Destroying a Near-Extremal Kerr-Newman Black Hole

PHYSICAL REVIEW D 84, 027501 (2011) Destroying a near-extremal Kerr-Newman black hole

Alberto Saa1,* and Raphael Santarelli2,† 1Departamento de Matema´tica Aplicada, UNICAMP, 13083-859 Campinas, SP, Brazil 2Instituto de Fı´sica ‘‘Gleb Wataghin’’, UNICAMP, 13083-859 Campinas, SP, Brazil (Received 19 May 2011; published 18 July 2011) We revisit here a previous argument due to Wald showing the impossibility of turning an extremal Kerr-Newman black hole into a naked singularity by plunging test particles across the black hole event horizon. We extend Wald’s analysis to the case of near-extremal black holes and show that it is indeed possible to destroy their event horizon, giving rise to naked singularities, by pushing test particles toward the black hole as, in fact, it has been demonstrated explicitly by several recent works. Our analysis allows us to go a step further and to determine the optimal values, in the sense of keeping to a minimum the backreaction effects, of the test particle electrical charge and angular momentum necessary to destroy a given near-extremal Kerr-Newman black hole. We describe brieﬂy a possible realistic scenario for the creation of a Kerr naked singularity from some recently discovered candidates to be rapidly rotating black holes in radio galaxies.

DOI: 10.1103/PhysRevD.84.027501 PACS numbers: 04.20.Dw, 04.70.Bw, 97.60.Lf

I. INTRODUCTION the horizon of a near-extremal black hole by overspinning or overcharging it with the absorption of test particles or There has been recently a revival of interest in the fields is nowadays a very active field of research and debate problem of turning a black hole into a naked singularity [1,2,7–16]. by means of classical and quantum processes, see, for The impossibility of destroying the event horizon by instance, [1,2] for references and a brief review with a plunging test particles into an extremal Kerr-Newman historical perspective. Such a problem is intimately related black hole is clear and elegantly summarized in Wald’s to the weak cosmic censorship conjecture [3,4]. Indeed, the argument [17], which we briefly reproduce here. We con- typical facility in covering a naked singularity with an sider test particles with energy E, electric charge e, and event horizon and the apparent impossibility of destroying orbital angular momentum L. The test particle approxima- a black hole horizon [5] have strongly endorsed the validity tion requires E=M 1, L=aM 1, and e=Q 1, assur- of the conjecture along the years, albeit it has started to be ing, in this way, that backreaction effects are negligible. challenged recently. The particle angular momentum L is assumed to be aligned The most generic asymptotically flat black hole solution with the black hole angular momentum J, and both Q and e of the Einstein equations we can consider is the Kerr- are assumed, without loss of generality, to be positive. Newman black hole, which is completely characterized According to the laws of black hole thermodynamics by its mass M, electric charge Q, and angular momentum (see, for instance, Sec. 33.8 of [18]) after the capture of a J ¼ aM. The distinctive feature of a black hole, namely, test particle, the black hole will have total angular momen- the existence of an event horizon covering the central tum aM þ L, charge Q þ e, and mass no greater than singularity, requires M þ E. In order to form a naked singularity, one needs M2 a2 þ Q2; (1) aM þ L 2 ðM þ EÞ2 < þðe þ QÞ2; M þ E (2) with the equality corresponding to the so-called extremal case. If (1) does not hold, the central singularity is exposed, which implies, for extremal black holes and in the test giving rise to a naked spacetime singularity, which should particle approximation, not exist in nature according to the weak cosmic censorship conjecture. All the classical results on the impossibility of QeM þ aL E< : destroying black hole horizons were obtained by consid- M2 þ a2 (3) ering extremal black holes. The first work arguing that it would be indeed possible to destroy the horizon of a near- However, in order to assure that the test particle be indeed extremal black hole is quite recent and it was due to plunged into the black hole, its energy must obey [18] Hubeny [6], which considered a Reissner-Nordstro¨m (a ¼ 0) black hole. The physical possibility of destroying Qer þ aL E E ¼ þ ; min 2 2 (4) rþ þ a *[email protected] †telsanta@ifi.unicamp.br with

