On particles collisions near rotating black holes
aa r X i v : . [ g r- q c ] O c t On Particles Collisions Near Rotating Black Holes
A. A. Grib , ∗ , Yu. V. Pavlov , , † A. Friedmann Laboratory for Theoretical Physics,Griboedov kanal 30/32, St. Petersburg 191023, Russia Russian State Pedagogical University (The Herzen University),Moyka-river emb. 48, St. Petersburg 191186, Russia Institute of Mechanical Engineering, Russian Acad. Sci.,Bol’shoy pr. 61, St. Petersburg 199178, Russia
Abstract.
Scattering of particles with different masses and energy in the gravitational field of rotatingblack holes is considered as outside as inside the black hole. Expressions for scattering energy of particlesin the centre of mass system are obtained. It is shown that scattering energy of particles in the centre ofmass system can obtain very large values not only for extremal black holes but also for nonextremal ones ifone takes into account multiple scattering. Numerical estimates for the time needed for the particle to getultrarelativistic energy are given.PACS numbers: 04.70.-s, 04.70.Bw, 97.60.Lf INTRODUCTION
In our publications [1] we came to the conclusionthat one can get very large energies in the centreof mass frame first mentioned in [2] for two particlecollision of particles with equal masses close to thehorizon of the rotating black hole (Active Galacticnucleus) if one considers multiple scattering. The ef-fect of getting very large energy depends on the valueof the angular momentum of one of the particles.The problem of the energy of collision of particles invicinity of black holes of different types now is in-tensively studied by different authors [3, 4, 5]. Herewe obtain similar formulas for particles of differentmasses in the field of Kerr’s black hole. To get verylarge energy one must have large time of rotatingof the particle around the black hole coming closerand closer to the horizon. We give some quantita-tive estimates for the time needed for a particle toobtain ultrarelativistic energy outside the horizon.Then we investigate the case of scattering inside thehorizon. The limiting formulas are obtained and itis shown that the collisions with infinite energy cannot be realized even in the singularity.The system of units G = c = 1 is used in the ∗ E-mail: andrei [email protected] † E-mail: [email protected] paper. THE ENERGY OFCOLLISIONS IN THE FIELDOF KERR’S BLACK HOLE
Let us consider particles falling on the rotatingchargeless black hole. The Kerr’s metric of the ro-tating black hole in Boyer–Lindquist coordinates hasthe form ds = dt − M r ( dt − a sin θ dϕ ) r + a cos θ − ( a cos θ + r ) (cid:16) dr ∆ + dθ (cid:17) − ( r + a ) sin θ dϕ , (1)where ∆ = r − M r + a , (2) M is the mass of the black hole, J = aM is an-gular momentum. In the case a = 0 the met-ric (1) describes the static chargeless black hole inSchwarzschild coordinates. The event horizon forthe Kerr’s black hole corresponds to the value r = r H ≡ M + p M − a . (3)The Cauchy horizon is r = r C ≡ M − p M − a . (4)For equatorial ( θ = π/
2) geodesics in Kerr’s met-1 ON PARTICLES COLLISIONS NEAR ROTATING BLACK HOLESric (1) one obtains ([6], § dtdτ = 1∆ (cid:20)(cid:18) r + a + 2 M a r (cid:19) ε − M ar L (cid:21) , (5) dϕdτ = 1∆ (cid:20) M ar ε + (cid:18) − Mr (cid:19) L (cid:21) , (6) (cid:18) drdτ (cid:19) = ε + 2 Mr ( aε − L ) + a ε − L r − ∆ r δ , (7)where δ = 1 for timelike geodesics ( δ = 0 forisotropic geodesics), τ is the proper time of the mov-ing particle, ε = const is the specific energy: theparticle with rest mass m has the energy εm in thegravitational field (1); Lm = const is the angularmomentum of the particle relative to the axis or-thogonal