscript zero. We further sup- pose as usual, that the metric functions A(t,r), B(t,r) and R(t,r) have the same time dependence in their per- κµ = 1 2rB0′ +B2 1 , (50) turbations. Therefore we consider the metric functions 0 (B r)2 B 0 − 0 (cid:18) 0 (cid:19) and material functions given by 1 A′ κP = 2r 0 B2+1 , (51) r0 (B r)2 A − 0 A(t,r)=A0(r)+ǫT(t)a(r), (41) 0 (cid:18) 0 (cid:19) 1 A′′ A′ B′ 1 A′ B′ B(t,r)=B0(r)+ǫT(t)b(r), (42) κP⊥0 = B2 A0 − A0B0 + r A0 − B0 ; (52) R(t,r)=R (r)+ǫT(t)c(r), (43) 0 (cid:20) 0 0 0 (cid:18) 0 0(cid:19)(cid:21) 0 µ(t,r)=µ (r)+ǫµ¯(t,r), (44) 0 P (t,r)=P (r)+ǫP¯ (t,r), (45) r r0 r P (t,r)=P (r)+ǫP¯ (t,r), (46) ⊥ ⊥0 ⊥ m(t,r)=m (r)+ǫm¯(t,r), (47) 0 Θ(t,r)=ǫΘ¯(t,r), (48) whereas from (13-16) we obtain for the perturbed quan- σ(t,r)=ǫσ¯(t,r), (49) tities T c ′′ 1 b ′ B′ 3 c ′ B 2 b c b κµ¯= 2 0 0 2κµ T , (53) − B02 "(cid:16)r(cid:17) − r (cid:18)B0(cid:19) −(cid:18)B0 − r(cid:19)(cid:16)r(cid:17) −(cid:18) r (cid:19) (cid:18)B0 − r(cid:19)#− 0 B0 T˙ c ′ b A′ 1 c 2 0 =0, (54) A B r − rB − A − r r 0 0 (cid:20)(cid:16) (cid:17) 0 (cid:18) 0 (cid:19) (cid:21) T¨ c T a ′ A′ c ′ B2 b c b κP¯ = 2 +2 + r 0 +1 0 2κP T , (55) r − A20r rB02 "(cid:18)A0(cid:19) (cid:18) A0 (cid:19)(cid:16)r(cid:17) − r (cid:18)B0 − r(cid:19)#− r0 B0 T¨ b c T a ′′ c ′′ A′ B′ 1 a ′ κP¯ = + + + + 2 0 0 + ⊥ −A20 (cid:18)B0 r(cid:19) B02 "(cid:18)A0(cid:19) (cid:16)r(cid:17) (cid:18) A0 − B0 r(cid:19)(cid:18)A0(cid:19) A′ 1 b ′ A′ B′ 2 c ′ b 0 + + 0 0 + 2κP T ; (56) ⊥0 −(cid:18)A0 r(cid:19)(cid:18)B0(cid:19) (cid:18)A0 − B0 r(cid:19)(cid:16)r(cid:17)#− B0 and for the expansion (8) and shear (11) we have T˙ b c σ¯ = . (58) T˙ b c A0 (cid:18)B0 − r(cid:19) (cid:18) (cid:19) 6 The Bianchi identities (19) and (21) become with (41- Consideringthesecondlawofthermodynamicsandthe 46), for the static configuration same arguments as given in [2–4], we can express a rela- tionship between P¯ and µ¯ given by A′ 2 r P′ +(µ +P ) 0 + (P P )=0, (59) r0 0 r0 A0 r r0− ⊥0 P¯ =Γ Pr0 µ¯, (69) r 1 µ +P and for the perturbed configuration, 0 r0 1 b c where Γ is the adiabatic index which measures the µ¯˙ +(µ +P )T˙ +2(µ +P )T˙ =0,(60) 1 A 0 r0 B 0 ⊥0 r variation of pressure related to a given variation of 0 (cid:20) 0 (cid:21) density, thereby measuring the stiffness of the fluid. We and considerit constantthroughoutthe fluid distribution or, at least, throughout the region that we want to study. 1 a ′ A′ B P¯r′+(µ0+Pr0)T A +(µ¯+P¯r)A0 We recall that Γ1 coincides with the ratio of the specific 0 " (cid:18) 0(cid:19) 0 heats for perfect Maxwell-Boltzman gas [30–32]. c ′ 1 +2(P P )T +2(P¯ P¯ ) =0. (61) r0 ⊥0 r ⊥ − r − r (cid:16) (cid:17) (cid:21) By substituting (54) into (60) we can integrate it, and V. NEWTONIAN AND POST NEWTONIAN we find TERMS b c µ¯= (µ +P ) +2(µ +P ) T. (62) 0 r0 0 ⊥0 − B r Before dwelling to obtain the dynamical expansion– (cid:20) 0 (cid:21) free equation, and since we are using relativistic units, The total energy inside Σ(e) up to a radius r given by the following comments should be helpful in order to (18) with (41-43) and (47) becomes, identify the terms belonging to the Newtonian (N), post Newtonian(pN)andpostpostNewtonian(ppN)approx- r 1 m = 1 , (63) imations. These terms are considered for the instability 0 2 − B2 (cid:18) 0(cid:19) conditionsstemmingfromthe dynamicalequationinthe T b c N and pN approximations. m¯ = r c′ +(1 B2) . (64) −B2 − B − 0 2 Thus, for N approximation we assume. 