Super-Penrose process due to collisions inside ergosphere
aa r X i v : . [ g r- q c ] J un Super-Penrose process due to collisions inside ergosphere
O. B. Zaslavskii
Department of Physics and Technology,Kharkov V.N. Karazin National University,4 Svoboda Square, Kharkov 61022, Ukraine andInstitute of Mathematics and Mechanics, Kazan Federal University,18 Kremlyovskaya St., Kazan 420008, Russia ∗ If two particles collide inside the ergosphere, the energy in the centre of mass framecan be made unbound provided at least one of particles has a large negative angularmomentum (A. A. Grib and Yu. V. Pavlov, Europhys. Lett. 101, 20004 (2013)). Weshow that the same condition can give rise to unbounded Killing energy of debris atinfinity, i.e. super-Penrose process. Proximity of the point of collision to the blackhole horizon is not required.
PACS numbers: 04.70.Bw, 97.60.Lf
I. INTRODUCTION
Investigation of high energy collisions in the black hole background now attracts muchattention. It was stimulated by the observation that collision of two particles moving towardsa black hole can produce an indefinitely large energy E c.m. in the centre of mass frame [1].The same happens if particles move in opposite directions [2] - [4]. In this context, thereare two different issues, connected with obtaining (i) large energies in the centre of mass E c.m. and (ii) large Killing energies E of debris at infinity. It turned out that there areserious restrictions on E even in spite of large E c.m. since strong gravitational redshift almostcompensates the excess of energy [5] - [7]. Nonetheless, there exist scenarios, in which E is also significantly amplified or even unbounded (hence, extraction of energy from a blackhole is big) [8] - [10]. Such cases are called the super-Penrose process in [9], and we stick ∗ Electronic address: [email protected] to this terminology. However, the super-Penrose process near black holes has its own severerestrictions [11], [12].Up to now, all discussion in literature concerning the super-Penrose process in the blackhole background applied to collisions near the event horizon only. In the present paper weshow that there exists an alternative mechanism in which large E and E c.m. are compatiblewith each other. It is based on the Grib-Pavlov mechanism of collision. It was shown in[13] that if two particles collide inside the ergosphere of the Kerr metric and at least oneof particles has the large negative angular momentum L , the resulting E c.m. is also large.Later on, it was shown in [14] that this is a universal property of ergoregions of genericaxially symmetric rotating black holes. In both aforementioned papers, only the propertiesof E c.m. were considered. Below, we will see that for such a type of collision the super-Penrose mechanism is possible. It is worth stressing that, although in the scenarios underdiscussion L for initial particles is supposed to be large, their Killing energies E are finite.The similar combination (large L and modest E ) occurs also in some other scenarios of highenergy collisions - say, in the vicinity of magnetized black holes [15] - [17] . II. BASIC FORMULAS
Let us consider the metric ds = − N dt + g φ ( dφ − ωdt ) + dr A + g θ dθ . (1)We assume that all metric coefficient do not depend on t and φ . This gives rise to theconservation of the energy E = − mu and angular momentum L = mu φ . Here, m is theparticle’s mass, u µ = dx µ dt is the four-velocity, τ is the proper time. In what follows, werestrict ourselves by motion in the equatorial plane θ = π . For such a motion, one canalways redefine the radial coordinate to achieve N = A . Then, equations of motion read m dtdτ = XN , (2) X = E − ωL , (3) m dφdτ = Lg φ + ωXN , (4) m drdτ = σZ , Z = s X − N ( L g φ + m ) . (5)Here, σ = ± dtdτ >
0, whence (for N = 0) X > . (6)If two particles 1 and 2 collide to produce particles 3 and 4, the conservation of energyand angular momentum gives us E + E = E + E , (7) L + L = L + L . (8)The conservation of the radial momentum reads σ Z + σ Z = σ Z + σ Z . (9)It is implied that masses of all particles are fixed. Say, one can take m = m , m = m for the elastic collision or m = m = 0 for annihilation of two initial particles into gammaquanta. The quantities E , E , L and L are fixed. We can also fix, say, L . Then, threeequations (7) - (9) determine three unknowns E , E , L . III. SCENARIOS OF COLLISION
