Upper Bound on the Center-Of-Mass Energy of the Collisional Penrose

Physics Letters B 759 (2016) 593–595

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Physics Letters B

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Upper bound on the center-of-mass energy of the collisional Penrose process ∗ Shahar Hod a,b, a The Ruppin Academic Center, Emeq Hefer 40250, Israel b The Hadassah Institute, Jerusalem 91010, Israel a r t i c l e i n f o a b s t r a c t

Article history: Following the interesting work of Bañados, Silk, and West (2009) [6], it is repeatedly stated in the physics Received 24 April 2016 literature that the center-of-mass energy, Ec.m, of two colliding particles in a maximally rotating black- Received in revised form 3 June 2016 hole spacetime can grow unboundedly. For this extreme scenario to happen, the particles have to collide Accepted 13 June 2016 at the black-hole horizon. In this paper we show that Thorne’s famous hoop conjecture precludes this Available online 16 June 2016 extreme scenario from occurring in realistic black-hole spacetimes. In particular, it is shown that a new Editor: M. Cveticˇ (and larger) horizon is formed before the infalling particles reach the horizon of the original black hole. As a consequence, the center-of-mass energy of the collisional Penrose process is bounded from above Emax ∝ 1/4 by the simple scaling relation c.m /2μ (M/μ) , where M and μ are respectively the mass of the central black hole and the proper mass of the colliding particles. © 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction that the center-of-mass energy, Ec.m, of two colliding particles in a maximally rotating black-hole spacetime can grow unboundedly. It is worth emphasizing again that, for the center-of-mass en- The collisional Penrose process, a collision between two parti- ergy of the colliding particles to diverge, the particles have to cles which takes place within the ergosphere of a spinning Kerr collide exactly at the black-hole horizon. The main goal of the black hole, has attracted the attention of physicists ever since present paper is to reveal the fact that Thorne’s famous hoop the pioneering work of Piran, Shaham, and Katz more than four conjecture [9] precludes this extreme scenario from occurring in decades ago [1,2]. Interestingly, it was shown [1,2] that the colli- realistic black-hole spacetimes. In particular, below we shall show sion of two particles in the black-hole spacetime can produce two that a new (and larger) horizon is formed before the infalling parti- new particles, one of which may escape to inﬁnity while carrying cles reach the horizon of the original black hole. As a consequence, with it some of the black-hole rotational energy [3–5]. the center-of-mass energy of the colliding particles in the black- The collisional Penrose process has recently gained renewed hole spacetime cannot grow unboundedly. interest when Bañados, Silk, and West [6] have revealed the interesting fact that the center-of-mass energy of the two colliding 2. The center-of-mass energy of two colliding particles in the particles may diverge if the collision takes place at the horizon of a black-hole spacetime maximally rotating (extremal) black hole. In particular, it was sug- gested [6] that these extremely energetic near-horizon collisions We shall explore the maximally allowed center-of-mass energy may provide a unique probe of the elusive Planck-scale physics associated with a collision of two particles that start falling from [7,8]. rest at inﬁnity towards a maximally rotating (extremal [10]) Kerr The intriguing discovery of [6], according to which black holes black hole of mass M and angular momentum J = M2 [6,8,11–13]. may serve as extremely energetic particle accelerators, has sparked The geodesic motions of the particles in the black-hole spacetime an enormous excitement in the physics community. In particu- are characterized by conserved energies lar, following [6] it is repeatedly stated in the physics literature E1 = E2 = μ (1) and conserved angular momenta * Correspondence to: The Ruppin Academic Center, Emeq Hefer 40250, Israel. E-mail address: [email protected]. L1 = l1 · Mμ ; L2 = l2 · Mμ . (2) http://dx.doi.org/10.1016/j.physletb.2016.06.028 0370-2693/© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 594 S. Hod / Physics Letters B 759 (2016) 593–595

