Gr-Qc] 3 Nov 2014 I’Avzef O.9,Is ,(00 147–151

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arXiv:1004.0913v2 [gr-qc] 3 Nov 2014 i’avZEF o.9,is ,(00 147–151. (2010) 3, iss. 92, vol. ZhETF, v Pis’ma o rtclrttn lc oes htisdimension- its that is so hole momentum black hole angular black the nature. less in if rotating that occur critical claimed not a 11] does not [10, it of that criticized sense authors was of the The [9] ro- in energy of the 11] the result [10, for one) The in critical that the shown unlimited. is was is it it scattering (if [9] hole made in black were Recently tating hole, black 8]. the [7, escape in ac- can particles into energy that result high taking the with with hole, process, black Penrose the rotating count the of ergosphere Greisen-Zatsepin- 6]. [5, the cosmology to in will limit hole up Kuzmin black analogously the energy to due close particles loose these photons or- Then with Unification interaction hole. Grand black to the the of escape can energy der high par- very and of one dark positive ticles the [4] the with with process one other the particles, Penrose energy, two negative in on decay that can [3] particle matter shown also collisions.” in was scattering It Grand be the can to and horizon energies the Unification dark to close of accelerated particles are cos- superheavy matter a which [1] as in acts in hole supercollider black then mic “This holes that idea black the rotating discussed we are ac- If nuclei galactic photons. visible (neutrinos), mat- tive into leptons dark quarks, particles converting — neutral of particles superheavy group processes by AUGER the formed to ter the in due by particles [2]) recently energy (see ultrahigh observed of rays sources cosmic the be can clei EPLtes 00 o.9,N.3 p 125-129. pp. 3, No. 92, Vol. 2010, Letters, JETP ntecnr fms ytmcnb tl niie in unlimited still be can system mass of centre the in limited. is energy 1) is acltoso h cteigo atce nthe in particles of scattering the of calculations First nu- galactic active that hypothesis the put we [1] In nti ae eso htteeeg fscattering of energy the that show we paper this In nprilscliin ntevcnt frttn lc ho black rotating of vicinity the in collisions particles On ntepolmo etn iheegtcprilsescaping particles energetic escapin high particle getting the of of af problem energy energy the the of in process Extraction Penrose ob the ones. can to nonextremal system due mass for of also centre but the holes in particles of energy tering a hoeia hsc n srnm eatet h eznU Herzen The Department, Astronomy and Physics Theoretical cteigo atce ntegaiainlfil frotati of field gravitational the in particles of Scattering b nttt fPolm nMcaia niern,RS 1Bo 61 RAS, Engineering, Mechanical in Problems of Institute ntecliin ewe atce ntevicinity the in particles between collisions the On A but 1 6= A frttn lc holes black rotating of 0 = . .A Grib A. 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Yu. , [email protected] iae.TeeethrznfrteKr’ lc hole black Kerr’s the value for the horizon to coor- event corresponds Schwarzschild The in hole dinates. black chargeless static the oetm ntecase the In momentum. lc oei oe–idus oriae a h form the has rotating coordinates the Boyer–Lindquist of in metric hole Kerr’s black The hole. black chargeless nrylre hnteiiiloei atcecollisions wrong. is particle hole in black one the the initial to infinity close the at than getting conclusion of larger the impossibility energy the that particles about show of [11] calculations of energies Our larger infinity. plays getting the process at for Penrose of the field role that the important occurs in It collision metric. after Kerr’s energy the of scattering traction the valid. not of are limitations holes black the the nonextremal on for scattering) 11] 11] energy (multiple [10, 10, cases of [9, some of results in results that cal- the show we and reproduce part first energy, the this In culate scattering. multiple of cases the M where − h lc oeaeresolved. are hole black the gbakhlsi osdrd ti hw htscat- that shown is It considered. is holes black ng ( h oea nnt a elre Contradictions large. be can infinity at hole the g r anvr ag ausntol o xrmlblack extremal for only not values large very tain e scnie atce aln nterotating the on falling particles consider us Let h ytmo units of system The ex- the for results the obtain we part second the In stems ftebakhole, black the of mass the is e h olso sivsiae.I ssonthat shown is It investigated. is collision the ter .TEEEG FCLIINI THE IN COLLISION OF ENERGY THE 1. iest,4,Mia t eesug 916 Russia 191186, Petersburg, St. Moika, 48, niversity, 2 + so,VO,S.Ptrbr,197,Russia 199178, Petersburg, St. V.O., lshoy, a 2 ds cos 2 b = 2 r IL FBAKHOLES BLACK OF FIELD θ 1) dt = ) = ∆ 2 r dr − H ∆ 2 ≡ 2 r Mr + 2 G M a − dθ = r h erc()describes (1) metric the 0 = ( + 2 2 dt 2 Mr c + p − sue ntepaper. the in used is 1 = − a M a + 2 ( r cos 2 sin a 2 J − + 2 2 , 2 = a dϕ θ θ a 2 2 aM . sin ) ) 2 2 sangular is dϕ θ 2 , (3) (2) (1) 2 GRIB, PAVLOV