0 (cid:20) (cid:18) 0(cid:19) (cid:21) From the matching condition (30) with (45) we have, µ P , µ P . (70) 0 r0 0 ⊥0 ≫ ≫ P Σ=(e) 0, P¯ Σ=(e) 0. (65) r0 r For the metric coefficients, given in c.g.s. units, ex- panded up to pN approximation become For c=0, which is the case that we want to study, with 6 (54), (55) and (65) we have m m 0 0 A =1 G , B =1+ G , (71) T¨ αT Σ=(e) 0, (66) 0 − 2r 0 2r C C − where where is the gravitational constant and is the speed G C of light. α= A0 2 a ′+ rA′0 +1 c ′ Next,from(51)with(63)wehave(inrelativisticunits) B A A r (cid:18) 0(cid:19) "(cid:18) 0(cid:19) (cid:18) 0 (cid:19)(cid:16) (cid:17) A′ κP r3+2m B2 b c 1 0 = r0 0, (72) 0 , (67) A 2r(r 2m ) − r (cid:18)B0 − r(cid:19)(cid:21) c 0 − 0 Solutions of (66) include functions which oscillate (cor- and substituting into (59) it yields responding to stable systems) and those which do not (unstable ones). Since we are interested here in estab- κP r3+2m 2 P′ = r0 0 (µ +P )+ (P P ). lishing the range of instability, we confine our attention r0 − 2r(r 2m ) 0 r0 r ⊥0− r0 to the non oscillating ones, i. e. we assume that a(r), (cid:20) − 0 (cid:21) (73) b(r) and c(r) are such that on rΣ(e), αΣ(e) >0. Then, Writing (73) in c.g.s. units it becomes T(t)=−exp √αΣ(e) t , (68) −2κP r3+2m 2 P′ = C r0 0 (µ + −2P )+ (P P ). representing a system that st(cid:0)arts colla(cid:1)psing at t = r0 −G 2r(r 2 −2 m ) 0 C r0 r ⊥0− r0 −∞ (cid:20) − C G 0 (cid:21) when T( ) = 0 and the system is static, and goes (74) collapsing−,∞diminishing itsarealradius,whilet increases. Expanding (74) up to terms of −4 order we obtain C 7 µ m 2 κ P′ = 0 0 + (P P ) G 2 µ m2+P m r+ µ P r4 r0 −G r2 r ⊥0− r0 − 2r3 G 0 0 r0 0 2 0 r0 C (cid:16) κ (cid:17) G 4 2µ m3+2 P m2r+ κµ P m r4+ P2 r5 . (75) − 4r4 G 0 0 G r0 0 G 0 r0 0 2 r0 C (cid:16) (cid:17) Hence from (75) we have the following terms for the dif- showingthat the perturbedenergydensityofthe system ferent orders of approximations: stems from the static background anisotropy. This fact iseasilyunderstoodifwerecallthatundertheexpansion N order: terms of order 0; (76) freecondition,asitfollowsfrom(20),changesinµforany C pN order: terms of order −2; (77) given fluid element, depend exclusively on the pressure ppN order: terms of orderC−4. (78) anisotropy. C On the other hand, with (69) and (82) we have P c VI. THE DYNAMICAL EXPANSION-FREE P¯ =2Γ r0 (P P )T . (83) r 1 r0 ⊥0 EQUATION µ +P − r 0 r0 Two further useful relations we can obtain. One from Withtheequationssofarobtainedwecanbuildthedy- (59), namicalequationthatwespecializetotheexpansion-free condition which is the aim of our study. The obtention of the dynamical equation is done via (61). A′0 = 1 P′ + 2(P P ) , (84) The expansion-free condition Θ=0 implies from (57) A0 −µ0+Pr0 (cid:20) r0 r r0− ⊥0 (cid:21) and another from (50) and (63) b c = 2 , (79) B0 − r B0′ = κµ0r3−2m0. (85) B 2r(r 2m ) 0 0 which together with the adiabatic condition (54) pro- − duces, We want to study the instability conditions for the