High efficiency of collisional Penrose process requires heavy particle production
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High efficiency of collisional Penrose process requires heavyparticle production Kota Ogasawara, ∗ Tomohiro Harada, † and Umpei Miyamoto ‡ Department of Physics, Rikkyo University,Toshima, Tokyo 171-8501, Japan and RECCS, Akita Prefectural University, Akita 015-0055, Japan (Dated: August 28, 2018)
Abstract
The center-of-mass energy of two particles can become arbitrarily large if they collide nearthe event horizon of an extremal Kerr black hole, which is called the Ba˜nados-Silk-West (BSW)effect. We consider such a high-energy collision of two particles which started from infinity andfollow geodesics in the equatorial plane and investigate the energy extraction from such a high-energy particle collision and the production of particles in the equatorial plane. We analyticallyshow that, on the one hand, if the produced particles are as massive as the colliding particles,the energy-extraction efficiency is bounded by 2 .
19 approximately. On the other hand, if a verymassive particle is produced as a result of the high-energy collision, which has negative energy andnecessarily falls into the black hole, the upper limit of the energy-extraction efficiency is increasedto (2 + √ ≃ .
9. Thus, higher efficiency of the energy extraction, which is typically as large as10, provides strong evidence for the production of a heavy particle.
PACS numbers: 04.70.Bw,97.60.Lf ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION Ba˜nados, Silk, and West (BSW) pointed out that the center-of-mass (CM) energy of twocolliding particles can be arbitrarily large, if the collision occurs near the event horizon ofan extremal Kerr black hole and the angular momentum of either of the colliding particlesis finetuned to the critical value [1]. This is now called the BSW effect. See Harada andKimura [2] for a brief review and references therein for further details. Particle collisionwith high CM energy had already been noticed by Piran, Shaham, and Katz in the study ofenergy extraction from collisional events in the ergoregion, which is called collisional Penroseprocess [3, 4]. This process is typically as follows. We consider the reaction of particles 1and 2 into particles 3 and 4 in the ergoregion, where particle 3 escapes to the infinity afterthe collision, while particle 4 falls into the black hole possibly with negative energy due tothe existence of the ergosphere. If one defines an energy-extraction efficiency as η := energy of the escaping particletotal energy of the injected particles = E E + E , (1.1)the energy extraction ( η >
1) from the black hole is possible provided E < .
19 approximately. However, if a very massive particle is allowed to2e produced, which has negative energy and necessarily falls into the black hole, the upperlimit is increased to (2 + √ ≃ .
