Kinematic censorship as a constraint on allowed scenarios of high energy particle collisions
aa r X i v : . [ g r- q c ] D ec Kinematic censorship as a constraint on allowedscenarios of high energy particle collisions
Yu. V. Pavlov , , ∗ , O. B. Zaslavskii , , ∗∗ Institute of Problems in Mechanical Engineering, Russian Academy of Sciences,Bol’shoy pr. 61, St. Petersburg 199178, Russia; N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University,18 Kremlyovskaya St., Kazan 420008, Russia; Department of Physics and Technology, Kharkov V.N. Karazin National University,4 Svoboda Square, Kharkov 61022, Ukraine
Abstract.
In recent years, it was found that the energy E c.m. in the centre of mass frame of two collidingparticles can be unbounded near black holes. If collision occurs exactly on the horizon, E c.m. is formallyinfinite. However, in any physically reasonable situation this is impossible. We collect different scenariosof such a kind and show why in every act of collision E c.m. is indeed finite (although it can be as large asone likes). The factors preventing infinite energy are diverse: the necessity of infinite proper time, infinitetidal forces, potential barrier, etc. This prompts us to formulate a general principle according to whichthe limits in which E c.m. becomes infinite are never achieved. We call this the kinematic censorship (KC).Although by itself the validity of KC is quite natural, its application allows one to forbid scenarios of collisionspredicting infinite E c.m. without going into details. The KC is valid even in the test particle approximation,so explanation of why E c.m. cannot be infinite, does not require references (common in literature) to the non-linear regime, backreaction, etc. The KC remains valid not only for free moving particles but also if particlesexperience the action of a finite force. For an individual particle, we consider a light-like continuous limit ofa time-like trajectory in which the effective mass turns into zero. We show that it cannot be accelerated toan infinite energy during a finite proper time under the action of such a force. As an example, we considerdynamics of a scalar particle interacting with a background scalar field.PACS numbers: 04.70.Bw, 97.60.Lf, 04.20.CvKey words: black holes, particle collisions
1. Introduction
Some of fundamental principles in physics have aform of prohibition. Say, impossibility to reach theabsolute zero of temperature constitutes the thirdlaw of thermodynamics. A similar statement inblack hole physics implies that one cannot convert anonextremal black hole into the extremal one duringa finite number of steps. The principle of cosmic cen-sorship states that one cannot see a singularity fromthe outside. Investigations on particle collisions re-vealed one more such a principle that remained shad-owed until recently. This can be called ”kinematiccensorship” (KC): the energy in the centre of massframe of any colliding particles cannot be infinite.In usual laboratory physics this looks quite trivial. ∗ [email protected] ∗∗ [email protected] Indeed, in, say, flat space-time any energy gained orreleased in any process is always finite. And, start-ing with the total finite energy, one cannot obtainsomething infinite due to the energy conservation.The situation radically changed after findings madein [1, 2]. It was shown there that if two particles col-lide in the extremal Kerr background, under certainconditions the energy E c.m. in the centre of massbecomes unbounded when a point of collision ap-proaches the horizon r + :lim r → r + E c.m. ( r ) → ∞ . (1)This was found for head-on collisions in [1] andfor particles moving in the same direction towardsthe horizon, provided one of particles is fine-tuned,in [2]. In the latter case it is called the BSW effect.Later on, other versions of this process were foundin different contexts. In doing so, the input was fi-1 KINEMATIC CENSORSHIP AS A CONSTRAINTnite (particles with finite masses and energies) butthe output is formally infinite in some limiting sit-uations. The fact that each time (i) something pre-vents an infinite E c.m. and (ii) the nature of this”something” is completely different depending onthe concrete scenario, suggests that there exists arather general principle that