arXiv:0810.0079v1 [gr-qc] 1 Oct 2008 tropy G o rao lc hole, hori- black the a inner Moreover, of the area expose observers. would zon distant it to because singularities principle this by out 16, 15, therein). 14, references 13, and 12, 23] 11, 22, 10, 21, open 9, 20, an 8, 19, 7, still 18, 6, is 17, over [5, conjecture research e.g. this of (see of question flurry principle validity basic the the years, a despite the as However, envisaged relativ- nature. being general of is of cornerstones it the Moreover, and of com- [4] ity. one pervasive the become not on has are based it singularities is that conjecture wisdom The mon Einstein’s gravity. of hypothesis predictability of this the of theory validity preserving The for 4]. essential 3, is 2, Penrose [1, by ago cosmic years forward weak forty put the of (WCCC), essence conjecture to the invisible censorship is holes, This black of observers. distant inside hidden always are lapse h lcrccharge electric the i hc scaatrzdb he osre parameters: mass met- conserved gravitational three the Kerr-Newman by characterized the is by which ric described uniquely equa- are Einstein-Maxwell tions the of solutions stationary all 29], to to fail eventually remains horizon. processes task the candidate remove the such how principle advocates out censorship the find For cosmic unphysical. the black-hole be the of to remove expected to sight, is pro- first horizon any at reasons, seems, two which these (GSL) cess For law [24]. second thermodynamics generalized of the violate black-hole event would the black-hole horizon the of of loss destruction the the entropy, enormous for compensate to mechanism h etuto fabakhl vn oio sruled is horizon event black-hole a of destruction The col- gravitational in arise that singularities Spacetime codn oteuiuns hoes[5 6 7 28, 27, 26, [25, theorems uniqueness the to According = c S ¯ = BH h ) hrfr,wtotayovosphysical obvious any without Therefore, 1). = = oio sirftbe hsosrainsget htthe that phenomena. suggests quantum observation singula a This accoun into inner irrefutable. taken its is properly are horizon exposing effects conclude thereby quantum to when hole, that one show black lead the may effects spin abso quantum over the Ignoring consider we hole. particular, black In cons been hypothesis. have a which censorship situations are extreme singularities reanalyze impo we spacetime most Letter undesirable the the of one that still asserts maint is validity is whose phenomena conjecture, This gravitational describing the Einstein’s in relativity, relativity general of features ubiquitous A/ h nunilterm fHwigadPnoedemonstrate Penrose and Hawking of theorems influential The 2](euentrluisi which in units natural use (we [24] 4 Q lc-oeslto utsatisfy must solution black-hole A . M h nua momentum angular the , A sascae iha en- an with associated is , h upnAaei etr mqHfr420 Israel 40250, Hefer Emeq Center, Academic Ruppin The euno h unu omccensor cosmic quantum the of Return h aashIsiue euae 11,Israel 91010, Jerusalem Institute, Hadassah The Dtd oebr8 2018) 8, November (Dated: J and , hhrHod Shahar and inlperturbations, tional h relation the egto h ed n sgvnby given is and field, the of weight auaeterlto 1.A swl nw,teKerr- the known, well is which As ones with the metric (1). Newman are relation holes black the Extreme saturate hole. black the with where hr ( where bations, in[0 1.Oemydcmoetefil as field the decompose may equa- Teukolsky One the 31]. by [30, governed tion is spacetime Newman space- black-hole the in fields various time. propagation of the scattering to study and order should In one processes (1). such Eq. analyze condition, “super- extremality may the which being waves saturate” of from absorption horizon the black-hole by mech- eliminated physical the the protects into mo- which inquire angular we anism work large this and In energy small mentum. with particles a it into with hole. associated black therefore a is than rather it singularity and naked horizon, event an amncidx and index, harmonic h cnevd rqec ftemode, the of frequency (conserved) the in(2), tion fcnun entp 3,3] ope yaseparation a by coupled 33], [32, constant type Heun confluent of a onaycniin fprl non ae