1550-7998=2011=84(2)=027501(4) 027501-1 Ó 2011 American Physical Society BRIEF REPORTS PHYSICAL REVIEW D 84, 027501 (2011) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 a ¼ M2 2 cos; rþ ¼ M þ M a Q (5) (8) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi being the event horizon radius of the black hole. For an Q ¼ M2 2 sin; (9) extremal black hole, rþ ¼ M, and it is clear that (3) and (4) will not be fulfilled simultaneously, implying that one with 0 =2. In this way, near-extremal black holes cannot turn an extremal Kerr-Newman black hole into a are characterized by the triple ðM; ; Þ. For instance, near- naked singularity by plunging the test particle across its extremal Kerr and Reissner-Nordstro¨m black holes corre- event horizon. Wald presents also a similar argument for spond, respectively, to ðM; ; 0Þ and (M, , =2). After the case of dropping spinning uncharged particles into a some straightforward algebra, it is possible to show that Kerr (Q ¼ 0) black hole. Notwithstanding, de Felice M þ Asin2 2 and Yunqiang [19] showed that it would be indeed possible Emax ¼ A ; (10) to transform a Reissner-Nordstro¨m black hole in a 2 þ 2cos2 M Kerr-Newman naked singularity after capturing an electri- where only terms up to second order in ð=MÞ were kept, cally neutral spinning body. and The purpose of this Brief Report is to extend Wald’s original analysis [17] to the case of near-extremal Kerr- ðL=MÞ cos þ e sin A ¼ 0: (11) Newman black holes and show explicitly that it is indeed 1 þ cos2 possible to overspin and/or overcharge near-extremal black The event horizon for near-extremal black holes are holes by plunging test particles across their event horizon located at rþ ¼ M þ , which implies that the condition while keeping backreaction effects to a minimum. All the (4) assuring that the particle is to be captured reads recently proposed mechanisms to destroy a near-extremal ð2 þ sin2ÞA 4B 2 black hole by using infalling test particles are accommo- E E ¼ A B ; dated in our analysis. Furthermore, we determine the opti- min M 2 þ 2cos2 M mal values, in the sense that they keep backreaction effects (12) to a minimum, of the electrical charge and angular mo- 2 mentum of the incident test particle in order to destroy a where, again, only terms up to ð=MÞ were kept and near-extremal Kerr-Newman black hole with given mass, 2ðL=MÞ cos þ esin3 B ¼ 0: charge, and angular momentum. We show also that it is not ð1 þ cos2Þ2 (13) strictly necessary to plunge the particles across the black hole horizon, but they can be thrown from infinity and It is clear that for the extremal case ( ¼ 0) we have proceed toward the black hole following a geodesics, Wald’s result Emax ¼ Emin, implying that (7) and (12) minimizing in this way any backreaction effect associated cannot be fulfilled simultaneously. However, for >0 it with the specific mechanism to release the particle near, or is indeed possible for a test particle to obey (7) and (12). push it against, the black hole horizon. As an explicit The intersection of Emax and Emin in the ð; "Þ plane example, we consider some recently discovered candidates corresponds to the straight line to be rapidly rotating black holes in radio galaxies and 1 þ cos2 show how it would be possible to create Kerr naked singu- 2 cos þ "sin3 ¼ ; (14) larities from them with minimal backreaction effects. 2 where ¼ L=M and " ¼ e=. No naked singularity is II. NEAR-EXTREMAL KERR-NEWMAN formed in the region above this line. BLACK HOLES We call near-extremal a Kerr-Newman black hole for Optimal test particles which In all derivations done so far, we have used the test particle approximation. The idea of minimizing the values 2 ¼ M2 a2 Q2 > 0; 1: M (6) of E, L, and e necessary to destroy the black hole is more than a simple requirement of consistence. It helps to assure For a near-extremal black hole, the condition (1) for the that backreaction effects are negligible and, consequently, creation of a naked singularity by absorbing a test particle that it will not be possible to restore the black hole event with energy E, electric charge e, and orbital angular horizon by means of any subdominant physical process. momentum L implies that The first, and maybe the more natural, criterium of opti- QeM þ aL M3 2 mality we can devise here is to require minimal test particle E