expansion-freefluiduptothepNapproximation,sincein- T˙ r2c ′ 2 =0, (80) termediatecalculationsareratherlongweshallputthem r3B0 (cid:18)A0(cid:19) in an Appendix. Considering the N approximation of the dynamical implying equation (98) by using (100) and (104-106) and that A0 terms with Pr0/µ0 being of ppN order it reduces to c=k , (81) r2 21 m where k is a constant. With (79) we have for (62) 3κµ0+κ|Pr′0|r+2 αΣ(e) − r2 r0 =4κ(5Pr0−2P⊥0), (cid:18) (cid:19) (86) c µ¯ =2(P P )T , (82) or, using(24) and rearrangingterms r0 ⊥0 − r κ P′ r4+ αΣ(e)m r2 = 2κ(5P 2P )r3+ κ 7 r µ r2dr µ r3 (87) 18| r0| 9 0 9 r0− ⊥0 6 ZrΣ(i) 0 − 0 ! where we assume P′ <0. For the pN approximation we have r0 8 21 m m 3κµ0+κ|Pr′0|r+2 αΣ(e) − r2 r0 +2κ|Pr′0|m0+καΣ(e)Pr0r2−κµ0 3 r0 −2αΣ(e)rm0 (cid:18) (cid:19) (cid:16) (cid:17) 5 m 2 m 0 0 +6 αΣ(e) − r2 r =4κ(5Pr0−2P⊥0)+2κ(Pr0−2P⊥0) r . (88) (cid:18) (cid:19)(cid:16) (cid:17) Let us first consider the N approximation. The first ob- In other words, for n < 0, the range of instability servationtobemadeisthatintheabsenceofasinglepa- increases as the absolute value of n increases. rametersuchasΓ theassessmentoftheinstabilityrange 1 2. n 0. depends in a rather complicated way on different struc- ≥ tural properties of the fluid, such as pressure anisotropy In this case the positivity of the last term in (87) and the radial profile of the energy density. is, again, assured in the region Indeed, for the onset of instability we need (87) to be 1/(n+3) 7 satisfied, and since the two terms at the left of (87) are r>rΣ(i) 4 n . (95) positive, then instabilities may develop only if so is the (cid:18) − (cid:19) combination of the two terms at the right of (87). For Therangeofinstabilitydecreaseswithn,vanishing that to happen it will be sufficient that Pr0 > (2/5)P⊥0 for n 4. andthatthelasttermin(87)bepositive. Letusexplore ≥ this possibility in some detail. 3. n= 3. − Thus, let us consider an energy density profile of the Inthisparticularcase,thepositivityofthethelast form µ = βrn where β is a positive constant, and n is term in (87) is assured in the region 0 alsoa constantwhose value rangesin the interval < n < . In this case the last term in (87) becom−es∞(for r >rΣ(i)e1/7 ≈rΣ(i)1.15 (96) ∞ n= 3) 6 − Ofcourse,ifPr0 <(2/5)P⊥0 (forthesameenergyden- κβ rn+3 4 n 7 rΣ(i) 3+n , (89) sityprofile),themaximalregionofinstabilitydiminishes. 6(3+n) − − r AtpN orderthe situationis essentially the same, with (cid:20) (cid:16) (cid:17) (cid:21) the relativistic effects at first order taken into account. whereas for the n= 3 case we obtain ObservethatΓ doesnotplayanyroleintheNandpN − 1 κβ r orders. In other words at N order, no matter how stiff 7log 1 . (90) is the material, the system will be unstable as long as 6 (cid:20) (cid:18)rΣ(i)(cid:19)− (cid:21) (86)issatisfied. Thisisatvariancewiththeresultinthe Then the following possibilities arise: non-vanishing expansion case when it appears that the range of instability is defined by Γ <(4/3)+anisotropic 1. n 0, n= 3. 