unifies all so differentparticular cases. We want to point out the follow-ing subtlety. The results on unbounded E c.m. wereobtained in the test-field approximation when self-gravitation and backreaction were neglected. As forsufficiently large E c.m. this can be no longer true,there is temptation to ascribe formally infinite E c.m. to the test-field approximation with the expectations(confirmed in some simple models, e.g., see [3]) thatin the non-linear regime E c.m. will become finite.However, it turns out that even in the test-particleapproximation infinite E c.m. are forbidden, althougharbitrarily large but finite E c.m. are possible. Thusthe limit under discussion turns out to be unreach-able in each case.It is worth noting that in [4] several objectionswere pushed forward against the BSW effect, in par-ticular — impossibility to reach the extremal stateand backreaction. Meanwhile, it was explained [5]that for nonextremal black holes the BSW effectdoes exist, although in a somewhat different setting.It was also shown in [6] for extremal black holesand in [7] for nonextremal ones that account of afinite force does not spoil the BSW effect. There-fore, the reasons why infinite E c.m are, nonetheless,unreachable, should have more fundamental natureand explanations should be done just within the test-particle approximation, without attempts to ascribethem to some factor neglected in this approximation.In the present work, we do not perform new con-crete calculations. Instead, we collect a number ofresults, already obtained earlier, and make new qual-itative generalizations based on them.The fact that an infinite E c.m. cannot occur inany act of collisions, is more or less obvious in theflat space-time. In a curved space-time, the issue isnot so trivial since formally infinite E c.m. was ob-tained with finite initial energies of colliding parti-cles. And, although from physical grounds the KCis more or less obvious, it would be of interest togive a proof of the KC. However, in the present ar-ticle we do not pretend for such a proof in an ar-bitrary curved space-time. Instead, we enumerateseveral typical situations and trace which conditionsprecisely prevent collisions with infinite E c.m. . Thecorresponding factors turn out be rather diverse. Wealso demonstrate that the KC works as some regu- lator that imposes constraint on possible scenariosand enables us to reject some of them in advance,even without going into details.Another separate issue is behavior of a single par-ticle that formally approaches the speed of light lo-cally in a curved background and becomes potentialsource of difficulties with infinite E c.m. . We suggestrather general and rigorous proof that this cannothappen under the action of finite forces. This is illus-trated by physically interesting example of a scalarparticle interacting with a background scalar field.We use the systems of units in which the speed oflight c = 1.
2. Can a massive particle beaccelerated until a speed oflight?
The question of the possibility or impossibility ofinfinite E c.m. reduces to the behavior of individualparticles and their Killing energy E . If E remainsfinite (equivalently, the velocity of a massive parti-cle is less than a speed of light), infinite E c.m. isimpossible. In the flat space-time, it is more or lessobvious, that any particle cannot reach the speedof light during a finite amount of the proper time,provided all the components of four-acceleration a µ are finite. Meanwhile, even in such a relatively sim-ple case, there are some subtleties. Because of theLorentz signature, it would seem that although sepa-rate components of a µ diverge, the scalar a = a µ a µ can, nonetheless, be finite. However, now we arguethat in the situation under discussion this is impos-sible.Statement: if a massive particle moving under theaction of some force reaches the speed of light duringthe finite proper time, the absolute value of acceler-ation a → ∞ in this point.To the best of our knowledge, the proof of the cor-responding statement is absent from literature, so wesuggest it below. We would like to stress that theissue under discussion is not exhausted by a simplefact that the square of the four-velocity is equal to − −