9, confirming the numerical result in Ref. [6].The organization of this paper is as follows. In Sec. II, we prepare for the analysis ofcollisional Penrose process, reviewing geodesic motions and near-horizon collision in the Kerrblack hole. In Sec. III A, we investigate the upper limits of the energy of escaping particleand of the energy-extraction efficiency in the case of produced particles as massive as thecolliding particles. In Sec. III B, we will see that how the upper limits will be significantlyincreased if we take the production of a very massive particles due to the BSW effect intoaccount. Section IV is devoted to conclusion. We adopt the geometrized unit in which c = G = 1. II. PRELIMINARIESA. Geodesics in the Kerr black hole
The spacetime metric of the Kerr black hole is given by g µν dx µ dx ν = − (cid:18) − M rρ (cid:19) dt − M ar sin θρ dtdϕ + ρ ∆ dr + ρ dθ + (cid:18) r + a + 2 M a r sin θρ (cid:19) sin θdϕ , (2.1)where ρ ( r, θ ) := r + a cos θ , ∆( r ) := r − M r + a and M and a (0 ≤ a ≤ M ) are themass and spin parameters, respectively. ∆( r ) vanishes at r ± := M ± √ M − a , and r = r + and r = r − correspond to the event horizon and Cauchy horizon, respectively.This spacetime is stationary and axisymmetric with Killing vectors ∂ t and ∂ ϕ . Theconserved energy E and angular momentum L of a particle with the four-momentum p µ are given by E = − g µν ( ∂ t ) µ p ν = − g tµ p µ and L = g µν ( ∂ ϕ ) µ p ν = g ϕµ p µ , respectively. Thecomponents of the four-momentum are given in terms of these conserved charges as (e.g. [9,10]) p t = 1∆ (cid:20)(cid:18) r + a + 2 M a r (cid:19) E − M ar L (cid:21) , p ϕ = 1∆ (cid:20) M ar E + (cid:18) − Mr (cid:19) L (cid:21) , (2.2)12 ( p r ) + V = 0 , V ( r ) = − M m r + L − a ( E − m )2 r − M ( L − aE ) r − E − m , (2.3)3here m ( ≥
0) denotes the mass of the particle, and the motion is assumed to be confinedin the equatorial plane, where θ = π/ p t > r → r + + 0 reduces to E − Ω H L ≥ , (2.4)where Ω H := a/ ( r + a ) is the angular velocity of the horizon. We call a particle a criticalparticle if it has a critical angular momentum E/ Ω H , for which the equality in Eq. (2.4) holds.Accordingly, we call a particle with L < E/ Ω H ( L > E/ Ω H ) a subcritical (supercritical)particle.In the rest of this paper, we only consider the extremal black hole a = M . In this case,the forward-in-time condition for general position r > r + is written as12 (cid:20)(cid:16) rM (cid:17) + rM + 2 (cid:21) E > ˜ L, (2.5)where ˜ L := L/M is a reduced angular momentum. (a) (b)FIG. 1. (a) The radial turning points are plotted for a massless particle with b = b ± ( r ). (b)The radial turning points are plotted for a massive particle with ˜ L = ˜ L ± ( r, E, m ), where we set E = m = 1. The negative energy particles are confined to the gray regions. B. Radial turning points of a geodesic particle
Here, we are concerned with a particle that comes from or escapes to the infinity, whichrequires the effective potential V in Eq. (2.3) is non-positive for large r . This requires4 ≥ m .For a massless particle ( m = 0) , solving V = 0 for the impact parameter b := L/E , weobtain b = b ± ( r ), where b + ( r ) := r + M, b − ( r ) := − (cid:18) r + M + 4 M r − M (cid:19) . (2.6)This means that a particle of which impact parameter b = b ± ( r ) has a turning point at r .The numerical plot of b = b ± ( r ) is given in Fig. 1(a). As r increases from M to infinity, b + ( r )begins with 2 M and monotonically increases to infinity. As r increases from M to 2 M , b − ( r )begins with 2 M and monotonically increases to infinity. As r increases from 