to the plane of movement.Let us find the energy E c . m . in the centre of masssystem of two colliding particles with rest masses m and m in arbitrary gravitational field. It can beobtained from( E c . m . , , , m u i (1) + m u i (2) , (8)where u i = dx i /ds . Taking the squared (8) and dueto u i u i = 1 one obtains E . m . m m = m m + m m + u i (1) u (2) i . (9)The scalar product does not depend on the choiceof the coordinate frame so (9) is valid in an arbi-trary coordinate system and for arbitrary gravita-tional field.We denote x = r/M , A = a/M , l n = L n /M ,∆ x = x − x + A and x H = 1 + p − A , x C = 1 − p − A . (10)For the energy in the centre of mass frame of two col-liding particles with specific energies ε , ε and an-gular momenta L , L , which are moving in Kerr’smetric one obtains using (1), (5)–(7), (9): E . m . m m = m + m m m − ε ε + 1 x ∆ x " l l (2 − x )+2 ε ε (cid:18) x ( x −
1) + A ( x +1) − A (cid:16) l ε + l ε (cid:17)(cid:19) − q ε x + 2( l − ε A ) − l x + ( ε − x ∆ x × q ε x + 2( l − ε A ) − l x + ( ε − x ∆ x . (11)It corresponds to results in [2] and [7] for the case ε = ε . Collisions of particles of equal masses withdifferent specific energies were considered in [8]. Writing the right hand side (11) as f ( x ) + ( m + m ) / m m , one obtains E c . m . ( r ) m + m = s f ( x ) −
1) 2 m m ( m + m ) . (12)This relation has maximal value for given r , specificparticle energies ε , ε and specific angular momenta l , l , if the particle masses are equal: m = m .To find the limit r → r H for the black hole witha given angular momentum A one must take in (11) x = x H + α with α → α . Taking into account A = x H x C , x H + x C = 2, after resolution of uncertainties in thelimit α → E . m . ( r → r H )2 m m = m + m m m − l H l H
4+ 18 (cid:20) ( l H + 4) l H − l l H − l + ( l H + 4) l H − l l H − l (cid:21) , (13)where l nH = 2 ε n x H A = 2 ε n A (cid:16) p − A (cid:17) . (14)is the limiting value of the angular momentum ofthe particle with specific energy ε n close to the hori-zon of the black hole. It can be obtained from thecondition of positive derivative in (5) dt/dτ >
0, i.e.going “forward” in time: l n < l nH (cid:18) x H + 12 α (cid:19) + o ( α ) , x = x H + α. (15)So close to the horizon one has the condition l n ≤ l nH .Note that for l = l H − β from (7) one gets (cid:18) drdτ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) r = r H = β x C x H > . (16)So there exists some region close to the horizonwhere one has particles moving with angular mo-mentum arbitrary close to the limiting value l = l H .In another form (13) is E c . m . ( r → r H ) m + m = (cid:20) m m ( m + m ) × ( l H l − l H l ) + 4( l H − l H + l − l ) l H − l )( l H − l ) (cid:21) / . (17)In special case ε = ε (for example for nonrelativis-tic on infinity particles ε = ε = 1) formula (13)can be written as E c . m . ( r → r H ) m + m == s m m ( m + m ) (4 + l H ) ( l − l ) l H − l )( l H − l ) . (18)RIB, PAVLOV 3For the extremal black hole A = x H = 1, l nH =2 ε n and the expression (13) is divergent when the di-mensionless angular momentum of one of the fallinginto the black hole particles l = l H = 2 ε . The scat-tering energy in the centre of mass system is increas-ing without limit (for case ε = 1 it was establishedin [2]). For example, if l = l H then one obtainsfrom Eq. (11) E . m . ( x )2 m m ≈ (2 ε − l )(2 ε − p ε − x − , x → . (19)Note that the small value of r − r H for the radialcoordinate of the point of the collision of particleswith high energy in the centre of mass frame does notmean small distance because the metrical coefficient g rr = − r / ∆ → ∞ .If A = 1 and l = l H = 2 ε then from (7) one gets (cid:18) drdτ (cid:19) = ( x − x (cid:0) ε + ( ε − x (cid:1) . (20)For ε ≥ A =1 and the limiting angular momentum l = 2 ε from (5), (7) one obtains dtdx = − M ε √ x ( x + x + 2)( x − p ε ( x + 2) − x . (21)So the time of movement in the vicinity of eventshorizon up to the point of collision with radial coor-dinate x f → x H = 1 is∆ t ∼ M ε ( x f − p ε − . (22)Taking into account (19) one obtains time of move-ment before collision with a given value of the en-ergy E in the centre of mass frame∆ t ∼ E m m M ε (2 ε − l ) p ε − x (2 ε − p ε − x ) . (23)For ε = ε = 1 , l = 0 one gets∆ t ∼ E m m M √ − ≈ · − MM ⊙ E m m s , (24) where M ⊙ is the mass of the Sun.So to have the collision of two protons with theenergy of the order of the Grand Unification onemust wait for the black hole of the star mass thetime ∼ s, which is large than the age of theUniverse ≈ s. However for the collision withthe energy 10 larger than that of the LHC one mustwait only ≈ s.Can one get the unlimited high energy of this scat-tering energy for the case of nonextremal black hole? THE ENERGY OF COLLISIONSFOR NONEXTREMAL BLACKHOLE
For a particle falling on the black hole from in-finity one must have ε ≥
1. In this section we con-sider the case ε = 1, when the particles falling intothe black hole are nonrelativistic at infinity. For-mula (7) leads to limitations on the possible valuesof the angular momentum of falling particles: themassive particle free falling in the black hole withdimensionless angular momentum A to achieve thehorizon of the black hole must have angular momen-tum from the interval − (cid:16) √ A (cid:17) = l L ≤ l ≤ l R = 2 (cid:16) √ − A (cid:17) . (25)Putting the limiting values of angular momenta l L , l R into the formula (18) one obtains the maximalvalues of the collision energy of particles freely fallingfrom infinity E maxc . m . ( r → r H ) m + m = (26)= s m m ( m + m ) (2 + √ A + √ − A ) (1 + √ − A ) √ − A . For A = 1 − ǫ with ǫ → E maxc . m . = 2 (cid:16) / + 2 − / (cid:17) √ m m ǫ / + O ( ǫ / ) . (27)So even for values close to the extremal A = 1 of therotating black hole E maxc . m . / √ m m can be not verylarge as mentioned in [9] for the case m = m . Sofor A max = 0 .
998 considered as the maximal possi-ble dimensionless angular momentum of the astro-physical black holes (see [10]), from (26) one obtains E maxc . m . / √ m m ≈ . r for particles with ε = 1 and angular momentum l = l H − δ . To do this one must put the left handside of (7) to zero and find the root. In the secondorder in δ close to the horizon one obtains l = l H − δ ⇒ x < x δ ≈ x H + δ x C x H √ − A . (28)The effective potential for the case ε = 1 defined bythe right hand side of (7) V eff ( x, l ) = − (cid:18) drdτ (cid:19) = − x + l x − ( A − l ) x (29)(see, for example, Fig. 1) leads to the following be- €€€€€€€€ r H - × V eff l R l H lx ∆ Figure 1: The effective potential for A = 0 .
95 and l R ≈ . l = 2 . l H ≈ .
76. Allowed zones for l = 2 . l = l H − δ large than l R and fall on the horizon. Ifthe particle falling from infinity with l ≤ l R arrivesto the region defined by (28) and here it interactswith other particles of the accretion disc or it de-cays into a lighter particle which gets an increasedangular momentum l = l H − δ , then due to (18) the scattering energy in the centre of mass system is E c . m . ≈ √ δ s m m ( l H − l )1 − √ − A (30)and it increases without limit for δ →
0. For A max =0 .
998 and l = l L , E c . m . ≈ . m/ √ δ .Note that for rapidly rotating black holes A = 1 − ǫ the difference between l H and l R is not large l H − l R = 2 √ − AA (cid:16) √ − A + √ A − A (cid:17) ≈ √ − √ ǫ , ǫ → . (31)For A max = 0 . l H − l R ≈ .