Geodesic trajectories that extend all the way from spatial infinity In the present study we shall use a weaker version of the hoop down to the black-hole horizon are characterized by angular mo- conjecture [21]. In particular, we conjecture that: A gravitating sys- menta in the bounded regime [14,6] tem of total mass M and total angular momentum J forms a black hole if its circumference radius rc is equal to (or smaller than) 2 2 −2(1 + 1 + J/M2) ≤ l ≤ 2(1 + 1 − J/M2). (3) the corresponding horizon radius rKerr = M + M − ( J/M) of the Kerr black-hole spacetime. That is, we conjecture that [22] In particular, it was shown in [6] that the center-of-mass energy of 2 2 the collision may diverge if one of the colliding particles (say 1) is rc ≤ M + M − ( J/M) =⇒ Black-hole horizon exists . (10) characterized by the critical angular momentum (for J/M2 = 1) In the context of the collisional Penrose process that we con- = l1 2 . (4) sider here, this version of the hoop conjecture implies that a new The test-particle approximation implies (and larger) horizon is formed if the energetic particles that fall towards the black hole reach the radial coordinate r = r , where μ hoop μ¯ ≡ 1 . (5) rhoop(E1, E2, L1, L2) is defined by the familiar functional relation of M the Kerr black-hole spacetime [see Eq. (10)] As shown in [6,8,11,12], the center-of-mass energy of the colli- r = M + E + E sion may diverge if the collision takes place at the horizon of the hoop 1 2 black hole. In particular, defining 2 2 + (M + E1 + E2) −[( J + L1 + L2)/(M + E1 + E2)] . rc − M xc ≡ (6) (11) M Taking cognizance of Eqs. (1), (2), (4), and (11), one finds as the dimensionless collision radius of the two particles, one finds 2 the leading-order divergent behavior [12] rhoop = M + 2(2 − l2)Mμ + 2μ + O (μ /M) (12) β±(2 − l ) for the radius of the new horizon. Assuming that the colliding par- E2 = 2 × 2 + c.m μ O (1) (7) ticles are not engulfed by an horizon, the relation (12) implies [see xc Eq. (6)] for the center-of-mass energy in the near-horizon xc 1region. min = − ¯ + ¯ As shown in [12], the dimensionless factor β± in (7) depends on xc 2(2 l2)μ 2μ (13) the sign of the radial momentum of the first (critical [15]) particle for the minimally allowed value of the dimensionless collision ra- + − ( for an outgoing particle and for an ingoing particle), on the dius [23]. value of the Carter constant Q 1 [16] which characterizes the polar Substituting (13) into (7) one finds that, in the collisional Pen- geodesic motion of that particle in the Kerr black-hole spacetime, rose process, the center-of-mass energy of the colliding particles is and on the polar angle θ which characterizes the collision point of bounded from above by the relation [24] the two particles in the black-hole spacetime [12]. Specifically, one finds [12] (2 − l2)β± Emax = μ × . (14) c.m − ¯ + ¯ + − ˜ 1/2 ˜ 1/2 2(2 l2)μ 2μ 2 2 (2 Q 1) 2(2 + Q 1) β+ = ; β− = , (8) − 1 + cos2 θ 1 + cos2 θ Assuming the strong inequality μ (2 l2)M [see Eq. (5) and Eq. (19) below], one can approximate (14) by where [12,17] − 1/4 max 2 l2 1/2 Q 1 E = μ × β± , (15) 0 ≤ Q˜ ≡ ≤ 2 . (9) c.m 2μ¯ 1 μ2 which yields the simple expression [25] Interestingly, as pointed out in [6,8,11,12], the center-of-mass Emax = · 1/4 3/4 energy (7) of the colliding particles diverges if the collision takes c.m 2γ M μ (16) → place at the horizon (xc 0) of the extremal black hole. for the maximally allowed center-of-mass energy of the colliding particles in the black-hole spacetime, where 3. The hoop conjecture and the upper bound on the ≡[ − 2 ]1/4 center-of-mass energy of the collisional Penrose process γ (2 l2)β±/32 . (17) What is the maximally allowed value of the dimensionless pre- In the previous section we have seen that the center-of-mass factor γ in (16)? Inspection of Eq. (8) reveals that the factor β± energy (7) of the colliding particles can diverge. As emphasized in is maximized if the collision takes place at the equatorial plane [6,8,11,12], this intriguing conclusion rests on the assumption that (θ = π/2) of the black hole while the first particle is on an out- ˜ the particles can collide exactly at the horizon of the extremal going trajectory with Q 1 = 0[see Eq. (9)], in which case one finds black hole. In the present section we shall show, however, that [see Eq. (8)] [26] Thorne’s famous hoop conjecture [9] implies that, due to the en- √ = + ergy carried by the infalling particles, a new (and larger) horizon is βmax 2(2 2). (18) formed before the particles reach the horizon of the original black In addition, taking cognizance of the fact that geodesic trajecto- hole. This fact implies, in particular, that the optimal collision [18] ries which extend all the way from spatial infinity down to the between the two particles cannot take place exactly at the horizon black-hole horizon are characterized by angular momenta in the of the original black hole. bounded regime (3), one finds (for J/M2 = 1) The hoop conjecture, originally formulated by Thorne more √ − = + than four decades ago [9], asserts that a gravitating system of to- (2 l2)max 2(2 2). (19) tal mass (energy) M forms a black hole if its circumference radius Substituting (18) and (19) into (17), one finds [27] rc is equal to (or smaller than) the corresponding horizon radius √ √ = 2 + 2 3/4 2 (20) rSch = 2M of the Schwarzschild black-hole spacetime [19–21]. γmax ( ) / . S. Hod / Physics Letters B 759 (2016) 593–595 595

4. Summary [8] M. Bejger, T. Piran, M. Abramowicz, F. Hakanson, Phys. Rev. Lett. 109 (2012) 121101. Following the important work of Bañados, Silk, and West [6], [9] K.S. Thorne, in: J. Klauder (Ed.), Magic Without Magic: John Archibald Wheeler, Freeman, San Francisco, 1972. it is repeatedly stated in the physics literature that the center-of- [10] As shown in [6,8], assuming the black hole to be a maximally-rotating (ex- mass energy of two colliding particles in an extremal (maximally tremal) one, allows one to maximize the center-of-mass energy of the colliding rotating) black-hole spacetime can diverge. For this extreme sce- particles. nario to happen, the particles have to collide exactly at the horizon [11] J.D. Schnittman, Phys. Rev. Lett. 113 (2014) 261102. of the black hole. [12] E. Leiderschneider, T. Piran, Phys. Rev. D 93 (2016) 043015. [13] We use natural units in which G = c = h¯ = 1. In this paper we have shown that Thorne’s famous hoop conjec- [14] J.M. Bardeen, W.H. Press, S.A. Teukolsky, Astrophys. J. 178 (1972) 347. ture [9] [and also its weaker version (10)] precludes this infinite- [15] It is worth mentioning that the first particle is characterized by the critical center-of-mass-energy scenario from occurring in realistic black- angular momentum (4). hole spacetimes. In particular, the hoop conjecture implies that, [16] B. Carter, Phys. Rev. 174 (1968) 1559. due to the energy carried by the infalling particles, a new (and [17] It was shown in [12] that a particle infalling from spatial infinity can reach the ≤ ≤ 2 larger) horizon is formed before the particles reach the horizon black-hole horizon provided 0 Q 2μ . [18] That is, the collision with the largest possible center-of-mass energy. of the original black hole. As a consequence, it was shown that [19] See A.M. Abrahams, K.R. Heiderich, S.L. Shapiro, S.A. Teukolsky, Phys. Rev. D 46 min = the optimal collision [18] takes place at [see Eq. (12)] rc (1992) 2452, and references therein. M + 2(2 − l2)Mμ + 2μ > M [28–31], which implies that the [20] See J.P. de León, Gen. Relativ. Gravit. 