2 M to infinity, b − ( r ) begins with negative infinity, monotonically increases to a local maximum − M at r = 4 M and monotonically decreases to negative infinity. Therefore, for 2 M < b < b + ( r ∗ ),the particle can escape to the infinity irrespective of the sign of the initial velocity, whichis shown by the yellow region in Fig. 1(a) and where we denote the radial coordinate ofcollision by r ∗ . On the other hand, for M < r ∗ < M and − M < b ≤ M , the particle canescape to the infinity only if it moves initially outwardly, which is shown by the blue regionof Fig. 1(a).For a massive particle ( m > V ( r ) = 0 for ˜ L , we obtain ˜ L = ˜ L ± ( r, E, m ), where˜ L ± ( r, E, m ) := − M E ± r ( r − M ) p E − m + 2 M m /rM ( r − M ) . (2.7)This means that a particle with E , m , and ˜ L = ˜ L ( r, E, m ) has a turning point at r . Thenumerical plot of ˜ L = ˜ L ± ( r, E, m ) is given in Fig. 1(b). As r increases from M to infinity,˜ L + ( r, E, m ) begins with 2 E and monotonically increases to infinity. As r increases from M to2 M , ˜ L − ( r, E, m ) begins with 2 E and monotonically increases to infinity. As r increases from2 M to infinity, ˜ L − ( r, E, m ) begins with negative infinity, monotonically increases to a localmaximum ˜ L max ( E, m ) ( <
0) at r = r max and monotonically decreases to negative infinity.Therefore, for 2 E < ˜ L < ˜ L + ( r ∗ , E, m ), the particle can escape to infinity irrespective of thesign of the initial velocity, which is shown by the yellow region of Fig. 1(b). On the otherhand, for M < r ∗ < r max and ˜ L max ( E, m ) < ˜ L ≤ E , the particle can escape to infinity onlyif it moves initially outwardly, which is shown by the blue region of Fig. 1(b).5 . Particle collision on the horizon Let us consider the reaction of two colliding particles, named particles 1 and 2, to twoproduct particles, 3 and 4. The local conservation of four-momenta can be written as p µ + p µ = p µ + p µ . (2.8)The t - and ϕ -components of Eq. (2.8) represent the conservations of energy and angularmomentum E + E = E + E and ˜ L + ˜ L = ˜ L + ˜ L , (2.9)respectively. The r -component represents the conservation of radial momentum σ | p r | + σ | p r | = σ | p r | + σ | p r | , (2.10)where σ i = sgn( p ri ) for i = 1 , , ,
4. Note that the mass and four-momentum of particle 4can be written in terms of those of the other particles using the momentum conservationsand identity m = − p µ p µ . From Eq. (2.3), we obtain | p ri | = 2 E i − ˜ L i , (2.11)where we have used the forward-in-time condition to open the square root.In the rest of this paper, we assume particle 1 to be critical ( ˜ L = 2 E ), particle 2 to besubcritical ( ˜ L < E ), particle 3 to escape to infinity, and particle 4 to fall into the blackhole with negative energy ( E < σ (2 E − ˜ L ) = σ (2 E − ˜ L ) (for σ = σ ) σ h E − ˜ L ) − (2 E − ˜ L ) i (for σ = − σ ) . (2.12)When we choose σ = −
1, several situations are possible depending on the values of σ and σ . We will see, however, that only a few situations among them are interestingfor our considerations. If σ = σ = 1, from Eq. (2.12), we obtain 2 E − ˜ L = 0, whichcontradicts our assumption. If σ = σ = −