04 so the possibilityof getting small additional angular momentum in in-teraction close to the horizon seems much probable.The probability of multiple scattering in the accre-tion disc depends on its particle density and is largefor large density. COLLISION OF PARTICLESINSIDE KERR BLACK HOLE
As one can see from formula (11) the infinite valueof the collision energy in the centre of mass systemcan be obtained inside the horizon of the black holeon the Cauchy horizon (4). Indeed, the zeroes of thedenominator in (11): x = x H , x = x C , x = 0.Let us find the expression for the collision energyfor x → x C . Denote l C = 2 εx C A = 2 εA (cid:16) − p − A (cid:17) , l nC = 2 ε n x C A . (32)Note that for ε = 1 the Cauchy horizon can becrossed by the free falling from the infinity parti-cle under the same conditions on the angular mo-mentum (25) as in case of the event horizon and l L < l C ≤ l R ≤ l H .To find the limit r → r C for the black hole witha given angular momentum A one must take in (11) x = x C + α and do calculations with α → l − l C )( l − l C ) > , (33)then E . m . ( r → r C )2 m m = m + m m m − l C l C
4+ 18 (cid:20) ( l C + 4) l C − l l C − l + ( l C + 4) l C − l l C − l (cid:21) , (34)RIB, PAVLOV 5This formula is similar to (13) if everywhere H ↔ C .If ( l − l C )( l − l C ) = 0 , (35)for example, l = l C , then E c . m . √ m m ≈ s l − l C ) ( ε x C + x H ) x C ( x H − x C )( x − x C ) , x → x C . (36)If ( l − l C )( l − l C ) < , (37)then E c . m . √ m m ≈ s x H ( l − l C )( l C − l ) x C ( x H − x C )( x − x C ) , x → x C . (38)It seems that the limits of (36) and (38) are in-finite for all values of angular momenta l , l (35)and (37). This could be interpreted as instabilityof the internal Kerr’s solution [11]. However, fromEq. (5) we can see dtdτ ( x → x C + 0) = (cid:26) + ∞ , if l > l C , −∞ , if l < l C . (39)That is why the collisions with infinite energy cannot be realized (see also [11]).For the particle falling to singularity in the equa-torial plain of the Kerr’s black hole with A = 0 dtdτ ( x →
0) = (cid:26) + ∞ , if l < εA , −∞ , if l > εA . (40)In case l = εA, A = 0 the righthand side of (7) formassive particle is negative for x → l − ε A )( l − ε A ) >
0. Then from (11) onegets E c . m . ( r → m + m == s m m ( m + m ) ( l − l + ( ε − ε ) A ) ( l − ε A )( l − ε A ) . (41)That is why collision of particles with infinite energyin the centre of mass frame is impossible even insingularity. REFERENCES [1] A. A. Grib and Yu. V. Pavlov, JETP Lett. ,125 (2010); arXiv:1004.0913. [2] M. Banados, J. Silk, and S. M. West, Phys. Rev.Lett. , 111102 (2009).[3] S.-W. Wei, Y.-X. Liu, H. Guo, and C.-E. Fu,Charged spinning black holes as particle accel-erators, arXiv:1006.1056;S.-W. Wei, Y.-X. Liu, H.-T. Li, F.-W. Chen,Particle collisions on stringy black hole back-ground, arXiv:1007.4333.[4] O. B. Zaslavskii, Acceleration of parti-cles as universal property of rotating blackholes, arXiv:1007.3678; Acceleration of par-ticles by nonrotating charged black holes,arXiv:1007.4598.[5] P.-J. Mao, R. Li, L.-Y. Jia, J.-R. Ren,Kaluza-Klein black hole as particles accelera-tors, arXiv:1008.2660.[6] S. Chandrasekhar, The Mathematical Theoryof Black Holes (Oxford Univ. Press, Oxford,1983).[7] A. A. Grib and Yu. V. Pavlov, On particle col-lisions near Kerr’s black holes, arXiv:1007.3222[8] T. Harada and M. Kimura, Collision of anISCO particle around a Kerr black hole,arXiv:1010.0962.[9] E. Berti, V. Cardoso, L. Gualtieri, F. Pretorius,and U. Sperhake, Phys. Rev. Lett. , 239001(2009).T. Jacobson and T. P. Sotiriou, Phys. Rev. Lett. , 021101 (2010).[10] K. S. Thorne, Astrophys. J. , 507 (1974).[11] K. Lake, Phys. Rev. Lett. , 211102 (2010);104