tthe at waves ingoing purely of conditions boundary cal h indices the s Ψ o clrprubtos W hl ecfrhomit henceforth shall (We perturbations. scalar for 0 = r fgaiy h tlt fcasclgeneral classical of utility The gravity. of ory h yaiso aefil nterttn Kerr- rotating the in Ψ field wave a of dynamics The injecting by hole black a spin” “over to try may One o h cteigpolmoesol moephysi- impose should one problem scattering the For lm wy idnisd fbakhls nthis In holes. black of inside hidden lways omccnosi rnil sintrinsically is principle censorship cosmic ie ytecsi esrhpprinciple. censorship cosmic the by ained − ( pinof rption tn pnqetosi eea relativity, general in questions open rtant ,r ,φ θ, r, t, a l drda oneeape otecosmic the to counterexamples as idered ,r ,φ θ, r, t, iyt itn bevr.Hwvr we However, observers. distant to rity s ψ ≤ A ≡ ,teitgiyo h lc-oeevent black-hole the of integrity the t, = and ( hta nietfrinwv may wave fermion incident an that ,m l, m aω J/M ± = ) [34]. ) r h oe-idus coordinates, Boyer-Lindquist the are ) 2 1 ≤ fermion htsaeiesnuaiisare singularities spacetime that S and , o asesnurn etrain,and perturbations, neutrino massless for byrda n nua qain,both equations, angular and radial obey l e steseicaglrmmnu of momentum angular specific the is M h parameter The . imφ 2 s m s M − s o rvt. ihtedecomposi- the With brevity.) for atce yaspinning a by particles = S steaiuhlhroi index harmonic azimuthal the is 2 Q lm − ± 2 ( o lcrmgei pertur- electromagnetic for 1 Q − θ ; 2 aω a − 2 a ) ≥ s 2 ψ 0 < lm s s , osntcontain not does 0 = ( scle h spin the called is l r stespheroidal the is ; ± aω o gravita- for 2 ) e − iωt , ω (1) (2) is 2 black-hole horizon and a mixture of both ingoing and It may therefore seem, at first sight, that fermion par- outgoing waves at infinity (these correspond to incident ticles of low-energy and high angular momentum are and scattered waves, respectively). Namely, not hindered from entering the black hole. This has led Richartz and Saa [23] to conclude that an incident iωy iωy e− + (ω)e as r (y ); fermion wave may over spin the black hole, thereby ex- ψ i(Rω mΩ)y → ∞ → ∞ (3) ∼ ( (ω)e− − as r r+ (y ) , posing its inner singularity to distant observers. T → → −∞ Everything in our past experience in physics tells us where the “tortoise” radial coordinate y is defined by that a black hole should defeat any attempt of destroying 2 2 2 2 2 dy = [(r +a )/∆]dr, with ∆ r 2Mr+Q +a . [The its event horizon. It seems every time we think we have 2 ≡ 2 − 2 1/2 zeroes of ∆, r = M (M Q a ) , are the black finally found a sophisticated way of violating the cosmic ± ± − − 2 2 hole (event and inner) horizons.] Here Ω = a/(r+ + a ) censorship conjecture, nature still has the final word. The is the angular velocity of the black hole. The coefficients cosmic censorship principle asserts that nature should al- (ω) and (ω) are the transmission and reflection am- T R ways provide a black hole with some physical mechanism plitudes for a wave incident from infinity. which would protect its integrity, thereby preventing one The transmission and reflection amplitudes satisfy from exposing the black-hole inner singularity. 2 the usual probability conservation equation (ω) + Where should we look for the physical mechanism 2 |T | (ω) = 1. The calculation of these scattering ampli- which may protect the cosmic censorship principle in the tudes|R | in the low frequency limit, Mω 1, is a common ≪ current case ? The absence of superradiance for fermion practice in the physics of black holes, see e.g. [35, 36] fields is a direct consequence the Pauli exclusion princi- and references therein. For boson fields (s = 0, 1, 2) ± ± ple (for fermion fields there can only be one quantum per one finds [22, 36] state). Being a quantum principle, it suggests that there may be some quantum phenomena which protects the in- 2 ω mΩ 2l+1 (ω) − (ATBH ω) , (4) tegrity of the black-hole horizon in the present gedanken |T | ∼ πT BH experiment. for the transmission probability, where T = (r Indeed, it has been shown by Unruh [40, 41] that BH + − r )/A is the Bekenstein-Hawking temperature of the the intrinsic parity non-conservation of neutrinos (i.e., − 2 2 black hole, and