1, we obtain 2 E − ˜ L = 2 E − ˜ L + 2 E − ˜ L ,which implies that particle 3 can be either critical ( ˜ E = 2 E ) or subcritical ( ˜ E < E ).Nevertheless, only the critical case is interesting since a subcritical ingoing particle cannotescape to infinity. If σ = − σ = 1, we obtain 2 E − ˜ L = 0 (particle 3 is critical). If6 = − σ = −
1, we obtain 2 E − ˜ L = 2 E − ˜ L , which implies particle 3 is subcritical, itis not interesting since a subcritical ingoing particle cannot escape to infinity again.When we choose σ = 1, only a few situations are interesting again. If σ = σ = 1, weobtain 2 E − ˜ L = (2 E − ˜ L ) + (2 E − ˜ L ), which implies that particle 3 can be eithercritical or subcritical. Since σ = 1 in the present case, particle 3 can escape to infinityeven if it is subcritical and outgoing, provided b or ˜ L satisfy b max , < b (massless case)or ˜ L max , ( E , m ) < ˜ L (massive case). If σ = σ = −
1, we obtain 2 E − ˜ L = 0, whichcontradicts our assumption. If σ = − σ = 1, we obtain 2 E − ˜ L = 2 E − ˜ L , which impliesthat particle 3 is subcritical. If σ = − σ = −
1, we obtain 2 E − ˜ L = 0.From the above considerations, we see the following four situations are interesting for theenergy efficiency. Case A: σ = − σ = 1 and particle 2 issubcritical. In this case, particle 2 must be created inside the ergoregion by some precedingprocess. In the Appendix, we will see that the upper limit of the energy extraction efficiencyis in case B is the same as in case A. Hence, we will focus on case A in Secs. III A and III B. D. Near-horizon and near-critical behaviors of a particle
We parameterize the radial position of near-horizon collision r ∗ as r ∗ = M − ǫ , < ǫ ≪ , (2.13)and the near-critical angular momentum as˜ L = 2 E (1 + δ ) , | δ | ≪ . (2.14)Then, we assume that δ can be expanded in powers of ǫ as δ = δ (1) ǫ + δ (2) ǫ + O ( ǫ ) . (2.15)Under the assumption that particle 3 escapes to the infinity, ˜ L ≤ ˜ L + ( r ∗ , E , m ) has tohold, which implies δ ≤ E − p E + m E ! ǫ + E p E + m − E − m E p E + m ! ǫ + O ( ǫ ) . (2.16)The forward-in-time condition for the near-horizon and near-critical particle implies δ < ǫ + 74 ǫ + O ( ǫ ) . (2.17)Therefore, the forward-in-time condition is always satisfied.7 . Expansion of | p ri | by ǫ Let us consider the series expansion of the radial momentum in powers of ǫ for eachparticle. Since we have assumed particle 1 to be critical and particle 2 to be subcritical, | p r | and | p r | are expanded as | p r | = q E − m ǫ − E p E − m ǫ + O ( ǫ ) , (2.18) | p r | = (2 E − ˜ L ) − E − ˜ L ) ǫ + (3 E − ˜ L )( E − ˜ L ) − m E − ˜ L ) ǫ + O ( ǫ ) . (2.19)If particle 3 is near critical and particle 4 is subcritical, the expansions of | p r | and | p r | aregiven by | p r | = q E (cid:2) − δ (1) ) − (cid:3) − m ǫ − E (cid:2) − δ (1) − δ (2) )(1 − δ (1) ) (cid:3)q E (cid:2) − δ (1) ) − (cid:3) − m ǫ + O ( ǫ ) , (2.20) | p r | = (2 E − ˜ L ) − h E − ˜ L ) + 2 E (1 − δ (1) ) − E i ǫ + " (2 E − ˜ L )2 − E (2 δ (1) − δ (2) ) − ( E + E − E ) + m E − ˜ L ) ǫ + O ( ǫ ) . (2.21)If particle 3 is subcritical and particle 4 is near critical, the expansion of | p r | and | p r | areobtained by exchanging subscripts 3 and 4 in Eqs. (2.20) and (2.21). If both particles 3 and4 are subcritical, the expansion of | p r | and | p r | are given by | p r | = (2 E − ˜ L ) − E − ˜ L ) ǫ + (3 E − ˜ L )( E − ˜ L ) − m E − ˜ L ) ǫ + O ( ǫ ) , (2.22) | p r | = (2 E − ˜ L ) − (2 E − ˜ L ) + 2( E − E + E + ˜ L − ˜ L ) ǫ + ( E − E + E + ˜ L − ˜ L )( E + 3 E − E − ˜ L + ˜ L ) + m E − ˜ L ) − (2 E − ˜ L )] ǫ + O ( ǫ ) . (2.23) III. ENERGY-EXTRACTION EFFICIENCYA. Case for m = O ( ǫ ) We focus on case A, where σ = σ = − σ = − O ( ǫ ) and O ( ǫ ) termsof radial momentum conservation Eq. (2.10) yield σ q E − m + 2 E − E (1 − δ (1) ) = σ q E [4(1 − δ (1) ) − − m , (3.1)8nd σ E p E − m + (3 E − ˜ L )( E − ˜ L ) − m E − ˜ L ) = σ E [1 − δ (1) − δ (2) )(1 − δ (1) )] p E [4(1 − δ (1) ) − − m + (2 E − ˜ L )2 − E (2 δ (1) − δ (2) ) − ( E + E − E ) + m E − ˜ L ) , (3.2)respectively.When we choose σ = 1, Eq. (3.1) implies B − E (1 − δ (1) ) = σ q E [4(1 − δ (1) ) − − m , (3.3)where B := 2 E + p E − m ( > − δ (1) = B + E + m B E , (3.4)which implies δ (1) , max − δ (1) = ( B − p E + m ) B E ≥ . (3.5)Substituting Eq. (3.4) into the left-hand side of Eq. (3.3), we obtain B − E + m B = 2 σ q E [4( δ (1) − − − m . (3.6)This implies E ≤ ˜ λ := p B − m ( E ≥ ˜ λ ) for σ = 1 ( σ = − σ = −
1, we need δ (1) ≥ δ (1) ≥ E − B E + B + m ≤ . (3.7)Inequality (3.7) is satisfied by˜ λ − ≤ E ≤ ˜ λ + , ˜ λ ± := 2 B ± q B − m , (3.8)where the discriminant D of Eq. (3.7) has to satisfy D/ B − m ≥
0. Since particle3 escapes to infinity, it is marginally bound or unbound ( E ≥ m ). m ≤ ˜ λ + has to besatisfied so that Eq. (3.8) and E ≥ m have an intersection. The relation m ≤ ˜ λ + issatisfied if m ≤ √ B , which is equivalent to D/ ≥
0. Therefore, if D/ ≥ E ≥ m always have an intersection.˜ λ and ˜ λ + are the upper limits on E for σ = 1 and σ = −
1, respectively. Since ˜ λ + islarger than ˜ λ , we concentrate on the case of σ = −
1. The maximum of ˜ λ + is given by˜ λ + , max = (2 + √ E , (3.9)9here we have assumed m = m = 0.Next, we consider the O ( ǫ ) terms in the radial momentum conservation. E = ˜ λ + canbe realized when δ (1) = 0. Substituting δ (1) = 0, σ = 1, and σ = − m , we obtain m = − E − ˜ L ) h E p E − m + ˜ λ q λ − m + 2 δ (2) ˜ λ + q λ − m (cid:18) λ + − q λ − m (cid:19) i +( E + m ) − ( E + E − ˜ λ + ) . (3.10)We need δ > δ (1) = 0, we need δ (2) ≥
0. This implies the first term on the right-hand side of Eq. (3.10) is negative. Thelower limit of E is then given by E ≥ ˜ κ := 12 (cid:20) (˜ λ + − E ) − m (˜ λ + − E ) (cid:21) . (3.11)Since we assume that particle 2 comes from infinity, it must be marginally bound or unbound( E ≥ m ). Therefore, we have to compare ˜ κ with m . If ˜ κ ≥ m , i.e., ( √ − λ + − E ) ≥ m , the lower limit of E is ˜ κ . Thus, we find η ≤ ˜ λ + E + ˜ κ = 2˜ λ + (˜ λ + − E )˜ λ − E − m =: f ( m ) . (3.12)One can see that f ( m ) defined above begins with 2˜ λ + / (˜ λ + + E ) and monotonically increasesto (3 + 2 √ λ + (2 + √ E + (1 + √ λ + =: g (˜ λ + ) , (3.13)as m increase from 0 to ( √ − λ + − E ). Since g (˜ λ + ) is a monotonically increasingfunction of ˜ λ + , the maximum of g (˜ λ + ) is given by g (˜ λ + , max ) = 179 + 186 √ √ √ ≃ . . (3.14)If ˜ κ ≤ m , i.e., ( √ − λ + − E ) ≤ m , the lower limit of E is m . Thus, we find η ≤ ˜ λ + E + m , (3.15)and easily notice that the right-hand side monotonically decreases as m increase. For theabove reason, the maximum of the right-hand side is about 2 . η max ≃ .