1, the square of the four-momentum van-ishes.At first, let us consider the situation in the flatspace-time. Let in the Minkowski metric ds = − dt + η ik dx i dx k (2)a particle move along the trajectory x µ = x µ ( τ ),where τ is the proper time. Then, we have a µ = du µ dτ , (3)whence along a given trajectory from point 1 topoint 2 one obtains u µ = Z a µ dτ, (4)where on the trajectory a µ is a function of τ only. Ifall components of a are finite, u µ remains finite aswell, so the speed of light cannot be reached.Now, we assume that as τ → τ , v →
1. Here,the velocity is defined according to v = v i v i , v i = dx i /dt . The four-velocity has a standard form u µ = (cid:18) √ − v , v i √ − v (cid:19) = γ (1 , v i ) , (5)where γ = 1 / √ − v is the Lorentz factor.For the components of a µ we have a = ˙ γ , a i = d ( γv i ) /dτ . Here, dot denotes d/dτ . After some al-gebra, one can obtain a = ˙ γ γ + γ dv i dτ dv k dτ η ik . (6)Now, let us consider a concrete law of approachingto the speed of light near the value τ = τ : v i = (cid:0) v i (cid:1) − β i ( τ − τ ) n + . . . , (7)where β i are some constants. Then, v ≈ − α ( τ − τ ) n , (8)where α = β i (cid:0) v i (cid:1) . (9)Then, direct evaluation of different terms in (6) givesus a ≈ n τ − τ ) → ∞ . (10) Thus, not only separate components of the four-acceleration diverge but also the scalar product doesso.In the curved space-time, the situation looks quitesimilar from the physical point of view. However, itwould seem that rigorous proof becomes much morecomplicated. In particular, this happens because ofthe appearance of the Christoffel symbols Γ ρµν in theexpression for a µ . Fortunately, this is not so. Thekey point here is the existence of coordinate frame inwhich Γ ρµν = 0 along a given line. This was shownby Fermi [9]. See also textbook [10], Ch. VII.91.Then, (3) and (4) apply with the same result.Thus if the interval τ ≤ τ ≤ τ is finite, u µ re-mains finite for any finite a . Otherwise, a particlecan indeed reach the speed of light but by expensethat a diverges in a corresponding point. One canpass from the coordinate frame under discussion toany other one by a finite Lorentz boost, so a parti-cle’s speed is less than the speed of light in a newframe as well.The context under discussion can be considered asa rather unexpected practical application of Fermi’sfinding.To be more precise, we must make a reservation.There exists a situation in which a particle can in-deed reach a speed of light during a finite propertime with a finite or even zero acceleration but thisis due to incompleteness and/or singular characterof the frame. For example, this happens if a parti-cle falls in the Schwarzschild black hole (see, say, eq.(102.7) in [11]). However, this velocity is measuredby a static observer who becomes singular on thehorizon, so this is a consequence of impossibility ofusing such a frame on the horizon and beyond. Also,the boost between the corresponding frame and anyregular frame becomes singular as well.In the next Section, we will consider a concretephysical example in which such a situation occurs.
3. Explicit example: vanishingof the effective mass
In this Section, we consider a concrete physicalexample. We deal with a scalar particle interactingwith some background field ψ . Let the action havea simple form S = − Z ( m − qψ ) dτ, (11)where ψ is the background scalar field, q being thescalar charge of a particle, m is its mass. For such a KINEMATIC CENSORSHIP AS A CONSTRAINTsystem, the exact spherically symmetric static solu-tion describing a black hole was found [12]–[15]. Itsmetric formally coincides with that of the extremalReissner-Nordstr¨om black hole: ds = − f dt + dr f + r dω , (12)where f = (cid:0) − r + r (cid:1) , r + is the horizon radius. Letus consider pure radial motion. Then, one can ob-tain easily that m ∗ ˙ t = Ef (13) m ∗ ˙ r = − P, (14)the effective “mass” m ∗ ≡ m − qψ , the radial mo-mentum P = p E − m ∗ f , (15)the energy E = const.We choose q >
