Scanning the parameter space of collapsing rotating thin shells
SScanning the parameter space of collapsing rotating thin shells
Jorge V. Rocha ,(cid:93) and Raphael Santarelli ,(cid:92) Departament de F´ısica Qu`antica i Astrof´ısica, Institut de Ci`encies del Cosmos (ICCUB),Universitat de Barcelona, Mart´ı i Franqu`es 1, E-08028 Barcelona, Spain Departamento de F´ısica, Universidade Federal de S˜ao Carlos,P. O. Box 676, 13565-905, S˜ao Carlos, S˜ao Paulo, Brazil
Abstract
We present results of a comprehensive study of collapsing and bouncing thin shells with rota-tion, framing it in the context of the weak cosmic censorship conjecture. The analysis is based on aformalism developed specifically for higher odd dimensions that is able to describe the dynamics ofcollapsing rotating shells exactly. We analise and classify a plethora of shell trajectories in asymp-totically flat spacetimes. The parameters varied include the shell’s mass and angular momentum,its radial velocity at infinity, the (linear) equation-of-state parameter and the spacetime dimension-ality. We find that plunges of rotating shells into black holes never produce naked singularities,as long as the matter shell obeys the weak energy condition, and so respect cosmic censorship.This applies to collapses of dust shells starting from rest or with a finite velocity at infinity. Noteven shells with a negative isotropic pressure component (i.e., tension) lead to the formation ofnaked singularities, as long as the weak energy condition is satisfied. Endowing the shells with apositive isotropic pressure component allows the existence of bouncing trajectories satisfying thedominant energy condition and fully contained outside rotating black holes. Otherwise any turningpoint occurs always inside the horizon. These results are based on strong numerical evidence fromscans of numerous sections in the large parameter space available to these collapsing shells. Thegeneralisation of the radial equation of motion to a polytropic equation-of-state for the matter shellis also included in an appendix. (cid:93) [email protected] (cid:92) [email protected] a r X i v : . [ g r- q c ] J un canning the parameter space of collapsing rotating thin shells While the study of spherical gravitational collapse of stars leading to black holes (BHs) has a longhistory dating back to the work of Oppenheimer and Snyder [1], full investigations of collapsingmatter carrying angular momentum only became possible with the advent of numerical relativity inthe ’80s [2, 3]. Of course, the reason for this historic delay is that the introduction of rotation inthree spatial dimensions typically breaks spherical symmetry, thus increasing the complexity of theproblem.Besides uncovering the fate of realistic stars, such studies have an important bearing on the weakcosmic censorship conjecture [4], which is still an issue of intense debate. In essence, this assertsthat any curvature singularities forming from generic collapse of physically reasonable matter remainhidden inside black hole horizons. In other words, the conjecture forbids the development of nakedsingularities, preventing quantum gravity spacetime regions to be accessed by asymptotic observers.Axisymmetric simulations of stellar collapse have shown that whether or not a BH forms dependsstrongly on the amount of angular momentum [5]. The endpoint of a full collapse process is expectedto be a stationary spacetime, which in four dimensions —and assuming departures from vacuum arenegligible— must belong to the Kerr family of solutions [6], parametrised by mass M and angularmomentum J . When J/M ≤ J/M > J/M > J/M ≤
1) regime is reached and themerger always produces a final Kerr black hole.There is a long history of assessments of cosmic censorship based on stress-testing the stability ofvacuum black hole horizons under the absorption of test particles. The first such study was the seminalwork of Wald [10], showing that 4D extremally rotating black holes cannot be over-spun with testparticles. Those with sufficiently large angular momentum to raise the joint BH+particle system abovethe Kerr bound are simply scattered by the black hole, leaving the horizon untouched. This picturehas been extended to higher dimensions [11] with similar results; Refs. [12, 13] specifically concernspacetimes with equal angular momenta, which will be the focus of the present paper. Test fieldsinteracting with extremal black holes were also shown to comply with weak cosmic censorship [14].The consideration of near-extremal BHs —as opposed to exactly extremal— opened a narrow windowof opportunity to overspin 4D black holes with point particles, by neglecting back-reaction effects [15].However, a proper account of the self-force was argued to restore the validity of the conjecture [16].Very recently, this was confirmed to be the case, by incorporating crucial self-force effects of secondorder in the angular momentum of a body falling into the black hole [17]. It should also be mentionedthat some notable violations of cosmic censorship do occur in higher-dimensional general relativity [18,19, 20], but these are of a completely different nature: they are a consequence of instabilities afflictingextended BH horizons. Therefore, such objects are not expected to result generically from gravitationalcollapse.The topic of gravitational collapse in the presence of rotation has hardly been explored with an-alytic methods. Some early partial results were obtained by employing approximations, such as theassumptions of adiabatic collapse, or slow rotation [21, 22]. Alternatively, Ref. [23] made progressby focusing attention on the dynamics only in a neighbourhood of the equatorial plane. Clearly,the restriction to two spatial dimensions bypasses the main technical hurdle, since it allows rotatingconfigurations while avoiding any dependence on angular coordinates. This was used to study gravi-tational collapse in (2 + 1)-dimensional spacetimes for the case of thin rings of matter [24, 25] and forinhomogeneous disks of dust [26]. In these contexts, naked singularities never arise from collapses withrotation, as long as matter obeys the weak energy condition, i.e., weak cosmic censorship is observed.Nevertheless, it is in fact possible to tackle the problem of gravitational collapse with rotation—without restricting to lower dimensions— in a fairly simple way [27]. The idea relies on the consid- canning the parameter space of collapsing rotating thin shells D = 5 , , , ... ), allowing rotating geometries to depend on a single coordinate — for this reason theyare referred to as cohomogeneity-1 geometries. This results from an enhanced symmetry, in practicemaking them resemble spherically symmetric spacetimes. The formalism for rotating thin matter shellsin 5D was developed in Ref. [27], and more recently has been extended to higher odd dimensions [28].A similar idea was first used in Ref. [29] to study critical collapse in 5D vacuum gravity, although thatinvestigation was restricted to non-rotating spacetimes.The goal of the present paper is to perform a comprehensive study of exact collapses of rotatingthin shells. Within the framework of Refs. [27, 28] we can assess the effect of rotation on gravitationalcollapse and consequently on cosmic censorship. Since the full evolution of the thin shell spacetime isobtained by a simple integration of a radial effective potential that is known exactly , we can easily scana large parameter space. The parameters we vary include the proper mass of the shell, the angularmomentum, the radial velocity of the shell at infinity, a linear equation-of-state parameter and thespacetime dimensionality.The output of this parameter scan is presented in the final section of the paper, alongside withample discussion of the results. Nonetheless, it is convenient to highlight here our main findings: • We observe no violation of the weak cosmic censorship conjecture. In the context of equalangular momenta spacetimes, collapses of thin shells onto rotating black holes can destroy thehorizon only if the matter shell violates the weak energy condition. This is also true for theless-restrictive null energy condition . • Full plunges of rotating dust shells —for which the isotropic component of the pressure vanishes,although the rotation induces a nonzero anisotropic pressure— always violate the dominantenergy condition before hitting the singularity. • Still considering rotating dust shells, the dominant energy condition can be satisfied if thereis a turning point, which can either be inside the horizon or else if there is no horizon at all.The former case corresponds to a two-world orbit in which the shell crosses a black hole horizonand later emerges from a white hole into a different universe. The latter case describes a shellbouncing off a naked singularity. • The consideration of a nonzero isotropic pressure component opens up more possibilities (butthe shell must approach the speed of light at infinity). In this case there is an interestingcompetition between centrifugal forces and pressure, which allows bounces with turning pointsoutside a black hole horizon, while obeying the dominant energy condition.Our approach offers some advantages over previous work concerning the gravitational collapseof rotating shells. Contrary to Ref. [22] (for a textbook exposition, see section 3.10 of [31]), wecan follow the shell’s entire trajectory exactly. The point is that in descriptions adopting a smallrotation approximation the solutions cease to be valid near the ergosphere, and therefore they cannotsay anything about full collapse to a black hole (or a naked singularity). Moreover, compared toRefs. [24, 25] which focused on the shell’s trajectory, we also take particular care in checking whetherenergy conditions are satisfied during the orbit, since this plays an important role in the formulationof cosmic censorship.The framework adopted to construct rotating thin shell spacetimes also faces a few limitations.The most obvious one is that the consideration of cohomogeneity-1 rotating geometries restricts us toodd spacetime dimensions. Secondly, the matching of two such backgrounds to obtain the thin shell isperformed in the simplest possible way, guaranteeing that the angular symmetry group —which turnsout to be U (cid:0) D − (cid:1) — is preserved by the global spacetime. This limits our ability to construct rotatingthin shells to cases in which the interior region already has a black hole or a naked singularity; i.e., it Here we are considering the energy conditions from the viewpoint of the shell’s worldsheet. One can also assessthe energy conditions as derived from a stress-energy tensor on the full spacetime, but being localized on the timelikehypersurface of the shell. The two descriptions are related: the weak energy condition on the shell is equivalent to thenull energy condition on the full spacetime [30]. canning the parameter space of collapsing rotating thin shells onto black holes ornaked singularities and, therefore, are not suitable for studies of critical collapse. Finally, since boththe exterior and interior of the shell are taken to be stationary —namely Myers-Perry solutions [32]with all spin parameters set equal— clearly there will be no gravitational radiation, even though theshell simultaneously contracts/expands and rotates. Hence, these are curiously special spacetimes.Nevertheless, from the point of view of testing cosmic censorship this is precisely the most dangerousscenario, since gravitational radiation is typically much more efficient in dissipating angular momentumthan energy [33, 34].The remainder of the paper is organised as follows. We start by reviewing the framework employedto construct rotating thin shell spacetimes in Section 2. Then we analyse energy conditions in Section 3.In Section 4 we study general properties of the radial dynamics of rotating shells. Finally, the scan ofthe parameter space of collapsing rotating thin shells is performed in Section 5, where we also includethe discussion of the results. We relegate to Appendix A some general bounds obtained from the weakand dominant energy conditions. In Appendix B we present the generalisation of the radial equationof motion to shells with polytropic equations-of-state.