We assume that particle 3 moves outward right after collision and escapes, so σ = +1.By assumption, particle 2 has a large negative angular momentum L = − | L | . In general,eq. (9) is quite cumbersome algebraically. As our goal is just to demonstrate the existenceof the super-Penrose process, we will make several simplifications. We assume that L = L b, L = aL , (10)where b and a are numbers. Then, the conservation of the angular momentum entails that L = L (1 + b − a ) . (11)In general, this still leads to rather bulky algebraic expressions in (9). We restrict ourselvesby the case a = 1 + b , so L = 0. This is quite sufficient for our purpose - to demonstratethe existence of the super-Penrose process. We are interested in the scenario in which E islarge and has the order | L | , so we put E = | L | y + O (1), (12) y >
0. (13)Correspondingly, E = − | L | y + O (1), (14)so collision must occur inside the ergosphere where negative energies are allowed. Then, X ≈ ωb | L | , (15) X ≈ ω | L | , (16) X ≈ | L | y , (17) X ≈ | L | [ ω (1 + b ) − y ]. (18)Here, condition (6) for particle 1 requires b >
0. It is satisfied automatically for particle 2.It is also satisfied for particle 3, provided (13) holds true. For particle 4 it gives us ω (1 + b ) − y >
0. (19)By substitution into (9), we have in the leading order in L the equationΩ c ( σ + σ b )) − y = σ p [( y − ω c (1 + b )] + (1 + b ) (Ω c − ω c ), (20)where subscript ”c’ means that the corresponding quantity is taken in the point of collision,Ω = r g g φ , g = − N + g φ ω . (21)As inside the ergosphere g >
0, the quantity Ω is real. This is again the point where theproperties of the ergoregion come into play.If eq. (20) has a positive root y and for this root the forward-in time condition (6) issatisfied, the super-Penrose process does occur.Taking the square of (20), one can find that b Ω c ( ε −
1) + y [(1 + b ) ω c − Ω c ( σ + bσ )] = 0, (22)where ε = σ σ . (23)As we are interested in the existence of the root y = 0, we must take ε = − . (24)Then, y = 2 b Ω c ( b + 1) ω c + Ω c σ ( b −
1) , (25) X | L | = ω c (1 + b ) − y = V ( b + 1) ω c + Ω c σ ( b −
1) , (26)where V = ω c (1 + b ) + ω c Ω c σ ( b − − b Ω c . (27)Taking into account that according to (21), ω > Ω, it is seen that for any b >
0, thedenominator in (26) is positive. For the numerator we have the condition
V >
0. We canrewrite (27) as V = 2 b ( ω c − Ω c ) + ω c [ ω c (1 + b ) + σ Ω c ( b − . (28)As ω > Ω , it is clear from (28) that indeed V > σ to avoid fake roots after taking the square.This sign must coincide with that of the left hand side of (20). It is straightforward to checkthat signσ = sign [ ω c σ (1 − b ) − Ω c (1 + b )]. (29)For example, for σ = − b <
1, we must take σ = −
1. However, if, say, σ = − b > ω > Ω b +1 b − , we have σ = +1.After collision, particle 3 moves away from a black hole. In doing so, there are no turningpoints for it. Indeed, for this particle L = 0 and Z = E − N g φ m , (30)where we took into account the mass term which was discarded before in (5) as smallcorrection. Here, E = O ( L ) is large, the second term is finite, so indeed Z >
0. Asfar as particle 4 is concerned, it falls into a black hole, if σ = −
1. If σ = +1, it movesfrom a black hole to the turning point and bounces back. Particle 4 cannot escape to theasymptotically flat infinity since its energy is negative.Thus we succeed in the sense that the unbounded energy E is obtained. For this purpose,it is necessary in our scenario with L = 0 that particles 1 and 2 move in the oppositedirections before collisions according to (23), (24). IV. ENERGY IN THE CENTRE OF MASS
To evaluate the energy of the centre of mass E c.m. , one can use the known formula (see,e.g. eq. 19 of [14] in which one should put θ = const ). It is more convenient to apply it tothe pair of particles 3 and 4 than to the original ones 1 and 2 since now L = 0. Then, itfollows from the aforementioned formula that E c.m. = X X − σ Z Z N . (31)Taking into account (5), (17) and (18) we obtain in the main approximation E c.m. = L µN , (32) µ = y { [ ω c (1 + b ) − y ] − σ p [( ω c (1 + b ) − y ] − (1 + b ) ( ω c − Ω c ) } . (33)Obviously, µ > σ . When L → ∞ , the energy E c.m. → ∞ as well. V. COLLISIONS NEAR THE BOUNDARY OF ERGOSPHERE