1 ≤ 6 − term (see [4] for details). A similar remark applies for In this case (89) will be positive if the pN order. As mentioned before the fact that Γ does not enter 7 1/(n+3) 1 r >rΣ(i) 4 n . (91) into (86) and (88) becomes intelligible when we recall (cid:18) − (cid:19) that the expansion-free collapse (close to the Newtonian regime)proceedswithout“compression”ofthefluid. Ac- Thus the maximal range of instability decreases cordingly the stiffness of matter is irrelevant for the on- from its value for n close to 3, for which we have | | setofinstabilities,thelaterbeingdependentonthelocal (see case 3 below) anisotropyofpressureandenergydensityinhomogeneity. r >rΣ(i)e1/7 ≈rΣ(i)1.15 (92) (unItdesrhotuhledebxepaonbssioernv-efrdeethcaotndinititohne) pthroeceensserogfycdoellnaspitsye to mayincreaseordecreasedepending onthe differencebe- 1/3 tween the velocity of the inner and the outer boundary 7 r>rΣ(i) 4 ≈rΣ(i)1.20 (93) surface. Thisdiferenceinturnisdeterminedbythelocal (cid:18) (cid:19) anisotropyofpressureandtheradialprofileoftheenergy corresponding to n = 0 (the incompressible fluid). density. On the other hand as n increases from n = 3 | | | | the maximal region of instability will embrace the VII. CONCLUSIONS whole fluid, since 1/(3+n) 7 We have seen so far that in the study of dynamical lim =1 (94) n→−∞ 4 n instability, under the expansion-free condition, at, both, (cid:18) − (cid:19) 9 theNewtonianandpostNewtonianregimes,therangeof describes the instability range of the cavity (keeping the instability is defined by the local anisotropy of pressure expansion-free condition). We observe that the pertur- andtheenergydensityradialprofile,butnotbytheadia- bation of the cavity itself produces a similar equation to baticindexΓ . Thisimpliesthatintheabovementioned T(t)asin (66)andit imposes a further constraintto the 1 regimes the stiffness of the fluid, measured by Γ1, which system αΣ(e) =αΣ(i). generally plays the central role in the definition of the It is worth mentioning that the role played by the instability range, is irrelevant here. This fact strength- anisotropy and the energy density inhomogeneity, as it ens further the relevance of local anisotropy of pressure followsfromthe previoussection,is fully consistentwith and energy density inhomogeneity in the structure and the results obtained from the study on the influence of evolution of self-gravitating objects. those two factors on the active gravitational (Tolman) Itshouldbestressedthatanypossiblemodelisfurther mass [33], presented in [34]-[36]. constrained by physical requirements such as positivity of energy density, sound speed less than light (energy Indeed, for the Tolman mass m [33] interior to a T density greater than pressure), and stability of local os- sphereofradiusr,uptopNapproximation,thefollowing cillations modes. expression may be obtained in the static or quasi-static The association of a cavity with the expansion-free case(see equation(47)in[34],orequation(58)in[35]or fluid distribution implies that the abovepresentedstudy equation (50) in [36] and the discussion therein) mT =(mT)Σ(e)(cid:18)rΣr(e)(cid:19)3+κr3ZrrΣ(e) "(Pr0−P⊥0)1r − 21r4 ZrΣr(i) µ′0r˜3dr˜#dr, (97) where (71) has been used. for naked singularity formation. The obvious relevance From the above it is evident that µ′ < 0 and P > of such an effect strengthens further the interest of the 0 r0 P increase the Tolman mass. That result, together problem discussed here. ⊥0 with the fact that the