19 (1987) 289; center-of-mass energy of the colliding particles in the black-hole H. Andreasson, Commun. Math. Phys. 288 (2009) 715. [21] S. Hod, Phys. Lett. B 751 (2015) 241, arXiv:1511.03665. spacetime is bounded from above by the simple scaling relation [22] It is worth mentioning that in [21] it was proved that the Thorne hoop con- [see Eqs. (16) and (20)] [32] jecture [9] (or its weaker version [21]) must be invoked in order to avoid Emax M 1/4 a possible violation of the generalized second law of thermodynamics [see c.m J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973)] in gedanken experiments which = γmax · . (21) 2μ μ involve near-extremal black holes. [23] That is, the two colliding particles cannot reach the near-horizon region x < For the case of two protons colliding in a supermassive Kerr c xmin without being engulfed by the new (and larger) horizon (12) which is 9 c black-hole spacetime of 10 solar masses, the bound (21) implies formed according to the hoop conjecture [9] [and also according to its weaker E max 14 c.m 10 TeV for the maximally allowed center-of-mass energy version (10)]. of the collision. Interestingly, this energy, though being finite, is [24] Note that, due to the finite proper masses of the infalling particles, the di- still many orders of magnitude larger than the maximally available mensionless critical angular momentum (4) receives a small correction term of order O (μ/M). Hence, one should replace l1 = 2 → l1 = 2[1 + O (μ/M)] in center-of-mass energy in the most powerful man-made accelera- Eq. (12). This, in turn, implies that the r.h.s of Eq. (14) should be multiplied by tors [33]. the correction factor 1 + O (μ/M). [25] It is worth emphasizing that the expression (16) for the maximally allowed Acknowledgements center-of-mass energy of the colliding particles can be approached (from below) arbitrarily close when the radial location of the collision approaches (from above) arbitrarily close to xmin [see Eq. (13)]. This research is supported by the Carmel Science Foundation. c [26] This maximally allowed value of the dimensionless factor β± corresponds to Iwouldlike to thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and ˜ the values δ = 0, 1 =+1, Q 1 = 0, and θ = π/2in equation (15) of [12]. Alona B. Tea for helpful discussions. [27] It is worth emphasizing that the expressions (14) and (16) for the maximally allowed center-of-mass energy of the colliding particles are valid for generic References trajectories, whereas the value (20) for γmax is obtained for a collision at the equatorial plane [12]. [28] That is, outside the horizon of the original black hole. [1] T. Piran, J. Shaham, J. Katz, Astrophys. J. Lett. 196 (1975) L107. [2] T. Piran, J. Shaham, Phys. Rev. D 16 (1977) 1615. [29] It is worth emphasizing again that, according to the hoop conjecture, it is the [3] Note that the second particle has a negative energy as measured by asymptotic presence of the particles themselves within a sufficiently small radius that trig- observers, and it must therefore plunge into the black hole. gers the formation of a new horizon. [4] R. Penrose, Riv. Nuovo Cimento 1 (1969) 252; [30] It is important to emphasize that Berti et al. [31] have shown that the gravi- ∼ 2 · R. Penrose, G.R. Floyd, Nature 229 (1971) 177. tational radiation emitted by the infalling particles is of order Erad (μ /M) − − = ∼− 2 · [5] It is worth mentioning that, in the original formulation of the Penrose process ln(2 l1), Since 2 l1 O (μ/M) (see [24]), one finds Erad (μ /M) [4], there is only one particle which disintegrates into two new particles within ln(μ/M), which implies that the radiated gravitational energy can be neglected − · the Kerr black-hole ergosphere. (that is, Erad μ) in the regime (μ/M) ln(μ/M) 1. [6] M. Bañados, J. Silk, S.M. West, Phys. Rev. Lett. 103 (2009) 111102. [31] E. Berti, V. Cardoso, L. Gualtieri, F. Pretorius, U. Sperhake, Phys. Rev. Lett. 103 [7] It is important to note that subsequent investigations [8] of the collisional Pen- (2009) 239001. rose process have shown that, while the center-of-mass energy of the colliding [32] It is important to emphasize that, had we used the original hoop conjecture particles may diverge as claimed in [6], the maximal energy which is radiated [9] instead of its weaker version (10), we would have found that the center- to infinity by an escaping particle (after the collision) is only one order of mag- of-mass energy of the colliding particles is bounded from above by a bound nitude larger than the initial energy of the colliding particles (this energy gain which is stronger than our bound (21). reflects the extraction of rotational energy from the spinning Kerr black hole). [33] See, https://en.wikipedia.org/wiki/List-of-accelerators-in-particle-physics.