19, which isrealized σ = 1, σ = σ = σ = − m = m = 0, and δ (1) = 0.10ext, let us consider the case of σ = −
1. Equation (3.1) implies A − E (1 − δ (1) ) = σ r E h − δ (1) ) − i − m , (3.16)where A := 2 E − p E − m ( > σ = 1 case applies. B → A := 2 E − q E − m , ˜ λ → λ := q A − m , ˜ λ ± → λ ± := 2 A ± q A − m , ˜ κ → κ := 12 (cid:20) ( λ + − E ) − m ( λ + − E ) (cid:21) . However, the value of the upper limit is different. Since λ + is larger than λ , we concentrateon the case σ = −
1. The maximum of λ + is given by λ + , max = (2 + √ − √ E , (3.17)where we have assumed m = E and m = 0. We have seen that E = λ + can be realized δ (1) = 0. Substituting δ (1) = 0 and σ = σ = − m ,we obtain m = − (cid:16) E − ˜ L (cid:17) h − E p E − m + λ p λ − m + 2 δ (2) λ + p λ − m (cid:18) λ + − q λ − m (cid:19) i +( E + m ) − ( E + E − λ + ) . (3.18)Again, we need δ > δ (1) = 0, we need δ (2) ≥
0. In this case, we can prove the first term on the right-hand side of Eq. (3.18) isnegative (See Ref. [5, Appendix B]). The lower limit of E is given by E ≥ κ := 12 (cid:20) ( λ + − E ) − m ( λ + − E ) (cid:21) . (3.19)From the discussion similar to that in the case of σ = 1, the upper limit of the energyextraction efficiency is also obtained as η max = 2 + √ √ ≃ . , (3.20)which is realized σ = σ = σ = σ = − E = m , m = 0, δ (1) = 0, and m /m ≃ . . Case for m = O ( ǫ − / ) In general, the CM energy of particles 1 and 2 is given by E = − ( p µ + p µ )( p µ + p µ ) . (3.21)For example, in the original BSW process [1], in which ˜ L = 2 E , ˜ L < E and σ = σ = −
1, the leading term of the CM energy is E ≃ E − ˜ L )(2 E − p E − m ) ǫ . (3.22)In such a high-energy collision, the masses of the product particles (particles 3 and 4) canbecome large with the following restriction m + m ≤ E cm . (3.23)Since E cm for the collision between a critical particle and a subcritical particle is proportionalto ǫ − / for both a rear-end collision and head-on collision, we assume m = O ( ǫ − / ) . (3.24)Then, we assume that particle 4 is very massive as m = µ ǫ + ν , (3.25)where µ ( >
0) and ν are constants. In fact, we can write m in terms of the quantities ofthe other particles, using the momentum conservation, as m = − ( p µ + p µ − p µ )( p µ + p µ − p µ ) = E + m + 2( p µ + p µ ) p µ (3.26)where we have assumed ˜ L = 2 E (1 + δ ). This implies µ = 2(2 E − ˜ L ) h E − E (1 − δ (1) ) + σ q E − m − σ q E [4(1 − δ (1) ) − − m i . (3.27)The expansion of radial momentum of particle 4 is given by | p r | = (2 E − ˜ L ) − (cid:20) E − ˜ L ) + 2 E (1 − δ (1) ) − E + µ E − ˜ L ) (cid:21) ǫ + F ,ǫ ǫ + O ( ǫ ) , (3.28)12here F ,ǫ := (2 E − ˜ L )2 − E (2 δ (1) − δ (2) ) − ( E + E − E ) + ν E − ˜ L )+ [ E − E + ˜ L − E (1 − δ (1) )] µ (2 E − ˜ L ) − µ E − ˜ L ) . (3.29)This implies that the O ( ǫ ) and O ( ǫ ) terms of the radial momentum conservation yieldEq. (3.27) and σ E p E − m + (3 E − ˜ L )( E − ˜ L ) − m E − ˜ L ) − F ,ǫ = σ E [1 + 4(2 δ (1) − δ (2) )( δ (1) − p E [4( δ (1) − − − m , (3.30)respectively.If we define C := 2 E + σ q E − m − µ E − ˜ L ) > , (3.31)we can discuss the upper limit of E in the way similar to that in Sec. III A. As we havealready seen, we need σ = − δ (1) ≥ E . Setting δ (1) ≥ E − CE + C + m ≤ . (3.32)This inequality is satisfied by¯ λ − ≤ E ≤ ¯ λ + , ¯ λ ± := 2 C ± q C − m . (3.33)The maximum value of ¯ λ + is given by¯ λ + , max = (2 + √ E − (2 + √ µ E − ˜ L ) , (3.34)where we have assume m = m = 0 and σ = 1. E = ¯ λ + is realized when δ (1) = 0.Substituting δ (1) = 0, σ = 1, and σ = − ν , weobtain ν = − E − ˜ L ) h E p E − m + ¯ λ q λ − m + 2 δ (2) ¯ λ + q λ − m (cid:0) λ + − q λ − m (cid:1)i + ( E + m ) − ( E + E − ¯ λ + ) + 2( E − E + ˜ L − ¯ λ + ) µ (2 E − ˜ L ) − µ (2 E − ˜ L ) . (3.35)13n Sec. III A, we have used the above equation with µ = 0 and ν = m ≥ E . However, in the present case, the sign of ν is not restricted to be positiveif µ >
