Ultrahigh energy particle collisions near many-dimensional black holes: general approach
aa r X i v : . [ g r- q c ] N ov Ultrahigh energy particle collisions near many-dimensional blackholes: general approach
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 moving towards a black hole collide near the horizon, their energyin the centre of mass frame can grow unbounded. This is the so-called Banados - Silk- West (BSW) effect. Earlier, it was shown that in the 3+1 space-time this effect hasa universal nature. We show that for a wide class of many-dimensional black holes(including, say, the Myers-Perry black hole) this is also true. The suggested analysisis general and does not require special properties of the metric like separability ofvariables for geodesics, etc.
PACS numbers: 04.70.Bw, 97.60.Lf
I. INTRODUCTION
In 2009, Ba˜nados, Silk and West made an interesting observation [1]. It turned out thatif two particles collide near the extremal Kerr black hole, the energy in the centre of massframe can grow unbounded (the BSW effect, called after the names of its authors). Lateron, the BSW effect was extended to generic extremal and nonextremal stationary axially-symmetric black holes, charged static black holes, black holes in the magnetic field, etc. Oneof separate issues provoking special interest is the possibility of generalization to the many-dimensional case. It was demonstrated in [2] for the five-dimensional Kerr metric and in [3]for many-dimensional Myers-Perry black hole [4] that the BSW effect does exist. It required ∗ Electronic address: [email protected] rather developed studies, especially in [3], where very subtle properties of motion in theMyers-Perry black hole background were exploited, including the separability of variables[4] - [6].One may ask: whether the BSW effect for a Myers-Perry black hole arises due to somespecial properties of this very space-time? If so, one would be led to incremental inspectionof different models step by step. However, it is worth reminding that it was shown earlierthat for 3+1 black holes the BSW effect is of universal nature [7], [8] and is due to theexistence of the horizon as such, independently of a particular model. Therefore, a naturalsupposition arises that the situation is similar in a many-dimensional case. In the 3+1space-time, the BSW effect requires the existence of so-called critical trajectories with fine-tuned parameters. Then, it is produced by collision between such a fine-tuned (”critical”)particle and generic (”usual”) one. However, both the overall approach and the formulationof critical condition are so different in [2], [3] and in [7], [8], that the issue of universality isnot obvious in advance.The aim of the present paper is to show that there exists deep continuity between 3+1and higher-dimensional cases. And, the approach to the BSW effect can be formulatedin such a way that, without relying on special properties of space-time like separability ofvariables, etc., one can obtain the critical condition and the effect under discussion as directgeneralization of the 3+1 case.Throughout the paper we use units in which fundamental constants are G = c = 1. II. BASIC EQUATIONS
We consider the metric ds = − dt + dr A ( r, θ ) + ρ dθ + n X i =1 g i ( { α k } , θ, r ) dφ i + B ( r, θ )[ dt − n X i b i ( { α k } , ) dφ i ] + n X i =1 A i ( { α k } , θ, r ) dα i .(1)Here, the variable θ is singled by analogy with the 3+1 case and to make comparison tothe many-dimensional Myers-Perry black hole [3] more convenient. We assume that all themetric coefficients do not depend on t and φ i , i = 1 , ...n . Although the metric (1), withthese restrictions, is not the most general one, it includes physically important space-times,in particular, the aformentioned Myers-Perry black hole. On the horizon, N = 0.The metric can be rewritten in the form which is more familar, especially from the 3+1context: ds = − N dt + n X i =1 h ij ( dφ i − ω i dt )( dφ j − ω j dt ) + dr A + g θ dθ + n X i =1 A i dα i . (2)The relaitonship between two forms of the metric are given by obvious formulas − N + X k h ij ω i ω j = B − Bb i = h ij ω j , Bb i b j = h ij . (4)The inverse metric is equal to g = − N , g rr = A , g θθ = 1 g θ , g i = − ω i N , g α k α k = 1 A k , (5) g ik = h ik − ω i ω k N , (6)where h ij h jk = δ ik .As the metric does not depend on time, the energy E = − mu is conserved ( m is a mass),the velocity u α = dx α dτ , where τ is the proper time. In a similar way, there are conservedangular momenta corresponding to variables φ i , mu i = L i . Then, the equaitons of motionfor geodesics read (dot denotes differentiation with respect τ ): m ˙ t = XN , (7) X = E − X k ω k L k , (8) m ˙ φ i = h ij L j + ω i XN . (9)Using the normalization condition u α u α = −