A =4π(r+ + a ) is its surface area. massless neutrinos have only one helicity) would lead to The transmission probability, Eq. (4), implies that vacuum polarization effects about a spinning black hole. those modes for which the frequency ω and the azimuthal What is the physical cause for this polarization effect? In quantum number m are related by ωergosphere), the spin-orbit coupling becomes strong fore finds that only modes for which enough to create negative energy orbits (as seen from in- finity) [42]. The phenomena of vacuum polarization in ω>mΩ , (5) the Kerr spacetime is a direct consequence of the possi- can be absorbed by the black hole. Thus, it is impos- bility of decay into such negative-energy orbits [40, 41]. sible to increase the black-hole spin without increasing The vacuum polarization effect is the wave analog of its mass simultaneously. This fact guarantees that the the Penrose process [43], in which rotational energy can black-hole condition, Eq. (1), would still be respected be extracted from a rotating black hole. The process after the absorption of the mode [22, 39]. One there- utilizes the existence of retrograde particle orbits in the fore concludes that the incident mode cannot remove the ergosphere of rotating black holes, for which the energy, black-hole horizon. Cosmic censorship is therefore re- as it would have been measured by an observer at in- spected! finity, is negative [44]. Such orbits can not come out to We have just learned that, thanks to the superradiance infinity. However, the negative-energy particles can in- phenomena the black-hole is protected from being over duce changes in the energies of other particles, which do spinned by an incident bosonic mode. It should be em- come out to infinity. phasized, however, that the same cannot be said if the The spontaneous polarization of the vacuum around incident mode is of a fermion type. It is a well-known rotating black holes involves the creation of two modes. fact that there is no superradiance effect for fermion fields One of these modes is co-rotating with the black hole [23, 40, 41, 45, 46, 47]. That is, the superradiant term and is characterized by a positive energy as measured at ω mΩ is absent from the expression of the transmis- infinity, while the other one is counter rotating, having sion− probability of fermion fields. For fermion fields one negative energy. The positive-energy particle can escape simply finds [23, 36] to infinity, while the negative-energy particle must fall into the black hole. An observer at infinity may detect (ω) 2 (AT ω)2l+1 . (6) the emitted positive-energy particle. He also measures a |T | ∼ BH 3 decrease in the rotational energy of the black hole, caused hole with no reflection is given by [50] by the infall of the retrograde, negative-energy particle 2 x 1 into it. He therefore concludes that by the spontaneous p(0 1) = (ω) (1 + e− )− , (9) | |T | polarization process rotational energy was extracted from the black hole [43, 44]. where It is worth noting that, the fact that the energy- x (ω mΩ)/T , (10) momentum tensor of neutrinos can have negative values ≡ − BH inside the ergosphere [40, 41, 45, 46, 47] is in imme- and (ω) 2 is given by Eq. (6). |T | diate contradiction with the positive energy condition. The absorption probability (9) implies that the claim This energy criterion is commonly assumed to be valid of Ref. [23] that the probability to over spin a black hole for classical matter distributions. Thus, the spontaneous is given by (ω) 2 [as defined in Eq. (6)] is actually creation of particles in the Kerr spacetime is obviously a erroneous. The|T correct| probability, given by Eq. (9), is quantum phenomena. x 1 actually suppressed by a factor of (1+ e− )− . It is clear It should be emphasized that the quantum polariza- that in order to over spin the black hole one must con- tion effect of neutrinos in the Kerr spacetime exists even sider an incident mode characterized by ω