This section reviews material covered in Refs. [27, 28], where more details can be found. Therefore,we will only summarise the main points, which will also serve to fix important notation. Our focushere is on asymptotically flat spacetimes, but note that the analysis can be easily extended to includea nonvanishing cosmological constant. We will work in geometrised units, for which the gravitationalconstant and the speed of light are set to unity, G = c = 1.We take the interior and exterior spacetimes (indicated with subscripts − and + on all associatedquantities, respectively) to be EAM Myers-Perry solutions in D = 2 N + 3 dimensions, with N aninteger. This family of stationary geometries has enhanced symmetry and their line element dependsessentially only on a radial coordinate [35, 36]: ds ± = g µν dy µ dy ν = − f ± ( r ) dt + g ± ( r ) dr + h ± ( r ) [ dψ + A a dx a − Ω ± ( r ) dt ] + r (cid:98) g ab dx a dx b , (2.1)where g ± ( r ) = (cid:18) − M ± r N + 2 M ± a ± r N +2 (cid:19) − , (2.2) h ± ( r ) = r (cid:18) M ± a ± r N +2 (cid:19) , Ω ± ( r ) = 2 M ± a ± r N h ± ( r ) , f ± ( r ) = rg ± ( r ) h ± ( r ) . (2.3)Here, M ± and a ± are the mass and spin parameters, respectively. This form of writing the metricrelies on the description of the constant t - and r -slices —which are topologically (2 N + 1)-spheres— asa S bundle over the complex projective space CP N . The x a are the 2 N coordinates on CP N , whichis endowed with the standard Fubini-Study line element (cid:98) g ab dx a dx b , and A = A a dx a is the associatedK¨ahler potential. Explicit expressions for (cid:98) g ab and A a can be obtained iteratively in N [37, 38] butwe shall not require them. The coordinates y µ cover the whole manifold and run over { t, r, ψ, x a } .The S fiber is parametrised by the angular coordinate ψ , with periodicity 2 π . The metrics (2.1) arevacuum solutions of the Einstein equations. The largest real root of g − ± indicates an event horizonwhose spatial sections have the geometry of a homogeneously squashed (2 N + 1)-sphere. When theparameters M ± and a ± are such that the function g − ± does not have zeroes, the associated spacetimecorresponds to a naked singularity, with the curvature diverging at r = 0.The next step is to match two spacetimes with line elements of the form (2.1) across a timelikehypersurface Σ defined by the parametric equations t = T ( τ ) and r = R ( τ ), where τ is the propertime of an observer comoving with the hypersurface. This does not mean that more involved matching surfaces cannot avoid this feature, but then we would encounterthe same difficulties found in the construction of 4D rotating thin shells. canning the parameter space of collapsing rotating thin shells g (+) ij = g ( − ) ij ≡ g ij . This has two immediate consequences:since the parameter τ is taken to be the proper time, this implies a relation between ˙ T and ˙ R , f ( R ) ˙ T − g ( R ) ˙ R = 1 , (2.4)where an overdot stands for d/dτ . In addition, the first junction condition imposes a rigid relationshipbetween the parameters of the interior and exterior geometries, h + ( R ) = h − ( R ) ≡ h ( R ) ⇒ M + a = M − a − . (2.5)This implies that a such a rotating cohomogeneity-1 exterior (with M + a + (cid:54) = 0) cannot be continuouslyjoined with a flat interior ( M − = 0) along a constant- r surface.The second junction condition fixes the form of the surface stress-energy tensor S ij that sourcesany possible divergences on Σ. S ij = − πκ N ([[ k ij ]] − g ij [[ k ]]) , (2.6)where k ij represents the extrinsic curvature and k = g ij k ij is its trace. Here we have introduced[[ C ij... ]] ≡ C (+) ij... − C ( − ) ij... (2.7)as a short-hand notation for the jump of any given tensorial quantity C ij... across Σ. We included adimension-dependent numerical prefactor on the right hand side, κ N , which will be fixed in Section 4.For N = 1 /
2, corresponding formally to standard four-dimensional gravity, we should recover κ / = 1.One might be tempted to assume ˙ T >
0. After all, this must be the case when a timelike shell isoutside the black hole event horizon or inside the Cauchy horizon. Note however, that between theevent horizon and the Cauchy horizon (where g ( R ) <
0) the derivative ˙ T can in fact change sign. Aswe will see, this overall sign does not affect neither the conservation equations nor the shell’s equationof motion.The various components of the extrinsic curvature were computed in [28]. The form of the stress-energy tensor S ij is then dictated by the second junction condition (2.6), S ij = ( ρ + P ) u i u j + P g ij + 2 ϕ u ( i ξ j ) + ∆ P R (cid:98) g ab dx a dy i dx b dy j , (2.8)where coordinates y i run over { τ, ψ, x a } . This stress-energy tensor describes an imperfect fluid. Here, u = u i ∂ i = ∂ τ is the normalised fluid velocity (assumed to be corotating with the shell), and ξ = ξ i ∂ i = h − ∂ ψ is a unit vector aligned with the S fiber (the direction that effectively incorporatesthe rotation of the spacetime). The quantity ϕ is commonly referred to as heat flow, and it can bethought of as an intrinsic momentum of the fluid, while ∆ P denotes the pressure anisotropy.More explicitly, the components of the stress-energy tensor are given by the following expressions: ρ = − [[ β ( R )]]8 πκ N R N +1 dd R (cid:2) R N h ( R ) (cid:3) , (2.9) P = h ( R )8 πκ N R N +1 dd R (cid:2) R N [[ β ( R )]] (cid:3) , (2.10) ϕ = − h ( R ) πκ N R [[Ω (cid:48) ( R )]] , (2.11)∆ P = [[ β ( R )]]8 πκ N dd R (cid:20) h ( R ) R (cid:21) . (2.12) From the point of view of the maximal analytic extension, the sign of ˙ T dictates which of the two allowed trajectoriesthe shell follows in the domain between the event horizon and the Cauchy horizon [41]. canning the parameter space of collapsing rotating thin shells β ± ≡ sign( ˙ T ) f ± (cid:113) g ± ˙ R . (2.13)From these expressions it follows immediately that in the absence of rotation both ϕ and ∆ P vanish, and so we recover a perfect fluid. Thus, the heat flow and the pressure anisotropy are inducedby the shell’s rotation.One can verify that such a stress-energy tensor is covariantly conserved. The conservation equa-tions, ∇ i S ij = 0, reduce to − dd R (cid:0) h R N ρ (cid:1) = (cid:18) N h R + h (cid:48) (cid:19) R N P + 2 N h R N − ∆ P , (2.14) ddτ (cid:2) ϕ R N h ( R ) (cid:3) = 0 . (2.15)The latter equation is automatically satisfied with ϕ given by (2.11) and taking into account thedefinitions of the metric functions (2.3). Similarly, Eq. (2.14) is obeyed with the energy density,pressure and pressure anisotropy prescribed by expressions (2.9), (2.10) and (2.12), respectively.Equation (2.15) simply expresses the conservation of the shell’s angular momentum during evo-lution. Indeed, the quantity within brackets is proportional to the jump in the angular momentumacross the shell, ϕ R N h ( R ) = − [[ M a ]]4 πκ N . (2.16)As for Eq. (2.14), it affords a clear interpretation: the (intrinsic) energy gained by the shell as it shrinksis accounted for by the work done by the pressure components. The shell’s surface area is proportionalto h ( R ) R N so a change in radius d R implies a change in area equal to (cid:0) N h R + h (cid:48) (cid:1) R N d R . Thisgives precisely the factor in front of the isotropic pressure component P . The factor appearing in frontof the pressure anisotropy ∆ P is instead 2 N h R N − , because this component only acts on the CP N coordinates x a , i.e., it is not sensitive to changes in area along the ψ direction. The standard energy conditions are generally specified as inequalities imposed on the stress-energytensor, when contracted with arbitrary timelike or null vectors [42]. In practice, it is useful to translatethis into explicit constraints on the stress-energy components. When applied to a perfect fluid, thisyields very simple inequalities to be satisfied by the energy density and pressure. For the case ofimperfect (viscous) fluids, such