0. Hence, we use Eq. (3.35) not for the estimate of the lower limit of E but for thedetermination of ν . Therefore, the upper limit of the energy-extraction efficiency is givenby η max = ¯ λ + , max E + E = (2 + √ E − (2+ √ µ E − ˜ L ) E + E . (3.36)From the above equation, we see that if E E ≫ µ , η max is approximately given by (2 + √ E / ( E + E ). Moreover, for E /E ≪
1, we find that the escaping massless particle hasthe energy that is nearly equal to (2 + √ ≃ . E cannot be exactly equal to (2 + √ E , for the following reason. FromEq. (3.34), E = (2 + √ E is realized when m = m = 0, δ (1) = 0 and µ = 0. However, µ = 0 implies m = O ( ǫ ) and hence ν must be positive. Substituting µ = 0 into Eq. (3.35)and requiring ν >
0, we obtain the lower limit of E , which is given by Eq. (3.11). In thiscase, we have already seen that the efficiency cannot reach (2 + √ ≃ . p r and p r have O ( ǫ / ) terms. Since there is no half-integer order terms on the left-hand sideof Eq. (2.10), O ( ǫ / ) term in the sum of p r and p r has to be zero. In fact, if one assumesparticle 3 to be very energetic so that E = ρ /ǫ + φ , where ρ ( >
0) and φ are constants,the O ( ǫ / ) terms in the radial momentum conservation yield σ q ρ [4(1 − δ (1) ) − − − δ (1) ) √ ρ = 0 . (3.37)There is no solution for ρ under the assumption of ρ >
0. By similar arguments, onecan conclude that there is no energy extraction, expect for the case where particle 4 is verymassive.
IV. CONCLUSION
We have studied particle collision and energy-extraction efficiency, where a critical particle(particle 1) and subcritical particle (particle 2) collide near the event horizon of an extremal14err black hole and then two particles are produced, one of which escapes to infinity (particle3) and another falls into the black hole (particle 4).There is an upper limit of the energy of particle 3, which is given by E , max = (2+ √ E .This is realized in the situation where particles 1 and 3 are massless, particle 3 is near critical,particle 1 is temporary outgoing and particles 2, 3 and 4 are temporarily ingoing. The energy-extraction efficiency, however, is bounded by 2 .
19 approximately, under the assumption thatparticle 4 has a mass of O ( ǫ ), where ǫ parametrizes the distance from the collision point tothe horizon.Since the CM energy of the near-horizon collision can be arbitrarily large, the collisioncan produce a heavy particle. From this viewpoint, we have considered the case in which themass of particle 4 is of O ( ǫ − / ). In this case, we found that the upper limit of the efficiencycan indeed reach (2 + √ ≃ .
9. Thus, if the efficiency as large as 10 is observed for acollisional Penrose process, this strongly suggests the production of a heavy particle as aresult of the collision of high CM energy.Finally, let us give an example of collision with a high energy-extraction efficiency. Weassume that particles 1 and 2 are protons, 3 is a photon, and 4 is some heavy particle andthat the collision happens in the vicinity of the horizon with ǫ = 10 − . Using m = m ∼ E cm ∼ p m m /ǫ ∼ GeV. If m ∼ GeV, we have µ ∼ m ǫ ∼ − GeV which is much smaller than E E . In this case, the efficiency is typically as large as10. Note added:
While completing the current paper, the authors found that a paper [11]studying a similar problem appeared in arXiv. It will be interesting to compare the resultin Ref. [11] with that in the current paper.