2 1 particular, it fails to over spin the extremal black hole. = (ω) exp[(ω mΩ)/T ]+1 − , lms |Tlms | { − BH } Suppose we have a near extremal black hole with mass (7) M and angular momentum J = M 2 1 (this is the and substituting TBH = 0 for the extremal limit, one “nearest extreme” black hole considered− in [23]). Of finds course, our analysis is meaningful only in the semiclas- sical regime, where M >> 1. In this limit, the tem- = (ω) 2 Θ(mΩ ω) , (8) lms |Tlms | − perature of the near-extremal black hole is given by T √2/4πM 2. Consider an incident mode with small where Θ(x) is the Heaviside step function. BH frequency≃ and l = m =3/2. In this case x = 3πM/√2. The result Eq. (8) implies that even cold (TBH = 0) Taking cognizance of Eq. (9) one finds that− the proba- spinning black holes emit fermion particles with frequen- bility of such a ‘dangerous’ mode to be absorbed by the cies below mΩ. As indicated by Unruh [40, 41], this black hole is vacuum polarization effect serves to constantly decrease 2 3πM/√2 both the mass and the angular momentum of the black p(0 1) = (ω) e− . (11) hole. It should be emphasized that only fermion modes | |T | for which ω> 1 considered black hole loses angular momentum more rapidly than here. For example, for a black hole of mass M = 1gr, the x 1 133009 it loses its (squared) mass. One therefore concludes suppression factor (1 + e− )− is 10− [51]. The that the black hole is constantly pushing itself away number 10133009 is much larger than∼ the total number from the extremal limit [that is, the dimensionless ratio of particles in the whole universe. It is therefore clear (r+ r )/r+ increases with time due to the spontaneous that the absorption probability of a dangerous mode is − − emission of neutrinos]. practically zero. The important point to realize is that the Pauli ex- Moreover, such dangerous absorptions (with extremely clusion principle would prevent one from beaming low- low probability) are not cumulative– most of the incident energy neutrinos (characterized by ωgeneral relativity and (1979). quantum physics. [32] A. Ronveaux, Heun’s differential equations. (Oxford Uni- versity Press, Oxford, UK, 1995). [33] C. Flammer, Spheroidal Wave Functions (Stanford Uni- ACKNOWLEDGMENTS versity Press, Stanford, 1957). [34] The functions S(θ; aω) are the spheroidal wave functions This research is supported by the Meltzer Science [30, 33]. In the aω ≪ 1 limit they become the familiar Foundation. I thank Jacob D. Bekenstein, Tsvi Piran and spin-weighted spherical harmonics with the correspond- ing angular eigenvalues sAlm = l(l + 1) − s(s + 1) + Liran Shimshi for stimulated discussions. I also thank O(a2ω2). Yael Oren for helpful discussions. [35] S. 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(N.Y.) 83, 548 (1974). was a non-rotating one (as assumed in [21]), the field [11] W. A. Hiscock, Ann. Phys. (N.Y.) 131, 245 (1981). would “ignite” its rotation, such that the propagation of [12] B. S. Kay and R. M. Wald, Classical and Quantum Grav- the field itself is actually taking place on a slowly rotating ity 4, 893 (1987). perturbed spacetime. [13] J. D. Bekenstein and C. Rosenzweig, Phys. Rev. D 50, However small, these higher-order backreaction effects 7239 (1994). turn out to be a crucial ingredient of the analysis. In par- 5

ticular, we have demonstrated [22] that a more complete [47] A. Gould, Phys. Rev. D 35, 449 (1987). analysis of the gedanken experiments (in which backreac- [48] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975). tion effects are properly taken into account) reveals that [49] Note that one can still send high-energy neutrinos (char- they do not violate the WCCC. acterized by ω>mΩ) into the black hole. However, such [40] W. Unruh, Phys. Rev. Lett. 31, 1265 (1973). high-energy modes would certainly push the black hole [41] W. Unruh, Phys. Rev. D 10, 3194 (1974). away from the extremal limit. [That is, they will increase [42] B. Carter, Phys. Rev. 174, 1559 (1968). the ratio (r+ − r−)/r+.] Thus, high-energy particles pose [43] R. Penrose, Riv. Nuovo Cimento 1, 252 (1969). no challenge to the cosmic censorship conjecture. [44] T. Piran and J. Shaham, Phys. Rev. D 16, 1615 (1977). [50] J. D. Bekenstein, Phys. Rev. D 15, 2775 (1977). [45] R. G¨uven, Phys. Rev. D 16, 1706 (1977). [51] It is worth emphasizing that the pre-factor |T (ω)|2 is also [46] S. M. Wagh and N. Dadhich, Phys. Rev. D 32, 1863 very small, see Eq. (6). (1985).