conditions have been worked out in Ref. [43]. We are unaware of anyexplicit energy conditions in the literature concerning the sort of anisotropic fluids we consider in thiswork. Therefore, we shall derive them in this section.The energy conditions are most conveniently expressed in terms of the eigenvalues of the stress-energy tensor (2.8). These are obtained as the coefficients λ n , with n = 0 , . . . , N + 1, such thatdet[ S ij − λ n g ij ] = 0 , (3.1)and they are given by [28] λ = P − ρ − (cid:115)(cid:18) P + ρ (cid:19) − ϕ , (3.2) λ = P − ρ (cid:115)(cid:18) P + ρ (cid:19) − ϕ , (3.3) λ α = P + ∆ P , α = 2 , . . . , N + 1 . (3.4)Some comments are in order: canning the parameter space of collapsing rotating thin shells • Firstly, the 2 N eigenvalues λ α are degenerate and it can be checked that their associated eigen-vectors are all spacelike. • One needs ( P + ρ ) − ϕ ≥ real eigenvalues λ and λ (as well as thecorresponding eigenvectors). Otherwise, there are only 2 N spacelike eigenvectors and the stress-energy tensor is of type IV. In this case, it cannot even satisfy the weak energy condition [44]. • In the limiting case ( P + ρ ) − ϕ = 0 there is —in addition to the 2 N spacelike eigenvectorsassociated with the eigenvalues λ α — a double null eigenvector. This yields a type II stress-energytensor. • When ( P + ρ ) − ϕ > N + 1 spacelike eigenvectors and 1 timelikeeigenvector. The stress-energy tensor is of type I. The timelike eigenvector is the one associatedwith λ as long as ρ + P ≥
0, otherwise it is the one associated with λ . However, in the lattercase it follows immediately that the condition W EC below cannot be satisfied (with λ and λ interchanged). Therefore, only in the case ρ + P ≥ W EC ≡ − λ ≥ , W EC ≡ λ − λ ≥ , W EC α ≡ λ α − λ ≥ ,W EC t ≡ P + ρ ≥ , W EC r ≡ ( P + ρ ) − ϕ ≥ . (3.5)In the non-rotating case, the heat flow ϕ and the pressure anisotropy ∆ P both vanish and the weakenergy condition reduces to the familiar relations: ρ ≥ ρ + P ≥
0. In coordinate-invariant terms,the WEC requires that the double contraction of the stress-energy tensor with any timelike vector isnonnegative. The less restrictive null energy condition (NEC) possesses a similar coordinate-invariantdefinition but one considers instead null vectors, which amounts to simply omitting the first inequalityin (3.5).The more physical dominant energy condition (DEC), which is typically obeyed by ordinary clas-sical matter, imposes, in addition to (3.5), the following inequalities:
DEC ≡ − λ − λ ≥ , DEC α ≡ − λ α − λ ≥ . (3.6)In the non-rotating limit we again retrieve the well-known relations for perfect fluids, namely ρ ≥ − ρ ≤ P ≤ ρ .Some general bounds on parameters derived from these energy conditions are presented in Ap-pendix A. Equations (2.9–2.12) determine the various components of the matter stress-energy tensor as a functionof the shell’s radial location and velocity. In order to close the system of equations one must specifyan equation-of-state (EoS) relating the different components. Here we adopt a linear EoS by takingthe isotropic pressure to be proportional to the energy density, P = P ( ρ ) = wρ . (4.1)The extension of our study to a polytropic EoS is possible. The expressions become increasinglyinvolved, so we relegate them to Appendix B.Inserting Eqs. (2.9) and (2.10) into relation (4.1) we can easily integrate the equation to obtain[[ β ( R )]] = − m N +12 N w R N (1+ w ) h ( R ) w , (4.2)where m is an integration constant with dimensions of mass. We can now plug in the expressions for β ± , which depend on the shell’s velocity as defined in (2.13). The resulting equation can be cast in canning the parameter space of collapsing rotating thin shells R + V eff ( R ) = 0 , (4.3)where the effective potential V eff is given explicitly by V eff ( R ) = 1 + 2 M a R N +2 − M + + M − R N − (cid:18) M + − M − m (cid:19) (cid:18) R N m (cid:19) N +1 N w (cid:18) M a R N +2 (cid:19) w − − (cid:16) m R N (cid:17) N +1 N w (cid:18) M a R N +2 (cid:19) − w . (4.4)From its definition (4.3), classically allowed motion of the shell is restricted to radii satisfying V eff ( R ) ≤ V eff ( R ) = 0.Recall that ˙ R = d R /dτ so, to obtain the velocity (squared) as seen by an asymptotic observer,one must convert from the shell’s proper time τ to T . This is accomplished by using (2.4), and theresulting radial potential becomes (cid:98) V ≡ − (cid:18) d R d T (cid:19) = R h ( R ) g ( R ) − V eff g ( R ) − − V eff . (4.5)In particular, when the shell approaches a horizon —where g − vanishes— the asymptotic observersees it slowing down to zero velocity, as expected. Also, a diverging d R /dτ when the shell is takento infinity acquires a sound physical meaning: an asymptotic observer sees the shell approaching thespeed of light, d R /d T →
We start by considering the case of rotating shells composed of dust. By ‘dust’ we mean that thematter does not experience any isotropic pressure. In terms of the EoS parameter, this translates into w = 0.Now, in this case Eq. (4.2) evaluates to[[ β ( R )]] = β + − β − = − m R N . (4.6)The jump in the gravitational energy across the shell is then given by∆ M = M + − M − = m β + + β − ) (cid:18) M a R N +2 (cid:19) . (4.7)This is easily derived by noting that ( β + − β − )( β + + β − ) = β − β − = ( g − − g − − ) R /h ( R ) and itshows that ∆ M ≥ m is positive. We can equaly express this as∆ M = Em , (4.8)where the energy per unit proper mass is defined by E ≡ ( β + + β − )2 (cid:18) M a R N +2 (cid:19) . (4.9)Although not apparent from this expression, Eq. (4.8) shows that E must be a constant of motion.By squaring the relation (4.7) and inserting the definition of β ± , one can can express the totalADM mass in terms of the remaining quantities, M + = M − + m (cid:114) M a R N +2 (cid:114) R + 2 M a R N +2 − M − R N − m R N (cid:18) M a R N +2 (cid:19) . (4.10) canning the parameter space of collapsing rotating thin shells a = 0), one easily recognises the different contributions to the change ∆ M in the total energy due to the shell: the square root term gives the relativistic kinetic energy of theshell (including rest mass), while the negative contribution proportional to m represents the bindingenergy (see Ref. [31], section 3.9). As expected, the presence of a black hole in the interior affectsthe total energy, yielding additional binding. We see that when rotation is included there are extracontributions to both the kinetic and the binding energy.When the shell is taken to infinity, Eq. (4.10) reduces to M + (cid:39) M − + m (cid:112) R , which alsoshows that ∆ M ≥ m for these dust shells, i.e., E ≥
1. Thus, we conclude that (for w = 0) E = (cid:112) R (cid:12)(cid:12)(cid:12) R→∞ , (4.11)and having E > intrinsic energy of the shell, as seen by an observer comoving with the shell, can be computedas an integral over the volume of the shell, E = (cid:90) π dψ (cid:90) d N x (cid:112) det γ αβ n i u j S ij , (4.12)where γ αβ is the induced metric on a τ = const. surface, n i = (1 , ,(cid:126)
0) is a unit normal to this surfaceand u j = ( ∂ τ ) j is the unit timelike vector (the comoving observer velocity). Thus we obtain E = 2 π Vol( CP N ) h ( R ) R N ρ . (4.13)Inserting the result for the energy density, Eq. (2.9), one finds E = (2 N + 1) A N +1 πκ N m (cid:20) N N + 1 2 M a R N +2 (cid:21) , (4.14)where A N +1 is the area of a unit 2 N + 1-dimensional sphere. Thus, we conclude that the rotationcontributes to the shell’s intrinsic energy. We can now fix the N -dependent factor so that the intrinsicenergy E precisely matches the rest mass m at infinity: κ N = (2 N + 1) A N +1 π = (2 N + 1) π N N + 1) . (4.15)For N = 1 / D = 4) we indeed get E| R→∞ = m with κ / = 1.Observe that, in the presence of rotation, the shell’s intrinsic energy E is not conserved, in contrastwith E . In fact, we already saw the origin of the non-conservation of E : it can be traced back to thework done by the pressure, see Eq. (2.14). (Even in the case w = 0 there is a nonvanishing pressureanisotropy component ∆ P ). The crucial difference between E and Em is that the former does notinclude the energy stored in the gravitational field. Next consider a = 0 and w (cid:54) = 0, in which case we must take into account the w -dependence of theradial potential, which is derived from Eq. (4.2). The difference between the exterior and interiorgravitational masses is now given by∆ M = M + − M − = m β + + β − ) (cid:16) m R N (cid:17) N +12 N w , (4.16)and the total ADM mass is M + = M − + m (cid:16) m R N (cid:17) N +12 N w (cid:114) R − M − R N − m R N (cid:16) m R N (cid:17) N +1 N w . (4.17)For large R (and assuming w ≥