ACKNOWLEDGMENTS
The authors would like to thank T. Igata, M. Kimura, T. Kobayashi, K. Nakao, M. Patiland S. Yokoyama. This work was supported by JSPS KAKENHI Grant Numbers 26400282(TH) and 15K05086 (UM). 15 ppendix A: Collision between a critical particle and an outgoing subcritical par-ticle
Here we consider case B, where σ = 1 and particle 2 is subcritical. We can divide thiscase into the following three subcases: B1: σ = σ = 1 ( σ and σ can be either 1 or − σ = σ = σ = 1. Both particles 3 and 4 are subcritical, and2 E − ˜ L = (2 E − ˜ L ) + (2 E − ˜ L ). B3: σ = σ = 1, σ = −
1. Particle 3 is subcritical,and 2 E − ˜ L = 2 E − ˜ L , which implies particle 4 to be critical.For case B1, the O ( ǫ ) terms in the radial momentum conservation yield σ q E − m − E + 2 E (1 − δ (1) ) = σ q E [4( δ (1) − − − m . (A1)This is obtained from Eq. (3.1) after replacing σ and σ with − σ and − σ , respectively.The equation obtained from the O ( ǫ ) terms of the radial momentum conservation is alsothe same as Eq. (3.2) after the above replacement. Therefore, η max in this case is given bythat in Sec. III A after changing the signs σ i ( i = 1 , , , σ requires the argument on the turning point, but one can finally see that this prescription isvalid.For case B2, the O ( ǫ ) terms in the radial momentum conservation equations yield σ q E − m − E = 0 . (A2)This equation has no solution for E under the assumption E > m ≥
0. Thus, thereis no energy extraction in this case.For case B3, the O ( ǫ ) terms of the radial momentum conservation equations yield2 E − σ q E − m − E (1 − δ (1) ) = q E [4( δ (1) − − − m . (A3)Equation (A3) implies B − E (1 − δ (1) ) = q E [4( δ (1) − − − m (A4)for σ = − − p B − m ≤ E <
0. From the forward-in-time condition and theargument on the turning point for a near-critical particle with negative energy, the restrictionon δ (1) is obtained as δ (1) ≥ − p E + m E . (A5)16quaring the both sides of Eq. (A4) and then solving it for E , we obtain E = 2(1 − δ (1) ) B + q B [4( δ (1) − − − m . (A6)This implies the lower limit of E is given by E , min = − (2 + √ E , (A7)where m = m = 0 and δ (1) = 3 / O ( ǫ ) terms of the radial momentumconservation equation yield − σ E p E − m + (3 E − ˜ L )( E − ˜ L ) − m E − ˜ L ) = E [1 − δ (1) − δ (2) )(1 − δ (1) )] p E [4( δ (1) − − − m + 2 E − ˜ L − E (2 δ (1) − δ (2) ) − ( E + E − E ) + m E − ˜ L ) . (A8)The lower limit of E can be realized only for m = m = 0 and δ (1) = 3 /
2. Because thedenominator of the first term on the right-hand side of Eq. (A8) becomes zero when m = 0and δ (1) = 3 /
2, we need 1 − δ (1) − δ (2) )(1 − δ (1) ) = 0, which is possible when δ (2) = 7 / η max = 1 − E , min E + E = (3 + √ E + E E + E < √ , (A9)where we have assumed m = m = 0, δ (1) = 3 /
2, and δ (2) = 7 / m = O ( ǫ − / ), the energy-extraction efficiency can be larger as in case A.For case B1, the result is the same as for case A after replacing σ i with − σ i ( i = 1 , , , ≃ .
9, only if a very massive particle is producedto fall into the black hole. For cases B2 and B3, the energy extraction is quite modest.Namely, the efficient energy extraction is not realized even when an subcritical outgoingparticles is considered, as long as its counterpart is critical. [1] M. Ba˜nados, J. Silk, and S. M. West, “Kerr Black Holes as Particle Accelerators to ArbitrarilyHigh Energy”, Phys. Rev. Lett. (2009) 111102 [arXiv:0909.0169 [hep-ph]].[2] T. Harada and M. Kimura, “Black holes as particle accelerators: a brief review,” Class. Quant.Grav. (2014) 243001 [arXiv:1409.7502 [gr-qc]].[3] T. Piran, J. Shaham, and J. Katz, “High efficiency of the Penrose mechanism for particlecollisions”, Astrophys. J. (1975) L107.
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