In the previous section, we mainly concentrated on the case when an escaping particle3 has the angular momentum L = 0. We found that the scenario with unbounded E arepossible, provided | L | is large enough. One can ask, whether this value is singled out andwhat changes if L = 0. Although, as is said above, formulas become in general cumbersome,there is a situation when analysis can be carried out analytically in a rather simple form.This is the case when collisions occur in the vicinity of the boundary of the ergoregion (seebelow).As before, we assume that all angular momenta are proportional to L . However, nowboth the coefficients a and b introduced in the beginning of Section III are free parameters.Then, instead of eqs. (17) and (18) we have X ≈ | L | [ y + ω (1 + b − a )], (34) X ≈ | L | [ ωa − y ] . (35)Equations (15) and (16) are still valid. Using the expression (5) and taking into account(10), (11) we can write the conservation of radial momentum (9) in main approximation inthe form( σ b + σ ) Ω = p y + 2 yω (1 + b − a ) + Ω (1 + b − a ) + σ p y − yωa + a Ω . (36)Here, we put σ = +1 for the escaping particle, as before. All quantities in (36) are takenin the point of collision. When a = 1 + b , we return to (20).It is sufficient to find at least one scenario with unbounded E . Let us choose σ = 1 , σ = − , b >
1, 0 < a < b . (37)By definition, g = 0 on the boundary of the ergoregion, so Ω = 0 according to (21).Let collision occur inside the ergoregion but very close to its boundary. Then, Ω →
0. Weexpect the existence of the solution of (36) in the form yω = Ω x , (38)where x = O (1). Although Ω is small, we imply that Ω | L | is still large enough to have E large according to (12). Now we can neglect in (36) terms y inside the radicals and obtainthe equation F ( b ) = b − f ( x ) ≡ p (1 + b − a )[2 x + (1 + b − a )] − √− xa + a , (39)where we chose σ = −
1. Then, x ≤ x max = a to guarantee that the expression inside thesecond radical is nonnegative.We want to show that the positive solution of this equation with x = O (1) does exist.It is seen from (39) that f (0) = 1 + b − a . Thus f (0) < F . Meanwhile, the function f ( x )is monotonically increasing. Therefore, if we achieve f ( a ) > F , it will mean that in someintermediate point 0 < x < x max the curve f ( x ) intersects the line of constant F , so thesolution exists. This condition is rendered as b − < p (1 + b − a )(1 + b ), (40)whence b > a − a . (41)This is quite compatible with (37), so the solution does exist. For instance, we can take b = 3, a = 2. Then, eq. (39) has the form1 = √ x + 1 − √ − x (42)that has a solution x = √ . VI. SUMMARY AND CONCLUSIONS
Thus we showed that there exist scenarios in which particles collide inside the ergoregionin such a way that not only (i) their energy in the centre of mass diverges, but also (ii) theKilling energy of one of particles escaping to infinity is unbounded. This suggests a moreeasy way of extracting energy since now (i) there is no problem with the redshift and timedelay [18], [19], (ii) there is no problem with fine-tuning typical of high energy collisionsnear the horizon [1]. The restrictions of the super-Penrose process indicated in [11], [12]are also irrelevant now. We analyzed in detail two situations: (a) escaping particle 3 has L = 0 and (b) collision occurs very closely to the boundary of the ergoregion. Meanwhile,it is clear from derivation that collisions with unbounded E can occur everywhere insidethe ergosphere.A separate interesting question that remained outside the scope of the present workis the conditions under which particles with unbounded negative L can occur inside theergosphere. It was pointed out in [13] that such values can be obtained as a result ofpreceding collision. However, more thorough inspection showed that the situation is notso simple since there are obstacles against such a scenario in that collisions of particleswith finite E and L cannot give rise to indefinitely large negative L (see Sec. VI of [21]for details). Therefore, other mechanisms should be relevant here to achieve large negative L (thermal fluctuations, variable or chaotic electromagnetic fields, etc.). They are model-dependent and need separate treatment. Meanwhile, the results of our work have generalmodel-independent character irrespective of the way the initial state is prepared.One can hope that the observation made in the present work can be of use for investigationof the role of collisional Penrose process in astrophysics [20]. Acknowledgments
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