0. Let us denote as r the point inwhich ψ = m/q , so m ∗ ( r ) = 0. It is implied that r > r + .We assume that ψ is a smooth function of r and weexamine the behavior of relevant quantities near thepoint r where the effective mass vanishes, m ∗ ( r ) =0: ψ = ψ ( r ) − C ( r − r ) + . . . , (16)where C is some constant. Then, m ∗ ≈ qC ( r − r ) (17)Then, ˙ r ≈ − ECq ( r − r ) , (18)for r ≥ r , τ ≤ τ r ≈ r + B p ( τ − τ ) , (19) B = s ECq . (20)Thus the proper time to reach r (say, from r > r )is finite.For the velocity V measured by a static observer,we have E = m ∗ √ f / √ − V , whence V = r − f m ∗ E . (21)Near r = r , V ≈ − C q f ( r )2 E ( r − r ) , (22)so V →
1. Thus a particle reaches the speed of light for afinite proper time but in one point r only.One can introduce a kinematic momentum accord-ing to p µ = m ∗ u µ . It is instructive to note thatalthough u µ u µ = − r , p µ p µ = − m ∗ → r → r . In this sense,we have light-like limit of a time-like particle. In thestandard case, for a particle of a given mass, thiswould lead to unbounded energy. However, in thepresent case E remains finite due to the fact that m ∗ → u µ u µ = − a α = u α ; β u β = qm ∗ [ ψ ; α + u α ( ψ ; β u β )] , (23)whence a ≡ a µ a µ = q m ∗ h µν ψ ,µ ψ ,ν , (24) h µν = g µν + u µ u ν , (25)where u µ is the four-velocity.In general ( h µν ψ ,µ ψ ,ν ) r = r = 0 and, moreover,this quantity can diverge. This happens in thepresent example. Then, it is seen that acceleration a → ∞ when r → r . In other words, a systemas a whole is singular although all geometric char-acteristics like the Kreschmann scalar are perfectlyregular in the point r . It is of interest to study fur-ther properties of such “intermediate” systems thatcombine regular and singular properties. For the ex-ample under discussion, one can check that Eq. (10)is reproduced exactly.
4. Black vs. white holes assources of high energycollisions
Now, we turn to concrete scenarios of high energycollisions. The first observations on such collisionsnear the horizon made in [1] predict formally un-bounded E c.m. that becomes infinite when r → r + .AVLOV, ZASLAVSKII 5However, the crucial point here consists in that ac-tually the corresponding scenario describes collisionsof particles moving in the opposite direction (head-on collision). This is especially clear from Eq. (2.57)of [18], where the issue under discussion was elab-orated in detail. In turn, this implies that one ofparticles moves not towards the horizon but awayfrom it. This condition cannot be realized in theimmediate vicinity of a black hole horizon (see fordetails Sec. IV A in [19]) and, rather, correspondsto a white hole. Collisions of particles near whiteholes were discussed in [20, 21]. Let particle 1 ap-pear from the inner region and collides with parti-cle 2 that comes from infinity or some finite distanceoutside. It is essential that collision cannot happenexactly on the horizon itself. Otherwise, any mas-sive particle 2 having a finite energy at infinity wouldbecome light-like. Instead, one can take a particle 2to be massless. However, either an observer 1 emitsor absorbs a photon of finite frequency or the fre-quency of particle 2 must be taken infinite from thevery beginning that deprives the scenario of physi-cal meaning [22]. Thus, nature of the white horizonprotects the energy E c.m. from being infinite.
5. Extremal black holes
Let us consider the metric ds = − N dt + g φ ( dφ − ωdt ) + dr A + g θ dθ (26)in which the coefficients do not depend on t and φ .We assume that the function η ≡ √ A/N is finiteon the horizon like in the Kerr-Newman metric. Bydefinition of an extremal black hole, near the hori-zon r + , N ∼ r − r + . (27)For equatorial motion, it follows from the geodesicequations that the proper time τ required for a par-ticle to travel from r to r < r towards the horizonequals τ = Z r r mη dr q ( E − ωL ) − N ( m + L g φ ) . (28)Here, E is the particle energy, L being its angularmomentum, m mass.The essential feature of the Ba˜nados-Silk-West ef-fect is that one of particles has E − ω H L = 0,where ω H is the value of ω on the horizon. Sucha particle is called critical. For small N we have E − ωL = O ( N ). Taking into account (27), we see that τ ∼ | ln( r − r + ) | diverges [5, 23, 24]. Itmeans that a fine-tuned particle never reaches thehorizon, so collision never occurs exactly on the hori-zon. It can happen close to it. Then, E c.m. is large(even unbounded) but finite in each concrete colli-sion. The same situation happens for collision ofradially moving particles in the background of theReissner-Nordstr¨om metric [25].