The Cauchy horizon is For the extremal black hole A = xH = 1, lH = 2 and the expression (10) is divergent when the dimensionless r = r M M 2 a2 . (4) C ≡ − − angular momentum of one of the falling into the black p hole particles l 2. The scattering energy in the centre The surface called “the static limit” is defined by the → of mass system is increasing without limit [9]. expression Let’s note, that to get the collision with infinite en- 2 2 2 ergy one needs the infinite interval of as coordinate as r = r0 M + M a cos θ . (5) ≡ − proper time of the free falling particle. Really, from p The region of space-time between the horizon and the Eqs. (6), (8) for a particle with dimensionless angular static limit is ergosphere. momentum l and specific energy ε = 1 falling from some For equatorial (θ = π/2) geodesics in Kerr’s met- point r0 = x0M to the point rf = xf M > rH one ob- ric (1) one obtains ([12], 61): tains for the coordinate time (proper time of the ob- § server at rest far from the black hole) 2 dt 1 2 2 2Ma 2Ma = r + a + ε L , (6) x0 dτ ∆ r − r √x x3 + A2x +2A(A l) dx ∆t = M − . (x x )(x x ) 2x2 l2x + 2(A l)2 dϕ 1 2Ma 2M Z H C = ε + 1 L , (7) xf − − − − dτ ∆ r − r p (11) For the interval of proper time of the free falling to the dr 2 2M a2ε2 L2 ∆ = ε2 + (aε L)2 + − δ , (8) black hole particle one obtains from (8) dτ r3 − r2 − r2 1 x0 where δ1 = 1 for timelike geodesics (δ1 = 0 for isotropic x3/2 dx ∆τ = M . (12) geodesics), τ is the proper time of the moving particle, 2x2 l2x + 2(A l)2 ε = const is the specific energy: the particle with rest xZf − − mass m has the energy εm in the gravitational field (1); p Lm = const is the angular momentum of the particle In extremal case (A = 1, l = 2) the integrals (11), (12) diverges for x x = 1 and ∆t M2√2(x 1)−1, relative to the axis orthogonal to the plane of move- f → H ≈ f − ∆τ M ln(x 1) /√2 for x 1. ment. ≈ | f − | f → From (7), (8) for the angle of the particle falling in We denote x = r/M, xH = rH /M, xC = rC /M, A = a/M, l = L /M, ∆ = x2 2x + A2. For equatorial plane of the black hole one obtains n n x − the energy in the centre of mass frame of two colliding x0 √x (xl + 2(A l)) dx particles with angular momenta L1,L2, which are non- ∆ϕ = − . 2 2 2 relativistic at infinity (ε1 = ε2 = 1) and are moving in (x xH )(x xC ) 2x l x + 2(A l) xZf − − − − Kerr’s metric using (1), (6)–(8) one obtains [9]: p (13)

2 If A = 0, then integral (13) is divergent for xf xH . Ec.m. 1 2 6 → −1 In extremal case (A = 1, l = 2) ∆ϕ √2(xf 1) 2 = 2x (x 1)+ l1l2(2 x) 2 m x∆x − − ≈ − " for xf 1. So before collision with infinitely large en- 2 → +2A (x + 1) 2A(l1 + l2) (9) ergy the particle must commit infinitely large number − of rotations around the black hole. (2x2 + 2(l A)2 l2x) (2x2 + 2(l A)2 l2x) . Can one get the unlimited high energy of this scat- − 1 − − 1 2 − − 2 q # tering energy for the case of nonextremal black hole? Formula (8) leads to limitations on the possible values of To find the limit r rH for the black hole with → the angular momentum of falling particles: the massive a given angular momentum A one must take in (9) particle free falling in the black hole with dimensionless x = xH +α with α 0 and do calculations up to the or- 2 → 2 angular momentum A being nonrelativistic at infinity der α . Taking into account A = xH xC , xH + xC = 2, (ε = 1) to achieve the horizon of the black hole must after resolution of uncertainties in the limit α 0 one → have angular momentum from the interval obtains