0) we get M + (cid:39) M − + m (cid:16) m R N (cid:17) N +12 N w (cid:112) R . (4.18) canning the parameter space of collapsing rotating thin shells M + ≥ M − as long as the constant m N +1 N w is non-negative. (A scenario with M + < M − would necessarily require m N +1 N w < E = A N +1 (2 N + 1) m πκ N (cid:16) m R N (cid:17) N +12 N w = m (cid:16) m R N (cid:17) N +12 N w . (4.19)We see that, if w >
0, the intrinsic energy of the shell vanishes as it approaches infinity. This mightseem suspicious at first sight, but it is in accordance with our earlier comments: the shell loosesintrinsic energy as R increases, and this goes into work done by the pressure. Finally, we arrive at the most general case, a (cid:54) = 0 and w (cid:54) = 0. Using Eq. (4.2), the difference betweenthe exterior and interior gravitational masses is now given by∆ M = M + − M − = m β + + β − ) (cid:16) m R N (cid:17) N +12 N w (cid:18) M a R N +2 (cid:19) − w/ . (4.20)Once again, from this we can obtain the total ADM mass, M + = M − + m (cid:16) m R N (cid:17) N +12 N w (cid:18) M a R N +2 (cid:19) − w (cid:114) R − M − R N + 2 M a R N +2 − m R N (cid:16) m R N (cid:17) N +1 N w (cid:18) M a R N +2 (cid:19) − w . (4.21)For large R the rotation terms are subdominant and we recover Eq. (4.18).The radial effective potential was already presented in (4.4) for the general case including bothrotation and isotropic pressure. A careful inspection of its expression reveals that for − N N +1 < w < w > M (cid:54) = 0 then the shell’s velocity approaches that of light at infinity: ˙ R ∼ R (2 N +1) w → ∞ , and aswe saw previously —converting to Boyer-Lindquist coordinates— this corresponds to d R /d T = 1. If w < − N N +1 , the shell can also be sent in from infinity, and in this case it approaches the speed oflight at a rate given by ˙ R ∼ (cid:16) R /m / (2 N )0 (cid:17) − N − (2 N +1) w → ∞ . (4.22)Such choices of negative w can result in either plunges or bounces, and energy conditions can besatisfied or violated depending on the parameters. For full plunges in D = 5 dimensions, the DECis always violated for sufficiently small values of R , as in the w ≥ w ≥
0. However, we notethat in higher dimensions, N ≥
2, there exist full plunges with w <
In this section we will present our results concerning the outcome of the numerical scan of the parameterspace describing collapses of rotating thin shells, with all independent angular momenta set equal.The first point to notice is that the dimensionality of the parameter space is quite large: we havea total of four continuous parameters to vary, { m , M a , E, w } , plus one discontinuous parameter,namely the spacetime dimensionality D = 2 N + 3. Even though we analyze different N valuesindependently, the dimensionality of the parameter space is too large to numerically explore entirelyand, for that matter, to efficiently represent in a single figure. In order to display our results we willtherefore present selected sections, by fixing values of D , w and E . These sections are representative canning the parameter space of collapsing rotating thin shells : • Full plunges (FP):
The shell has no turning point and collapses onto the singularity. • Two-world orbits (TWO):
The shell crosses a black hole horizon, has a turning point (there-fore avoiding the singularity) and then exits a white hole horizon into a distinct universe. Fromthe point of view of an asymptotic observer, the shell simply falls into the black hole, taking aninfinity time to do so. • True bounces (TB):
The shell has a turning point occurring outside any existing horizon (ifthere is one). This is a time-symmetric orbit and after reaching a minimum radius the shelldisperses back to infinity.In addition to this classification, we also evaluate explicitly whether the WEC and the DEC aresatisfied throughout the shell’s orbit. All the results presented in the following remain unaltered ifwe consider the NEC instead of the weak energy condition. The difference between them is just theinclusion or not of a single inequality (
W EC ≥ m and M a ) and evaluating the quantities ofinterest. The other —more efficient— strategy is to obtain directly the curves that separate differentregions in the phase space. For example, the lines marking the boundary between full plunges andbounces are computed by imposing that a local maximum of the effective potential takes the value V eff = 0. This corresponds to the critical configurations we are looking for: small changes in theparameters can raise or lower the potential barrier above or below zero, yielding a bounce or a plungetrajectory, respectively.The fact that we obtained fully consistent results with the two approaches serves as a good checkon our calculations.A word of caution is in order. In dimensions D ≥
7, and for some choices of the parameters,our shell evolutions can originate highly spinning black holes (nearly extremal) that are known tobe unstable [38, 46]. In these extreme cases, the resulting cohomogeneity-1 Myers-Perry black holecannot be expected to be the endpoint of the collapse. w = 0 , E = 1 , varying D As pointed out in [27], having the thin shell initially at rest at infinity requires that we consider w = 0and m = ∆ M = M + − M − , at least when restricting to linear EoS. In this case the effective potentialreduces to V eff ( R ) = 1 + 2 M a R N +2 − M − + m R N − (cid:18) M a R N +2 (cid:19) − − (cid:16) m R N (cid:17) (cid:18) M a R N +2 (cid:19) . (5.1)Note that there is a scale invariance in the problem: the shell’s equation of motion remainsunchanged when the masses M − and m are rescaled by a factor κ , while the radius R , the spin a and the proper time τ are rescaled by a factor κ / (2 N ) . As a consequence we can, without loss ofgenerality, set M − = 1. This allows us to reduce the dimensionality of the parameter space to just 2:these collapses depend only on m and M a , up to a trivial rescaling.In Appendix A it is shown that under these conditions ( w = 0 and m = ∆ M ) and imposing M a < ( M a ) max so that the interior spacetime has an event horizon, the WEC is satisfied (violated) This classification is similar to what is done in Ref. [45] for the orbits of test particles in five-dimensional rotatingblack hole spacetimes. canning the parameter space of collapsing rotating thin shells - - m M a N =
1, w =
0, E = FP1FP2 TWO1
TWO2
TB1TWO3TB2
AC D BFE - - m M a N =
2, w =
0, E = FP1FP2
TWO1
TWO2
TB1
TWO3
TB2 - - m M a N =
3, w =
0, E = FP1FP2 TWO1TWO2TB1TWO3TB2 - - m M a N =
4, w =
0, E = FP1FP2 TWO1TWO2TB1TWO3TB2
Figure 1: Plots of the { m , M a } parameter space fixing E = 1 and w = 0. We display results forfour different spacetime dimensions, namely D = 5 , , ,
11, corresponding to N = 1 , , ,
4. The blueregions (FP1) indicate full plunges satisfying the WEC (though all of them violate the DEC), whilethe green regions (FP2) correspond to full plunges violating the WEC. Light-purple regions (TWO1)indicate two-world orbits satisfying the WEC but not the DEC, and light-orange regions (TWO2)correspond to two-world orbits satisfying the WEC and the DEC. True bounces respecting both theWEC and the DEC are indicated by dark-orange domains (TB1). The light-gray regions (TWO3)correspond to two-world orbits that violate the WEC and those colored dark-gray (TB2) identify truebounces violating the WEC. The dashed black curve indicates the maximum value of