6. Nonextremal black holes,collisions outside
Now, there are no trajectories with infinite τ .However, another difficulty comes into play. Theallowed region of motion is characterized by V eff ≤ V eff is de- r €€€€€€€€ r + × V eff L cr Lr ∆ €€€€€€€€ r + Figure 1: The effective potential for motion in theequatorial plane of the Kerr black hole with a/M =0 .
95 and a critical particle with L cr ≈ . M m . Al-lowed zones for a particle with L = 2 . M m are col-ored.fined according to (see [26] for details) (cid:18) drdτ (cid:19) + V eff = 0 . (29)The critical particle cannot approach the horizon atall because of the potential barrier. One can let pa-rameters of a particle to differ slightly from thoseof the critical one. Then, such a near-critical parti-cle can approach the horizon and collide there withsome another one. In such a case E c.m. ∼ / √ δ where δ is the parameter that controls the deviationfrom the exact critical relationship [5]. Thus, E c.m. can be made as large as one likes but not infinite.And, collision of interest can occur within a narrowstrip near the horizon only [5, 23]:0 ≤ N ≤ N , (30) KINEMATIC CENSORSHIP AS A CONSTRAINTwhere N ∼ δ . For the Kerr metric (see Fig. 1) r + ≤ r ≤ r δ . When δ →
0, the region with high en-ergy outcome degenerates into the point, so collisionbetween two particles becomes impossible.
7. Nonextremal black holes,collisions inside
Now, both potential barrier and infinite propertime are not encountered in the problem. Formally,when a particle approaches the inner horizon r − ,lim r → r − E c.m. ( r ) → ∞ . (31)The example from the present Section is espe-cially instructive since it demonstrates the predic-tive power of the principle under discussion. Severalyears ago, an intensive discussion developed in lit-erature concerning the possibility of the analogue ofthe BSW effect inside a black hole. Firstly, it wasclaimed in [27] that such an analogue does exist.Later on, the author himself refuted this result [28],independently this was done in [29]. More weak ver-sion of this effect revived in [30]. Another examplesof the same kind (with predictions of infinite E c.m. )were suggested in [31] for the nonextremal Kerr met-ric and in [32] for the cosmological horizon. Theseresults are incorrect, as is explained in [30]. The rea-son consists in that relevant trajectories (that other-wise would have given infinite E c.m. ) do not intersecton the horizon.Meanwhile, such scenarios can be rejected at onceonly due to contradiction to our KC. This principleis able to reject the scenario described in [27, 31, 32]even without elucidating these details (which canbe found in aforementioned references). Indeed, (i)the result (31) is always formally valid for any two particles, (ii) Eq. (31) implies that an infinite E c.m. is reachable in a separate act of collision. But thisis what our principle forbids!
8. Scalar field and infiniteacceleration
Let a black hole be surrounded by a scalar fieldand one of colliding particles is minimally coupledto this field. The corresponding action describ-ing interaction of a scalar particle with the back-ground scalar field is described by Eq. (11). How-ever, now we change the sign of the particle’s scalar charge q . Then, there are no divergences for accel-eration outside the horizon. But, instead, anotherphenomenon comes into play. It is connected withthe immediate vicinity of the horizon. Let the scalarfield diverges near the horizon like ϕ ∼ N − β . If β <
1, the proper time of traveling to to the horizonis indeed finite [33]. Meanwhile, if one calculatesthe absolute value of the four-acceleration experi-ences by a particle under action of the scalar field, a ∼ N β − → ∞ near the horizon [33]. The gradi-ent of a diverges as well. Any particle having smallbut nonzero size will be teared to pieces, so collisionon the horizon becomes impossible.