2 2 1+ √1+ A = l l l =2 1+ √1 A . Ec.m.(r rH ) (l1 l2) − L ≤ ≤ R − → = 1+ − , (10) (14) 2m s 2xC (l1 lH )(l2 lH ) − − Putting the limiting values of angular momenta where lH =2xH /A. lL,lR into the formula (10) one obtains the maximal ONTHECOLLISIONSBETWEENPARTICLES 3 values of the collision energy of particles freely falling from the infinity but from some distance defined by (18) from infinity then due to the form of the potential it can have values of l = lH δ large than lR and fall on the horizon. inf 2m − E (r rH )= (15) If the particle falling from infinity with l lR arrives c.m. → √4 1 A2 ≤ − to the region defined by (18) and here it interacts with 2 1 A2 + 1+ √1+ A + √1 A other particles of the accretion disc or it decays into a − − . ×s 1+ √1 A2 lighter particle which gets an increased angular momen- − tum l = l δ, then due to (10) the scattering energy 1 H − For A =1 ǫ with ǫ 0 formula (15) gives: in the centre of mass system is − → inf 1/4 −1/4 m Ec.m.(r rH ) 2 2 +2 1/4 . (16) m 2(lH l2) → ∼ ǫ Ec.m. − (19) ≈ √δ s1 √1 A2 So even for values close to the extremal A = 1 of the − − inf rotating black hole Ec.m./m can be not very large as and it increases without limit for δ 0. For A = → max mentioned in [10, 11]. So for Amax = 0.998 considered 0.998 and l = l , E 3.85m/√δ. 2 L c.m. ≈ as the maximal possible dimensionless angular momen- Note that for rapidly rotating black holes A =1 ǫ − tum of the astrophysical black holes (see [13]), from (15) the difference between lH and lR is not large one obtains E max/m 18.97. c.m. ≈ Does it mean that in real processes of particle scat- √1 A l l =2 − √1 A + √1+ A A tering in the vicinity of the rotating nonextremal black H − R A − − holes the scattering energy is limited so that no Grand 2(√2 1)√ǫ, ǫ 0. (20) Unification or even Planckean energies can be obtained? ≈ − → Let us show that the answer is no! If one takes into ac- For Amax = 0.998, lH lR 0.04 so the possibility − ≈ count the possibility of multiple scattering so that the of getting small additional angular momentum in inter- particle falling from infinity on the black hole with some action close to the horizon seems much probable. The fixed angular momentum changes its momentum in the probability of multiple scattering in the accretion disc result of interaction with particles in the accreting disc depends on its particle density and is large for large and after this is again scattering close to the horizon density. then the scattering energy can be unlimited. Here we consider the model when the gravitation of The limiting value of the angular momentum of the the accretion disc is treated as some perturbation much particle close to the horizon of the black hole can be smaller than the gravitation of the black hole so it is not obtained from the condition of positive derivative in (6) taken into account. One must also mention that “par- dt/dτ > 0, i.e. going “forward” in time. So close to ticles” are considered as elementary particles and not the horizon one has the condition l<ε2xH /A which for macroscopic bodies. So their “large” energy is limited ε = 1 gives the limiting value lH . by a Planckean value and we neglect back reaction of From (8) one can obtain the permitted interval it on the Kerr’s metric of the macroscopic black hole. in r for particles with ε = 1 and angular momentum Electromagnetic and gravitational radiation of the par- l = l δ. To do this one must put the left hand side ticle surely can change the picture but one needs exact H − of (8) to zero and find the roots: calculations to see what will be the balance.

l2 l4 16(A l)2 x = ± − − . (17) 1,2 4 2. THE EXTRACTION OF ENERGY AFTER p THE COLLISION IN KERR’S METRIC In the second order in δ close to the horizon one obtains 2 2 Let us consider the case when there occurred for δ xC l = lH δ x . xH + . (18) 2 some r > rH interaction between particles with − ⇒ 4xH √1 A − masses m, specific energies ε1, ε2, specific angular mo- The effective potential defined by the right hand side menta L1,L2 falling into a black hole, so that two new of (8) leads to the following behaviour of the particle. particles with rest masses µ appeared, one of them (1) If the particle goes from infinity to the black hole it can moved outside the black hole, the other (2) moved in- achieve the horizon if the inequality (14) is valid. How- side it. Denote the specific energies of new particles ever the scattering energy in the centre of mass frame as ε1µ, ε2µ, their angular momenta (in units of µ) as i i given by (15) is not large. But if the particle is going not L1µ,L2µ, v = dx /ds — their 4-velocities. Consider 4 GRIB, PAVLOV particle movement in the equatorial plane of the rotat- in [11] is larger than the energy of initial particles as it ing black hole. must be in the case of a Penrose process [7, 8]. What is Conservation laws in inelastic particle collisions for the reason of this contradiction? Let us investigate the the energy and momentum lead to problem carefully. Note that if one neglects the states with negative i i i i m(u(1) + u(2))= µ(v(1) + v(2)). (21) energy in ergosphere energy extracted in the considered process cannot be larger than the initial energy of the Equations (21) for t and ϕ-components can be written pair of particles at infinity, i.e. 2m. The same limit 2m as for the extracted energy for any (including Penrose pro- m(ε1 + ε2)= µ(ε1µ + ε2µ), (22) cess) scattering process in the vicinity of the black hole