1, we obtain the equaiton for the radialcomponent . Then, we have m ˙ r = AN Z , (10) Z = X − N ( X k h ij L i L j + m + X k A k ˙ α k ). (11) III. COLLISION OF TWO PARTICLES
If two particles 1 and 2 collide, one can define their energy in the centre of mass framein the point of collision according to E c.m. = − ( m u µ + m u µ )( m u µ + m u µ ) = m + m + 2 m m γ , (12)where the Lorentz factor of relative motion equals γ = − u µ u µ . (13)Then, it follows from the equaitons of motion (7) - (11) that γ = X X − Z Z m m N − X k h ij L i L j − X k A k ˙ α k ˙ α k . (14)We do not specify the character of motion and only require that ˙ α k and ˙ α k be finite.In particular, there exist simplest geodesics with all α k = const .Let the point of collision be close to the horizon. In general, this does not lead to thegrowth of γ since the numerator in γ vanishes with the same rate as the denominator.However, there is an exception. Let us call a particle usual if X H = 0 and critical if X H = 0(subscript ”H” means that the quantity is calculated on the horizon). According to (8), thecondition of criticality reads E = X k ( ω k ) H L k . (15)Let the critical particle 1 and a usual particle 2 collide near the extremal horizon. Now,we need the behavior of X near the horizon. We restrict ourselves by the extremal case,simiularily to what was done in [3] for Myers-Perry black holes. Then, we can exploit theTaylor expansion near the horizon: ω k = ω kH − ω k N + O ( N ), (16)where ω does not depend on r . (For more general discussion of expansions of this kind fornonextremal and extremal horizons 3+1 dimensional see Ref. [9]. Extension of its resultsto the many-dimensional case is straightforward). By substitution into (8) and taking eq.(16) into account, we obtain that near the horizon X = CN + O ( N ), (17)where C = X k ω k L k .Then, it follows from (14) that near the horizon γ ≈ X DN m m , D = C − s C − ( X k h ij L i L j + m + X k A k ˙ α k ) H . (18)Thus for N → γ grows unbounded, and we have the BSW effect. IV. CONCLUSION
Thus we have shown in a very simple and direct manner that the general approach to theBSW effect developed earlier for 3+1 black holes [7], [8], works, with some modifications,also in a many-dimensional case. The key ingredients are preserved: (i) the presence of thehorizon, (ii) classification of particles according to which one of them should be critical,another one should be usual. In the particular case of the Myers-Perry black hole, thecondition of criticality (15) corresponds to eq. (2.30) of [3]. As long as we are interested inthe BSW effect as such, there is no need in examining new models one after another andelucidating complicated and subtle details of particles’ motion (which, of course, can be ofinterest by themselves). If the metric belongs to the class (1), (2), the existence of the BSWeffect follows from general grounds.The general character of our treatment enables us also to use (directly or with minimummodifications) some other results concerning properties of the BSW effect. For instance,classification of the possible types of the BSW scenarios suggested in Sec. 6 of Ref. [8] nowapplies and is insensitive to the number of dimensions. It includes as particular cases the3+1 Kerr metric (see Sec. IV A of Ref. [10]) and the Myers-Perry black hole [3].Actually, we showed that the persistence of the BSW effect can be considered as onemore manifestation of universality typical of black holes physics. This can have interestingphysical consequences like, for example, instability of extremal horizons [3] but this issuerequires separate treatment. [1] M. Ba˜nados, J. Silk and S.M. West, Phys. Rev. Lett. , 084036(2013) [arXiv:1310.4494]. [3] N. Tsukamoto, M. Kimura and T. Harada, Phys. Rev. D , 024020 (2014) [arXiv:1310.5716].[4] R. C. Myers and M. J. Perry, Ann. Phys. (N.Y.) , 304 (1986).[5] M. Vasudevan, K. A. Stevens, and D. N. Page, Classical Quantum Gravity , 1469 (2005)[arXiv:gr-qc/0407030].[6] R. Emparan and H. S. Reall, Living Rev. Relativity , 6 (2008) [arXiv:0801.3471].[7] O.B. Zaslavskii, Phys. Rev . D (2010) 083004 [arXiv:1007.3678].[8] O. B. Zaslavskii, JHEP, 2012, (2012) [arXiv:1209.4987].[9] I. Tanatarov and O. Zaslavskii, Phys. Rev. D (2012) 044019 [arXiv:1206.2580].[10] T. Harada and M. Kimura, Phys. Rev. D83