M a for whichthe external geometry possesses an horizon (and therefore corresponds to an extremal black hole).The horizontal dashed red line indicates a similar situation but for the interior geometry. The pointsmarked A–F in the first panel were chosen each one in a different region, and the respective plotsshowing the potential, as well as the WEC and DEC curves, are displayed in Fig. 2.if m > m < DEC , but DEC α is violated at sufficientlysmall R , which is necessarily explored by full plunges —though this occurs always inside the exteriorhorizon, when there exists one.In Fig. 1 we display our results for the scan of the parameter space { m , M a } , for the four lowestspacetime dimensionalities we can consider: D = 5 , , ,
11. Different regions are identified accordingto the classification above (full plunges, two-world orbits or true bounces), indicating also whetherthe geometries interior and exterior to the shell correspond to a black hole (below the red and blackdashed curves) or a naked singularity (above the red and black dashed curves). canning the parameter space of collapsing rotating thin shells m < M a , while for m > m = 0 divides the plots in two regions, and the right-hand half of the plots correspondsto physically reasonable matter content on the shell.Fig. 1 also provides visual confirmation of a point made above: the blue (FP1) and the orangeregions (TB1 and TWO2) never intersect, i.e., for these cases ( w = 0) full plunges always violate theDEC, even when the WEC is satisfied. This is more obvious in the first panel, D = 5, where domainsFP1 and TWO2 are separated by an intermediate light-purple region, TWO1. For higher dimensions, N ≥
2, it is clear that this region extends only up to a finite value of m , beyond which the full plungeregion FP1 and the two-world orbit region TWO2 become contiguous.There are two notable conclusions that can be inferred from Fig. 1, especially from the last threepanels. One is that true bounces only occur when the geometry exterior to the shell corresponds toa naked singularity, i.e., above the dashed black line. This will change when we consider shells with w (cid:54) = 0. Another is that full plunges satisfying the WEC always form an exterior horizon, while fullplunges violating the WEC ( m <
0) only occur when the interior geometry has a black hole. Themarginal orbits lying at the border of the full plunge regions FP1 and FP2 become tangent at onepoint to the black and red dashed curves, respectively. These special points indicate the appearanceof a turning point that coincides with the black hole horizon radius.In all panels, the physically more interesting quadrant is the lower-right one, where the WEC issatisfied and the geometry interior to the shell is dressed. For the case w = 0 we are considering, notrue bounces can be found in this quadrant. Note that as the dimensionality N increases, more of thisregion is covered by full plunges; for N = 4 only a small corner corresponds to TWOs, and the tinyregion where the DEC is satisfied occurs only for shells falling through near extremal black holes.At last, we comment on the implications of our results regarding cosmic censorship. From thispoint of view, the potentially dangerous situation is the one in which the shell fully collapses ontoa pre-existing black hole (below the red dashed line) and ends up with a naked singularity (abovethe black dashed line). This corresponds to the FP2 region, which necessarily violates the WEC.The domain TB2 also starts off with a black hole in the interior and an over-extreme exterior, and itdescribes the temporary appearance of a naked singularity, followed by the re-creation of the horizonafter the shell bounces. In any case, the WEC is also violated for TB2, since the two dashed linescross exactly at m = 0. In summary, these results are in accordance with the weak cosmic censorshipconjecture. D = 5 , w = 0 , varying E Having studied the effect of dimensionality on the space of collapses, we will now fix N = 1 for theremainder of the paper, i.e., we consider a five-dimensional spacetime. In this subsection we allow forthe infalling shell to start with finite velocity at infinity, by varying the energy parameter E >
1. Forconcreteness, we take w = 0, corresponding to shells with vanishing isotropic pressure component.For our calculations it is convenient to work with the energy parameter E , since it directly connectsthe ADM mass of the exterior spacetime M + with the proper mass of the shell m [see Eq. (4.8)].However, for presentation purposes it is useful to translate this into a more intuitive quantity, such asthe velocity of the shell at infinity, expressed as a fraction of the speed of light. The relation betweenthe two is straightforwardly obtained using (4.5) and (4.11), (cid:12)(cid:12)(cid:12)(cid:12) d R d T (cid:12)(cid:12)(cid:12)(cid:12) R→∞ = √ E − E . (5.2)Therefore, the previous case E = 1 indeed corresponds to shells starting from rest at infinity, but E = 1 . .
3% the speed of light.The first significant difference between this case and the previous subsection is that now the WECcan be violated even for m >
0, as can be seen in the last two panels of Fig. 3. For E = 10 all theshell trajectories in the parameter space scanned violate the WEC.As the energy parameter E is increased, the region corresponding to plunges with positive propermass m grows, as expected. Raising the value of E also shrinks the region where the DEC is satisfied. canning the parameter space of collapsing rotating thin shells - - R N = m = = =
0, E = V eff × WEC × WEC × WEC α × WEC t × WEC r × DEC × DEC α r h - r h + V eff A - - R N = m = = =
0, E = V eff × WEC × WEC × WEC α × WEC t × WEC r × DEC × DEC α r h - r h + V eff B - - R N = m = - = =
0, E = V eff × WEC × WEC × WEC α × WEC t × WEC r × DEC × DEC α r h - V eff C - - R N = m = - = =
0, E = V eff × WEC × WEC × WEC α × WEC t × WEC r × DEC × DEC α r h - V eff D - - R N = m = - = =
0, E = V eff × WEC × WEC × WEC α × WEC t × WEC r × DEC × DEC α r h - r h + V eff E - - R N = m = = =
0, E = V eff × WEC × WEC × WEC α × WEC t × WEC r × DEC × DEC α r h - r h + V eff F Figure 2: Plots of the radial potential V eff and the various energy conditions (3.5) and (3.6), corre-sponding to choices of parameters indicated by the points A–F in the first panel of Fig. 1. The variousWEC and DEC conditions are satisfied only when the respective graphs are positive. Panels A and B represent a full plunge and a two-world orbit, respectively, both with WEC satisfied and DEC α violated inside the horizons. Panels C and D correspond to a full plunge and a true bounce (withouthorizon in the exterior), respectively, both with WEC violated. Panel E represents a two-world orbitwith WEC violated, and panel F shows a two-world orbit in which both WEC and DEC are satisfied(note that DEC α would be violated only inside the turning point). The vertical dashed lines indicatethe locations of the horizons for the geometries exterior (black) and interior (red) to the shell. canning the parameter space of collapsing rotating thin shells - - m M a N =
1, w =
0, E = FP1FP2 TWO1
TWO2
TB1TWO3TB2 - - m M a N =
1, w =
0, E = FP1FP2 TWO1
TWO2
TB1TWO3TB2 - - m M a N =
1, w =
0, E = FP1FP2FP2 TWO3TWO1TB2 - - m M a N =
1, w =
0, E = FP2FP2 TWO3TWO3 TB2
Figure 3: Plots of the { m , M a } parameter space fixing N = 1 —corresponding to five-dimensionalspacetimes— and w = 0. We display results for four different choices of the energy parameter E =1 , . , ,
10. The top left panel is a repetition of the first panel of Fig. 1 but we include it here to easethe comparison. The remaining three values translate into the shell having finite velocity at infinity,given respectively by 55 . , .
