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

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Upper Bound on the Center-Of-Mass Energy of the Collisional Penrose Physics Letters B 759 (2016) 593–595 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 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 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 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 in- teresting 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 2 black hole of mass M and angular momentum J = M [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 to- wards 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.
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