Tolman mass may be interpreted Finally, let us mention that an extensions of these re- as the active gravitational mass, provide full support to sultsforf(r)gravitytheory,havebeenrecentlypresented the conclusions on the instability range obtained in the [37], [38]. previous section. It is worth noticing that if we keep the expansion-free conditionallalongthe collapse,then assoonasthe fluid approachesthecentretherewillbeablowupoftheshear Appendix scalar, as implied by (40). But it is shown in [15] that sufficiently strong shearing effects near the singularity delay the formation of the apparent horizon, implying Inspecting (83) we see that P¯ is of ppN order,as well r the appearance of a naked singularity. In other words as µ¯A′/A by considering (82) and (83), hence (61) to- 0 0 the expansion-free condition provides a simple scenario gether with (68) and (56) reduces to a ′ c ′ c 2 a ′′ c ′′ A′ B′ 1 a ′ κ(µ +P )r +2κ(P P )r 8κP + + 2 0 0 + 0 r0 (cid:18)A0(cid:19) r0− ⊥0 (cid:16)r(cid:17) − ⊥0r − B02 "(cid:18)A0(cid:19) (cid:16)r(cid:17) (cid:18) A0 − B0 r(cid:19)(cid:18)A0(cid:19) + 3A′0 B0′ + 4 c ′ 2αΣ(e) c =0.(98) A − B r r − A2 r (cid:18) 0 0 (cid:19)(cid:16) (cid:17)(cid:21) 0 From (55) with (66), (79) and (81) we have Thefirstthreetermsof(98)with(81)and(99)uptopN order become a ′ A A′ 2 2A′ = k 0 2κP B2+ 0 0 (cid:18)A0(cid:19) − r2 " r0 0 (cid:18)A0(cid:19) − rA0 2 3 B + (B02−1)−αΣ(e)(cid:18) 0(cid:19) #. (99) 10 a ′ c ′ c κkA 2 κ(µ +P )r +2κ(P P )r 8κP = 0 2κµ P rB2+2P′ + (5P P ) 0 r0 A r0− ⊥0 r − ⊥0r − r2 0 r0 0 r0 r r0− ⊥ (cid:18) 0(cid:19) (cid:16) (cid:17) (cid:26) 3 B 2 κk m 2 +(µ0+Pr0)r"r2(B02−1)−αΣ(e)(cid:18)A00(cid:19) #)=−r2 (cid:26)2κµ0Pr0r+(cid:16)1− r0(cid:17)(cid:20)2Pr′0+ r(5Pr0−P⊥0)(cid:21) m m 6 15 m 2 0 0 0 +(µ0+Pr0)r 6 r3 −αΣ(e) 1+3 r +µ0r r2 − 2 αΣ(e) r . (100) h (cid:16) (cid:17)i (cid:18) (cid:19)(cid:16) (cid:17) (cid:27) Now we calculate the following terms, a ′′ A′ B′ 1 a ′ A 1 B′ 2 A′ ′ 2A′ B′ + 2 0 0 + =k 0 2κP B2 0 2κP′ B2+ 0 0 0 A A − B r A r2 r0 0 r − B − r0 0 r A − rA B (cid:18) 0(cid:19) (cid:18) 0 0 (cid:19)(cid:18) 0(cid:19) (cid:20) (cid:18) 0(cid:19) (cid:18) 0(cid:19) 0 0 1 A′ 3 B′ 9 A B 2 A′ B′ 1 +r2A0(5−9B02) −r2B0(B02+1)+ r3(B02−1) +αΣ(e)kr20 A0 A0 + B0 − r , (101) 0 0 (cid:21) (cid:18) 0(cid:19) (cid:18) 0 0 (cid:19) and c ′′ A′ B′ 4 c ′ A A′ ′ A′ B′ 11A′ 3B′ + 3 0 0 + =k 0 0 0 0 0 + 0 , (102) (cid:16)r(cid:17) (cid:18) A0 − B0 r(cid:19)(cid:16)r(cid:17) r3 "(cid:18)A0(cid:19) − A0B0 − r A0 rB0# where (81) and (99) have been used. With (101) and (102) we can build 2 a ′′+ c ′′+ 2A′0 B0′ + 1 a ′ + 3A′0 B0′ + 4 c ′ 2αΣ(e) c −B02 "(cid:18)A0(cid:19) (cid:16)r(cid:17) (cid:18) A0 − B0 r(cid:19)(cid:18)A0(cid:19) (cid:18) A0 − B0 r(cid:19)(cid:16)r(cid:17)(cid:21)− A20 r A B′ 1 A A′ ′ A′ B′ =4κk 0 P′ +P 0 6k 0 0 0 0 r2 (cid:20) r0 r0(cid:18)B0 − r(cid:19)(cid:21)− B02r3 "(cid:18)A0(cid:19) − A0B0 (3B2+2)1A′0 1B B′ + 3 (B2 1) 2αΣ(e)k A′0 + B0′ . (103) − 0 rA − r 0 0 r2 0 − − A r2 A B 0 (cid:21) 0 (cid:18) 0 0(cid:19) By using (71), (84) and (85) we obtain the following ex- pressions up to pN order, A B′ 1 4κk P κ m 4κk 0 P′ +P 0 = P′ r0 + rµ P 0P′ , (104) r2 r0 r0 B − r r2 r0− r 2 0 r0− r r0 (cid:20) (cid:18) 0 (cid:19)(cid:21) (cid:18) (cid:19)