6% and 99 .
5% the speed of light. The color coding and region namesare the same as in Fig. 1.For E = 2 ,
10 it is not satisfied anywhere in the scanned space. Notice that the minimum of the blackdashed parabola moves to the right as E increases, while it always intersects the red dashed horizontalline ( M a = 1 /
2) at m = 0. As a result, the region corresponding to true bounces shrinks, and forvery large values of E it gets squeezed into a small interval around m = 0.Recall the region of interest for the cosmic censorship conjecture is below the red dashed line andabove the black dashed line, which gets smaller as E is increased. Once again, collapsing shells ontoblack holes result in naked singularities only in regions where the WEC is violated, in accordance withcosmic censorship. D = 5 , varying w and E ) For the more general case of matter shells with a nonvanishing isotropic pressure component, w (cid:54) = 0,it is not possible to have the shell starting from rest at infinity. Also there is no reason to impose m = ∆ M , so there is one more free parameter, namely E . So now we have a four-dimensionalparameter space: { m , M a , w, E } .It can be shown that the weak energy condition is violated for EoS parameter w > (2 N +1) − . This canning the parameter space of collapsing rotating thin shells R , seeAppendix A. It can also be shown that DEC α ≥ R unless w < − /N .In what follows we focus our attention on positive w , but in Appendix A we briefly consider the caseof negative w , which translates into a negative pressure P , i.e., a tension . This case is interestingbecause the tension of the shell will assist the gravitational collapse, creating what might seem to bemore favorable conditions to destroy the horizon and form a naked singularity. Nevertheless, there isonly a narrow window allowing for negative w shells to be thrown from infinity, while satisfying thedominant energy condition, − ≤ w < min (cid:26) − N N + 1 , − N (cid:27) . (5.3)The lower bound is derived from W EC t ≥
0, see Eq. (3.5). The upper bound comes from either V eff ( R → ∞ ) ≤ DEC α | R→ ≥
0, see Appendix A. In any case, we never observe naked singularityformation from the collapse of shells (satisfying the DEC) onto black holes.It is evident, from a glance at Eqs. (4.2) and (4.4), that a nonzero w easily leads to non-integerexponents of m in the radial potential and in the energy conditions. For this reason, here we restrictthe scanned region to m >
0. Otherwise, not even the weak energy condition could be satisfied withthe shell at infinity. Just like in the cases w = 0 , E > m > w = 0 . w = 0 . w = 0 .
1, with the choice of energy parameter E = 0 .
25, resultsin an uninteresting parameter space almost entirely filled by the TB1 region, with very small domainscorresponding to full plunges and two-world orbits. Nevertheless, the plots presented are sufficientto infer the trend followed when varying the EoS parameter w . The points G–J marked in somepanels of Fig. 4 were chosen as representatives of four different regions, and the respective plots of thepotential, as well as the WEC and DEC constraints, are shown below in Fig. 5.Compared to the previous cases of rotating ‘dust’ shells, the consideration of w (cid:54) = 0 brings abouttwo notable new features. One is that we can now have true bounces in the presence of both interiorand exterior horizons —for the previous cases with w = 0 true bounces were only allowed abovethe black dashed line, i.e., when there was no exterior horizon. Another remarkable difference is theexistence of a novel dark-purple region (TB3) which corresponds to a true bounce satisfying the WECbut violating the DEC.With a nonvanishing isotropic component of the pressure, there is the possibility of having, inaddition to a centrifugal potential barrier at R ∼ ( M a ) / (2 N +2) , also a pressure barrier at R ∼ m N ) E − N +1) w . [The position of these features in the radial potential can be straightforwardlyinferred from Eq. (4.4).] This property can be observed in Fig. 5, especially in panel H where thetwo maxima of the potential are more evident. It is exactly this new pressure barrier that permits atrue bounce outside the exterior horizon satisfying the DEC — the centrifugal barrier occurs insidethe horizon, and if we turned off the isotropic component of the pressure we would get a two-worldorbit instead of a true bounce. The differences between panels G and H are only a consequence ofchanging the proper mass parameter m . For some intermediate value one would find a marginalorbit, a true bounce on the verge of becoming a two-world orbit. Such trajectories are associated withthe nearly-vertical solid black line in the first panel of Fig. 4, between the TWO1 and TB1 regions.Fig. 4 allows us to infer some general behavior as the parameters w and E are varied. For the same w , increasing E shifts the regions to the right (to larger values of m ). The same effect is obtained byincreasing w for fixed E . Increasing the value of either E or w shrinks the domains where the WECand the DEC are satisfied. For E = 0 .
25 the DEC is satisfied in most part of the scanned region(dominated by TB1), but for E = 1 the DEC is not satisfied anywhere, and the WEC is just satisfiedin a small region (bottom-right corner).Just like in the previous cases, we find no shell trajectories that would correspond to violations ofcosmic censorship. Since we only plot the parameter space with m > canning the parameter space of collapsing rotating thin shells m M a FP1TWO1 TWO2
TB1
TB3 HG N =
1, w = = m M a N =
1, w = = FP1TWO1 TWO2
TB1
TB2TB3TWO3 m M a N =
1, w = = FP1TWO1 TWO2
TB1
TB2TB3TWO3 m M a N =
1, w = = FP1TWO1
TB1
TB3TB2TWO3 IJ m M a N =
1, w = = FP1TWO1TB2 TB3TWO3 m M a N =
1, w = = FP1TWO1TB2 TWO3
Figure 4: Plots of the { m , M a } parameter space fixing N = 1, corresponding to five-dimensionalspacetimes. All combinations with w = 0 . , . E = 0 . , . , canning the parameter space of collapsing rotating thin shells - - R N = m = = = = V eff × WEC × WEC × WEC α × WEC t × WEC r × DEC × DEC α r h - r h + V eff G - - R N = m = = = = V eff × WEC × WEC × WEC α × WEC t × WEC r × DEC × DEC α r h - r h + V eff H - - R N = m = = = = V eff × WEC × WEC × WEC α × WEC t × WEC r × DEC × DEC α V eff I - - R N = m = = = = V eff × WEC × WEC × WEC α × WEC t × WEC r × DEC × DEC α r h - r h + V eff J Figure 5: Plots of the radial potential V eff and the various energy conditions (3.5) and (3.6), cor-responding to choices of parameters indicated by the points G–J in the first and fourth panels ofFig. 4. Panel G represents a two-world orbit satisfying the WEC but violating the DEC once insidethe horizon. Panel H corresponds to a true bounce where both WEC and DEC are satisfied. Panel I represents a true bounce off a naked singularity, with WEC satisfied and DEC violated. Panel J illustrates a two-world orbit with WEC violated (and therefore also violating the DEC). Acknowledgement
We thank T´erence Delsate for initial collaboration on this project. We would also like to thankMasashi Kimura, Jos´e Lemos, Jos´e Nat´ario, Alberto Saa and Jorge Santos for useful comments andcorrespondence. JVR is grateful to
Universidade Federal de S˜ao Carlos and