m(L1 + L2)= µ(L1µ + L2µ), (23) was obtained in [11]. Let us show why this conclusion is incorrect. i.e. the sum of energies and angular momenta of col- If the angular momentum of the falling particles is liding particles is conserved in the field of Kerr’s black the same then (see (24)) one has the situation similar hole. The initial particles in our case were supposed to to the usual decay of the particle with mass 2m in two be nonrelativistic at infinity: ε1 = ε2 = 1 and therefore particles with mass µ. Due to the Penrose process in for the r-component from (8) one obtains ergosphere it is possible that the particle falling inside the black hole has the negative relative to infinity en- 2M 2M L2 2 1 ergy and then the extracted particle can have energy m 3 (a L1) + 2 − "r r − r − r larger than 2m. 2 The main assumption made in [11] is the supposi- 2M 2 2M L2 + 3 (a L2) + 2 (24) tion of the collinearity of vectors of 4-momenta of the r − r − r # r particles falling inside and outside of the black hole in 2M a2ε2 L2 ∆ limiting case A = 1, l1 = 2 (see (9)–(11) in [11]). The = µ ε2 + (aε L )2 + 1µ − 1µ − s 1µ r3 1µ − 1µ r2 authors of [11] say that these vectors are “asymptoti- cally tangent to the horizon generator”.  2 2 2 In the limiting case the expressions dt/dτ, dϕ/dτ of 2M a ε2µ L2µ ∆ ε2 + (aε L )2 + − − . the components of the 4-velocity of the infalling parti- − s 2µ r3 2µ − 2µ r2  cle (6), (7) go to infinity when r r , but dr/dτ goes → H  to zero. In spite of smallness of r-components in the ex- The signs in (24) are put so that the initial particles pression of the square of the 4-momentum vector they and the particle (2) go inside the black hole while parti- have the factor grr going to infinity at the horizon. For cle (1) goes outside the black hole. The values ε1µ,ε2µ example from (1), (8) it is easy to see that are constants on geodesics ([12], 61) so the problem of § evaluation of the energy at infinity extracted from the g (x)ur ur 2, x x . (26) rr (1) (1) →− → H black hole in collision reduces to a problem to find these So putting them to zero can lead to a mistake. To see if values. u and v are collinear it is necessary to put the coor- For the case when the collision takes place on the (1) (1) dinate r of the collision point to the limit rH = M and horizon of the black hole (r rH ) the system (22)– → resolve the uncertainties / and 0/0. For the falling (24) can be solved exactly ∞ ∞ particle ε = 1, l = 2 the expressions for components AL1µ 2m AL1µ m of the 4-vector u can be easily found from (6)–(8). For ε1µ = , ε2µ = ,L2µ = (L1+L2) L1µ. 2rH µ − 2rH µ − the particle outgoing from the black hole due to exact (25) solution on the horizon (25) one puts ε1µ = l1µ/2+ α, In general case the system of three Eqs. (22)–(24) where α is some function of r and l , such that α 0 1µ → for four variables ε1µ, ε2µ,L1µ,L2µ can be solved nu- when r r . Putting this ε into (6)–(8) one gets for → H 1µ merically for a fixed value of one variable (and fixed x = r/M 1 → parameters m/µ, L1/M, L2/M, a/M, r/M). The t ϕ v v α l example of numerical solution is µ/m = 0.3,l1 = (1) (1) 1µ = ϕ = + , (27) 2.2, l = 2.198, A = 0.99, x = 1.21, l = 16.35, l = ut u x 1 2 2 1µ 2µ (1) (1) − 1.69, ε1µ =7.215, ε2µ = 0.548. Note that the energy − − r 2 of the second final particle is negative and the energy v(1) 2α 2l1µα 3 1 = + + l2 . (28) of the first final particle contrary to the limit obtained ur − (x 1)2 x 1 8 1µ − 2 (1) s − − ONTHECOLLISIONSBETWEENPARTICLES 5

Due to the condition dt/dτ > 0 (movement forward in time) the necessary condition for collinearity is that both (27) and (28) must be zero, which is not true. The notion of the asymptotic behaviour of 4-vectors on the horizon (asymptotic collinearity) needs delicate mathematical analysis. One must find the limit of the norm of vectors. But this norm is defined by the scalar prod- uct of vector on itself and it contains finite terms of the type (26) neglected by the authors [11], when they put the hypothesis that the momentum of ejecta particle is λk. This hypothesis means that the r-components in (28) are taken as identically zero and (27) is enough for collinearity, however (26) and (28) show that this is incorrect. This leads to the conclusion that the con- siderations of the authors of [11] for scattering exactly on the horizon can not be used for the real situation of particle scattering close to the horizon.

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