027501-2 BRIEF REPORTS PHYSICAL REVIEW D 84, 027501 (2011) optimization problem [20], and the solution is known to The event horizon rþ is the outermost zero of and correspond to one of the points ð0;"Þ or ð; 0Þ, i.e., the Eq. (4) corresponds to the potential (16) evaluated on minimal energy E will be given either by Eð0;"Þ¼ r ¼ rþ.Forr !1one has V ! , as it is expected for =2sin2 or Eð; 0Þ¼=4. It is clear that Eð; 0Þ < any asymptotically flat solution. For a particle of energy E, Eð0;"Þ for any value of , suggesting that the best option the points for which VðrÞ¼E are return points and delimit to turn a black hole into a naked singularity would be to the classically allowable region for the particle motion. In plunge an uncharged particle, irrespective of the value order to assure that a particle with energy E thrown from of . However, we see from (14) that can increase infinity reaches the horizon, we need to have E>VðrÞ in considerably for small , despite Eð; 0Þ being a mini- the exterior region of the black hole. In some cases, one can mum. The minimization of Eð; "Þ does not guarantee the have that >Emin, i.e., the energy necessary for the minimization of L and e, risking the validity of the test incident test particle reach the horizon is smaller than its particle approximation. In order to avoid theses problems, rest mass. This is not a surprise in gravitational systems, we will require that 2 þ "2 be minimal for optimal test but, of course, this trajectory cannot start from infinity. To particles. Such a requirement corresponds with selecting t follow this trajectory, a particle must be released near the the nearest point of the straight line (14) to the origin in the horizon by some external mechanism. This is an extra and ð; "Þ plane. From simple trigonometry, we have that a test unnecessary complication in our analysis. The external particle with angular momentum L and electrical charge e mechanism could be subject to some backreaction or sub- obeying dominant physical effect that could eventually prevent the particle of entering the black hole. This can be avoided if eM sin3 ðQ=aÞ3 ¼ ¼ we adjust the particle rest mass properly. An explicit 2 (15) L 2 cos 2 þ 2ðQ=aÞ example can enlighten this point. Let us consider the case of a Kerr black hole (Q ¼ 0). is the optimal test particle to turn a near-extremal Kerr- (For a recent review on equatorial orbits in Kerr black holes Newman black hole with parameters ðM; ; Þ into a naked and naked singularities, see [21].) For a near-extremal Kerr singularity. black hole, the effective potential (16) can be written as It is clear from (15) that the optimal test particle to turn a near-extremal Kerr black hole ( ¼ 0) into a naked singularity should be electrically neutral (e ¼ 0). Moreover, VðrÞ¼V~ðrÞþOðð=MÞ2Þ; (17) from Eq. (14), we see that the particle must have L=M =2 Eð; 0Þ . The minimum particle energy is given by , where V~ðrÞ stands for the effective potential for the ex- assuring the validity of the test particle approximation in tremal Kerr black hole, which can be calculated from this case (E=M 1 and L=aM 1) provided the black Q ¼ 0 a ¼ M =M 1 (16) with and . Figure 1 depicts the effective hole be near-extremal ( ). For the case of a potential V~ðrÞ for different values of . Typically, Reissner-Nordstro¨m black hole ( ¼ =2), the optimal =2 the choice of min will allow the particle to test particle must be charged, with , and have reach the horizon when thrown from infinity with energy vanishing orbital angular momentum (L ¼ 0). The mini- E Emin. mum particle energy for this case is given by E ¼ =2. The validity of the test particle approximation and the minimization of any backreaction effect is assured also in 1e−05 this case. 9e−06 (e) 8e−06 III. THROWING PARTICLES FROM INFINITY 7e−06 (d) 6e−06 E min The expression for Emin given by Eq. (4) corresponds to 5e−06 ~ V(r)/M (c) the minimal energy that a test particle can have at the black 4e−06 hole horizon. On the other hand, the minimal energy that a 3e−06 (b) 2e−06 particle can have anywhere on the equatorial plane (a) 1e−06 outside a Kerr-Newman black hole is given by the effective 2 4 64 8 10 12 1 16 18 20 potential [18] r/M pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 0 VðrÞ¼ ; (16) FIG. 1. The effective potential (16) for a test particle with angular momentum L and rest mass around an extremal M L=M2 ¼ 105 2 2 2 2 2 Kerr black hole with mass . For this figure, where ¼ðLa þ QerÞðr þ a ÞLa, ¼ðr þa Þ =M ¼ 106 2:5 106 5 106 2 2 2 2 2 and (a), (b), (c), a , and 0 ¼ðLa þ QerÞ ðL þ r Þ, with 7:5 106 (d), and 105 (e). The dotted horizontal line corre- 2 being the particle rest mass and ¼ r 2Mrþ sponds to the value of the effective potential at the black-hole 2 2 a þ Q . horizon (Emin), which does not dependent on .