Universidade Estadualde Campinas , for their kind hospitality. JVR and RS would also like to thank FAPESP grants2013/09357-9 and 2016/01343-7 for funding a visit to ICTP-SAIFR in May of 2017, where part ofthis work was done. JVR acknowledges financial support from the European Union’s Horizon 2020research and innovation programme under the Marie Sk(cid:32)lodowska-Curie grant agreement No REGMat-2014-656882 and under ERC Advanced Grant GravBHs-692951. JVR was partially supported by theSpanish MINECO under Project No. FPA2013-46570-C2-2-P. canning the parameter space of collapsing rotating thin shells A Bounds on parameters from energy conditions
In order to satisfy the energy conditions (null, weak or dominant) we need at the very least that thediscriminant in Eqs. (3.2) and (3.3) be non-negative, i.e.,
W EC r ≥ (cid:18) m N +12 N w (cid:19) ≥ N + 1) M a ( √ M + − √ M − ) R N +1) w (1 + w ) R (cid:16) Ma R N +2 (cid:17) − w (cid:16) N + N Ma R N +2 (cid:17) , (A.1)whose compliance at large R implies an upper bound on the EoS parameter, w ≤ N + 1 . (A.2)Assuming this holds, a further constraint can be inferred by studying the large R behavior of theremaining conditions (3.5) and (3.6). One finds that W EC and DEC α are automatically positive as R → ∞ , while
W EC , W EC α , W EC t and DEC are positive as R → ∞ if and only if m N +12 N w ≥ . (A.3)The bounds obtained above are necessary conditions for the WEC and DEC to be satisfied. Par-ticularizing to the case of shells starting from rest at infinity ( w = 0 and E = 1) and collapsing ontoblack holes —as opposed to naked singularities— we can give sufficient conditions for the validity ofthe WEC. Consider first the constraint W EC r ≥
0. Inspection of (A.1) shows that it is satisfiedat both large and small R . The question is whether W EC r ≥ all values of R , which a fullycollapsing shell will necessarily explore. For w = 0 we can write (A.1) as m ( √ M + − √ M − ) ≥ N + 1) M a R (cid:16) Ma R N +2 (cid:17) (cid:16) N + N Ma R N +2 (cid:17) ≡ Λ( R , M a , N ) . (A.4)It can be easily shown that Λ, as a function of R , has a maximum value of the formΛ max ( M a , N ) = c N ( M a ) NN +1 , (A.5)where c N is a dimension-dependent constant that satisfies 1 < c N <
4. Requiring the interior geometryto have a horizon covering the singularity translates into an upper bound on the spin:
M a ≤ ( M a ) max ≡ (cid:34) NN + 1 (cid:18) N (cid:19) / ( N +1) M − (cid:35) N +1 N . (A.6)Thus, we see that the right hand side of (A.4) is bounded from above byΛ( R , M a , N ) ≤ Λ max ( M a , N ) < M − . (A.7)On the other hand, since we are assuming E = 1 the left hand side is bounded from below: m ( √ M + − √ M − ) = ( (cid:112) M + + (cid:112) M − ) ≥ (2 (cid:112) M − ) = 4 M − , (A.8)Thus, inequality (A.1) is satisfied for all R when w = 0 and E = 1.Moreover, given expressions (2.9–2.12) and (4.2), the positivity of W EC , W EC , W EC α and W EC t follows straightforwardly if m ≥
0; otherwise these conditions are violated. (The same thingholds for
DEC .) In conclusion, for the case w = 0, starting from rest at infinity and excludingover-extremal interior geometries, the shell’s matter respects the weak energy condition throughoutits entire motion if and only if m ≥ canning the parameter space of collapsing rotating thin shells - - R N = m = = = - = V eff × WEC × WEC × WEC α × WEC t × WEC r × DEC × DEC α r h - r h + V eff - - R N = m = = = - = V eff × WEC × WEC × WEC α × WEC t × WEC r × DEC × DEC α r h - r h + V eff Figure 6: Examples of a full plunge (left) and a true bounce (right), satisfying both the weak anddominant energy conditions, for N = 2 (i.e., a seven-dimensional spacetime) and with w = − .
9. Thisvalue was chosen as representative within the narrow interval (5.3) corresponding to tense (negative w ) shells that are classically allowed to come in from infinity and that satisfy the DEC at arbitrarilysmall radii.However, the status of the dominant energy condition differs from this. In particular, the condition DEC α = ρ − P − ∆ P + (cid:115)(cid:18) ρ + P (cid:19) − ϕ ≥ R , where the pressure anisotropy term dominates. Note thatRef. [47] suggested that arbitrarily small over-spinning objects can exist, at least as transients, butwithin our construction it does not seem possible to obtain arbitrarily small over-spinning dust shellssatisfying the dominant energy condition.Nevertheless, if the exterior geometry has an event horizon (and m ≥ R v , determined by DEC α | R = R v = 0, or equivalently (cid:18) M a R N +2 v (cid:19) (cid:18) N − N M a R N +2 v (cid:19) = 2( N + 1) R Nv m ( √ M + − √ M − ) m , (A.10)is smaller than the radius of the exterior horizon, r + h , which obeys1 − M + ( r + h ) N + 2 M a ( r + h ) N +2 = 0 . (A.11)Indeed, it is not hard to see that DEC α | R = r + h ≥ R v < r + h .There is a possibility that the dominant energy condition is satisfied even for full plunges, but thisrequires a negative isotropic pressure. Assuming that W EC r ≥ DEC α that dominates at small radii is DEC α | R→ ∝ − (1 + N w ) R − − (3+ w ) N . (A.12)Thus, for w < − /N there is a chance that the DEC is satisfied everywhere. This is indeed the case,as shown explicitly in Fig. 6. However, since the condition W EC t ≥ w ≥ −
1, it isclear that this scenario requires N ≥ canning the parameter space of collapsing rotating thin shells B Polytropic equation-of-state
Polytropic equations-of-state refer to non-linear relations between the pressure and energy density ofthe following form: P = wρ (cid:18) ργ (cid:19) /n , (B.1)where n is known as the polytropic index. In the limit n → ∞ one recovers the linear EoS consideredin the main text.Inserting expressions (2.9) and (2.10) into this relation we obtain dd R (cid:104)(cid:0) −R N [[ β ( R )]] (cid:1) − /n (cid:105) = wn (8 πγ ) /n (cid:0) dd R [ R N h ( R )] (cid:1) /n R N + N +1 n h ( R ) . (B.2)Upon integration, this gives [[ β ( R )]] = − πγ R N (cid:18) nw I ( R ) (cid:19) n , (B.3)where I ( R ) = (cid:90) d R (cid:0) dd R [ R N h ( R )] (cid:1) /n R N + N +1 n h ( R ) . (B.4)It turns out the integral (B.4) can be computed in terms of hypergeometric functions. The result is I ( R ) = n R − N +1 n N + 2 (cid:18) N M a R h ( R ) (cid:19) n (cid:18) R N +2 M a (cid:19) n × (cid:20) ( N + 1) F (cid:18) − N + 2(2 N + 2) n ; 1 + 12 n , − n ; 1 − N + 2(2 N + 2) n ; − R N +2 M a , − (2 N + 1) R N +2 N M a (cid:19) − (2 N + 1) F (cid:18) − N + 2(2 N + 2) n ; 12 n , − n ; 1 − N + 2(2 N + 2) n ; − R N +2 M a , − (2 N + 1) R N +2 N M a (cid:19)(cid:21) , (B.5)where F ( a ; b, c ; d ; x, y ) denotes the two-variable Appell hypergeometric function.Analogously to the case of a linear equation-of-state, we can again obtain the radial equation ofmotion for the shell in the form (4.3), but now the effective potential is given by V eff ( R ) = 12 (cid:0) g − ( R ) + g − − ( R ) (cid:1) − h ( R ) R (cid:18) πγ R N (cid:18) nw I ( R ) (cid:19) n (cid:19) − R h ( R ) (cid:0) g − ( R ) − g − − ( R ) (cid:1) (cid:16) πγ R N (cid:16) nw I ( R ) (cid:17) n (cid:17) = 1 − M − + M + R N + 2 M a R N +2 − (cid:18) M a R N +2 (cid:19) (cid:18) πγ R N (cid:19) (cid:18) nw I ( R ) (cid:19) n − (cid:18) M a R N +2 (cid:19) − (cid:18) πγ R N (cid:19) − (cid:18) nw I ( R ) (cid:19) − n (cid:18) M + − M − R N (cid:19) . (B.6)Having determined the radial potential, one can proceed to study the possible shell trajectories as theparameters are varied, along the lines of Section 5. Such an analysis is beyond the scope of this paperand merits a separate study on its own. References [1] J. R. Oppenheimer and H. Snyder, “On Continued gravitational contraction,” Phys. Rev. , 455(1939).[2] T. Nakamura, “General Relativistic Collapse of Axially Symmetric Stars Leading to the Formationof Rotating Black Holes,” Prog. Theor. Phys. , 1876 (1981). canning the parameter space of collapsing rotating thin shells , 891 (1985) Erratum: [Phys. Rev. Lett. , 97 (1986)].[4] R. Penrose, “Gravitational collapse: The role of general relativity,” Riv. Nuovo Cimento , 252(1969) [Gen. Relativ. Gravit. , 1141 (2002)].[5] M. Shibata, “Axisymmetric simulations of rotating stellar collapse in full general relativity: Cri-teria for prompt collapse to black holes,” Prog. Theor. Phys. , 325 (2000) [gr-qc/0007049].[6] R. P. Kerr, “Gravitational field of a spinning mass as an example of algebraically special metrics,”Phys. Rev. Lett. , 237 (1963).[7] A. M. Abrahams, G. B. Cook, S. L. Shapiro and S. A. Teukolsky, “Solving Einstein’s equationsfor rotating space-times: Evolution of relativistic star clusters,” Phys. Rev. D , 5153 (1994).[8] B. Giacomazzo, L. Rezzolla and N. Stergioulas, “Collapse of differentially rotating neutron starsand cosmic censorship,” Phys. Rev. D , 024022 (2011) [arXiv:1105.0122 [gr-qc]].[9] U. Sperhake, V. Cardoso, F. Pretorius, E. Berti, T. Hinderer and N. Yunes, “Cross section, finalspin and zoom-whirl behavior in high-energy black hole collisions,” Phys. Rev. Lett. , 131102(2009) [arXiv:0907.1252 [gr-qc]].[10] R. M. Wald, “Gedanken Experiments to Destroy a Black Hole,” Annals Phys. , 548 (1974).[11] M. Bouhmadi-L´opez, V. Cardoso, A. Nerozzi and J. V. Rocha, “Black holes die hard: can onespin-up a black hole past extremality?,” Phys. Rev. D , 084051 (2010) [arXiv:1003.4295 [gr-qc]].[12] J. V. Rocha, R. Santarelli and T. Delsate, “Collapsing rotating shells in Myers-Perry-AdS space-time: A perturbative approach,” Phys. Rev. D , 104006 (2014) [arXiv:1402.4161 [gr-qc]].[13] J. V. Rocha and R. Santarelli, “Flowing along the edge: spinning up black holes in AdS spacetimeswith test particles,” Phys. Rev. D , no. 6, 064065 (2014) [arXiv:1402.4840 [gr-qc]].[14] J. Nat´ario, L. Queimada and R. Vicente, “Test fields cannot destroy extremal black holes,” Class.Quant. Grav. , no. 17, 175002 (2016) [arXiv:1601.06809 [gr-qc]].[15] T. Jacobson and T. P. Sotiriou, “Over-spinning a black hole with a test body,” Phys. Rev. Lett. , 141101 (2009) Erratum: [Phys. Rev. Lett. , 209903 (2009)] [arXiv:0907.4146 [gr-qc]].[16] E. Barausse, V. Cardoso and G. Khanna, “Test bodies and naked singularities: Is the self-forcethe cosmic censor?,” Phys. Rev. Lett. , 261102 (2010) [arXiv:1008.5159 [gr-qc]].[17] J. Sorce and R. M. Wald, “Gedanken Experiments to Destroy a Black Hole II: Kerr-NewmanBlack Holes Cannot be Over-Charged or Over-Spun,” Phys. Rev. D , no. 10, 104014 (2017)[arXiv:1707.05862 [gr-qc]].[18] L. Lehner and F. Pretorius, “Black Strings, Low Viscosity Fluids, and Violation of Cosmic Cen-sorship,” Phys. Rev. Lett. , 101102 (2010) [arXiv:1006.5960 [hep-th]].[19] P. Figueras, M. Kunesch and S. Tunyasuvunakool, “End Point of Black Ring Instabilitiesand the Weak Cosmic Censorship Conjecture,” Phys. Rev. Lett. , no. 7, 071102 (2016)[arXiv:1512.04532 [hep-th]].[20] P. Figueras, M. Kunesch, L. Lehner and S. Tunyasuvunakool, “End Point of the UltraspinningInstability and Violation of Cosmic Censorship,” Phys. Rev. Lett. , no. 15, 151103 (2017)[arXiv:1702.01755 [hep-th]].[21] J. M. Cohen, “Gravitational Collapse of Rotating Bodies,” Phys. Rev. , 1258 (1968). canning the parameter space of collapsing rotating thin shells , 3151 (1974).[23] R. V. Wagoner, “Rotation and Gravitational Collapse,” Phys. Rev. , B1583 (1965).[24] J. Cris´ostomo and R. Olea, “Hamiltonian treatment of the gravitational collapse of thin shells,”Phys. Rev. D , 104023 (2004) [hep-th/0311054].[25] R. B. Mann, J. J. Oh and M. -I. Park, “The Role of Angular Momentum and Cosmic Cen-sorship in the (2+1)-Dimensional Rotating Shell Collapse,” Phys. Rev. D , 064005 (2009)[arXiv:0812.2297 [hep-th]].[26] C. Vaz and K. R. Koehler, “A Rotating, Inhomogeneous Dust Interior for the BTZ Black Hole,”Phys. Rev. D , 024038 (2008) [arXiv:0805.1908 [gr-qc]].[27] T. Delsate, J. V. Rocha and R. Santarelli, “Collapsing thin shells with rotation,” Phys. Rev. D , 121501(R) (2014) [arXiv:1405.1433 [gr-qc]].[28] J. V. Rocha, “Gravitational collapse with rotating thin shells and cosmic censorship,” Int. J.Mod. Phys. D , no. 09, 1542002 (2015) [arXiv:1501.06724 [gr-qc]].[29] P. Bizo´n, T. Chmaj and B. G. Schmidt, “Critical behavior in vacuum gravitational collapse in4+1 dimensions,” Phys. Rev. Lett. , 071102 (2005) [gr-qc/0506074].[30] J. Nat´ario, L. Queimada and R. Vicente, “Rotating elastic string loops in flat and black holespacetimes: stability, cosmic censorship and the Penrose process,” Class. Quant. Grav. , no. 7,075003 (2018) [arXiv:1712.05416 [gr-qc]].[31] E. Poisson, “A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics,” CambridgeUniversity Press (2004).[32] R. C. Myers and M. J. Perry, “Black Holes In Higher Dimensional Space-Times,” Annals Phys. , 304 (1986).[33] M. Shibata, H. Okawa and T. Yamamoto, “High-velocity collision of two black holes,” Phys. Rev.D , 101501 (2008) [arXiv:0810.4735 [gr-qc]].[34] D. Pollney, C. Reisswig, E. Schnetter, N. Dorband and P. Diener, “High accuracy binary blackhole simulations with an extended wave zone,” Phys. Rev. D , 044045 (2011) [arXiv:0910.3803[gr-qc]].[35] V. P. Frolov and D. Stojkovic, “Quantum radiation from a five-dimensional rotating black hole,”Phys. Rev. D , 084004 (2003) [gr-qc/0211055].[36] H. K. Kunduri, J. Lucietti and H. S. Reall, “Gravitational perturbations of higher dimensionalrotating black holes: Tensor perturbations,” Phys. Rev. D , 084021 (2006) [hep-th/0606076].[37] P. Hoxha, R. R. Martinez-Acosta and C. N. Pope, “Kaluza-Klein consistency, Killing vectors,and Kahler spaces,” Class. Quant. Grav. , 4207 (2000) [hep-th/0005172].[38] O. J. C. Dias, P. Figueras, R. Monteiro, H. S. Reall and J. E. Santos, “An instability of higher-dimensional rotating black holes,” JHEP , 076 (2010) [arXiv:1001.4527 [hep-th]].[39] W. Israel, “Singular hypersurfaces and thin shells in general relativity,” Nuovo Cim. B , 1(1966) [Erratum-ibid. B , 463 (1967)] [Nuovo Cim. B , 1 (1966)].[40] G. Darmois, “Les ´equations de la gravitation einsteinienne”, Chapitre V, M´emorial de SciencesMath´ematiques, fascicule XXV (1927). canning the parameter space of collapsing rotating thin shells , 2943 (2008)[arXiv:0804.0295 [hep-th]].[42] R. M. Wald, “General Relativity,” Chicago, USA: University Press (1984).[43] C. A. Kolassis, N. O. Santos and D. Tsoubelis, “Energy conditions for an imperfect fluid,” Class.Quant. Grav. , 419 (1991).[45] V. Diemer, J. Kunz, C. L¨ammerzahl and S. Reimers, “Dynamics of test particles in thegeneral five-dimensional Myers-Perry spacetime,” Phys. Rev. D , no. 12, 124026 (2014)[arXiv:1404.3865 [gr-qc]].[46] O. J. C. Dias, R. Monteiro and J. E. Santos, “Ultraspinning instability: the missing link,” JHEP , 139 (2011) [arXiv:1106.4554 [hep-th]].[47] K. i. Nakao, M. Kimura, T. Harada, M. Patil and P. S. Joshi, “How small can an over-spinningbody be in general relativity?,” Phys. Rev. D90