027501-3 BRIEF REPORTS PHYSICAL REVIEW D 84, 027501 (2011) IV. FINAL REMARKS will destroy the black hole. Hence, any test body thrown from infinity with angular momentum L=M2 ¼ 105 and We have shown that the Wald analysis [17] can be energy E such that Emin quasars [22] and radio galaxies [23]. These black holes 8 reaction effect that could prevent the formation of a naked are very massive, having typically M 10 M, and can singularity in this case. We close noticing that, probably, attain an angular momentum such that a=M * 0:9. Let us the Thorne limit a=M 0:998 [24] corresponds to the suppose we have one of these black holes with =M most realistic near-extremal astrophysical Kerr black 105. According to Sec. II, the capture of a test body hole. For such a case, =M 6% and we have analogous with L=M2 ¼ 105 is sufficient for the creation of a naked 3 results to the preceding example. However, in this case, singularity. For this case, Emin 10 M. A test body with 8 the validity of the test particle approximation could be mass comparable to the moon mass, 4 10 M, questioned and we could have appreciable backreaction will certainly be able to reach the horizon if thrown from effects. infinity with angular momentum L=M2 ¼ 105 (see Fig. 1). The minimal necessary energy for a test body to ACKNOWLEDGMENTS reach the horizon with such orbital angular momentum is 5 5 given by (12), Emin=M ¼ 10 ð1 10 Þ=2. On the other This work was supported by FAPESP and CNPq. The hand, with this angular momentum, any test body captured authors wish to thank V.E. Hubeny for enlightening 5 5 with energy E=M < Emax=M ¼ 10 ð1 10 =2Þ=2 conversations.

[1] G. E. A. Matsas and A. R. R. da Silva, Phys. Rev. Lett. 99, [12] M. Bouhmadi-Lopez, V. Cardoso, A. Nerozzi, and J. V. 181301 (2007). Rocha, Phys. Rev. D 81, 084051 (2010). [2] T. Jacobson and T. P. Sotiriou, J. Phys. Conf. Ser. 222, [13] T. Jacobson and T. P. Sotiriou, arXiv:1006.1763. 012041 (2010). [14] G. Chirco, S. Liberati, and T. P. Sotiriou, Phys. Rev. D 82, [3] R. M. Wald, in Black Holes, Gravitational Radiation and 104015 (2010). the Universe, Fundamental Theories of Physics, Vol. 100, [15] E. Barausse, V. Cardoso, and G. Khanna, Phys. Rev. Lett. edited by B. R. Iyer and B. Bhawal (Kluwer Academic, 105, 261102 (2010). Dorddrecht, 1999), pp. 69–85. [16] J. V. Rocha and V. Cardoso, Phys. Rev. D 83, 104037 [4] C. J. S. Clarke, Classical Quantum Gravity 11, 1375 (2011). (1994). [17] R. Wald, Ann. Phys. (N.Y.) 82, 548 (1974). [5] J. M. Cohen and R. Gautreau, Phys. Rev. D 19, 2273 [18] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (1979). (W. H. Freeman, San Francisco, 1973). [6] V.E. Hubeny, Phys. Rev. D 59, 064013 (1999). [19] F. de Felice and Y.Q. Yu, Classical Quantum Gravity 18, [7] M. Richartz and A. Saa, Phys. Rev. D 78, 081503 (2008). 1235 (2001). [8] S. Hod, Phys. Rev. Lett. 100, 121101 (2008); Phys. Lett. B [20] G. B. Danzig and M. N. Thapa, Linear Programming 2: 668, 346 (2008). Theory and Extensions (Springer, Berlin, 2003). [9] G. E. A. Matsas, M. Richartz, A. Saa, A. R. R. da Silva, [21] D. Pugliese, H. Quevedo, and R. Rufﬁni, arXiv:1105.2959. and D. A. T. Vanzella, Phys. Rev. D 79, 101502 (2009). [22] J. M. Wang, Y.M. Chen, L. C. Ho, and R. J. McLure, [10] T. Jacobson and T. P. Sotiriou, Phys. Rev. Lett. 103, Astrophys. J. 642, L111 (2006). 141101 (2009); 103, 209903(E) (2009). [23] Q. Wu, X. Cao, and D. X. Wang, Astrophys. J. 735,50 [11] E. Berti, V. Cardoso, L. Gualtieri, F. Pretorius, and U. (2011). Sperhake, Phys. Rev. Lett. 103, 239001 (2009). [24] K. S. Thorne, Astrophys. J. 191, 507 (1974).

027501-4