9. From black holes to nakedsingularities
Up to now we considered collisions in the back-ground of black holes that is the main subject ofour work since it is this case when KC is nontrivial.Meanwhile, we would like to make a short commenthow KC reveals itself for naked singularities. Al-though this case is much more simple, it is quiteinstructive.It was noticed in [34] that naked singularities canserve as accelerators in the two-step scenario. Oneof particles moves from infinity, reflects from the po-tential barrier and collides with the second particlefalling from infinity in the point where N is verysmall, i.e. on the verge of forming the horizon (butit does not form). Then, E c.m. ∼ N − c , where N c isthe value of N in the point of collision. By its verymeaning, N c cannot reach the value N c = 0 sincethis would mean formation of the horizon, so KCholds true.
10. Conclusion
We considered several completely different exam-ples in which it seemed to be ”obvious” that E c.m. could be infinite. However, we saw that in eachof the examples, some ”hidden” factor reveals itselfwhich acts to prevent E c.m. from being infinite. Thenature of these factors is quite different and dependsstrongly on the situation. This indeed points tothe validity of some fundamental underlying princi-ple called by us ”principle of kinematic censorship”.From a more practical point of view, this principle,by itself, cannot suggest some new scenarios of highenergy collisions. Rather, it can serve as some con-straint that enables us to separate forbidden sce-narios from physically possible ones and reveal theAVLOV, ZASLAVSKII 7factors (sometimes hidden) that make E c.m. impos-sible.Thus we formulated some new principle, suggestedarguments in its favour and checked it on concreteexamples. There are two qualitative lessons follow-ing from our analysis: (i) there is a crucial differ-ence between infinite and unbounded E c.m. (finitein each act of collision), (ii) the impossibility of in-finite E c.m. is inherent to any scenario in the testparticle approximation . There is no need to refer tosome not quite understandable factors which will betaken into account in future investigations of non-linear regimes!In addition to results concerning the propertiesof collisions, we gave proof for a curved backgroundthat under the action of a finite force, a trajectoryof a massive particle cannot become light-like. Inother words, a massive particle cannot turn into amassless one. This can be of some interest on itsown in other contexts [35]. Acknowledgments.
This work was supportedby the Russian Government Program of CompetitiveGrowth of Kazan Federal University. The work ofYu. P. was supported also by the Russian Foundationfor Basic Research, grant No. 18-02-00461-a. O. Z. isgrateful to Serguei Krasnikov and Alexey Toporen-sky for discussion of Sec. 2 and to Kirill Bronnikovfor interest to materials of Sec. 3.
References [1] T. Piran, J. Shaham, J. Katz,
High efficiency ofthe Penrose mechanism for particle collisions ,Astrophys. J. , L107 (1975).[2] M. Ba˜nados, J. Silk, S. M. West,
Kerr blackholes as particle accelerators to arbitrarily highenergy , Phys. Rev. Lett. , 111102 (2009).[3] M. Kimura, Ken-ichi Nakao, H. Tagoshi,
Accel-eration of colliding shells around a black hole:Validity of the test particle approximation inthe Banados-Silk-West process , Phys. Rev. D , 044013 (2011).[4] E. Berti, V. Cardoso, L. Gualtieri, F. Pretorius,U. Sperhake, Comment on “Kerr black holes asparticle accelerators to arbitrarily high energy” ,Phys. Rev. Lett. , 239001 (2009).[5] A. A. Grib, Yu. V. Pavlov,
On particles col-lisions in the vicinity of rotating black holes ,Pis’ma v ZhETF , 147 (2010), (JETP Let-ters , 125 (2010)). [6] I. V. Tanatarov, O. B. Zaslavskii, Ba˜nados-Silk-West effect with nongeodesic parti-cles: Extremal horizons , Phys. Rev. D , 044019 (2012).[7] I. V. Tanatarov, O. B. Zaslavskii, Banados-Silk-West effect with nongeodesic parti-cles: Nonextremal horizons , Phys. Rev. D , 067502 (2014).[8] Ya. B. Zel’dovich, I. D. Novikov, Relativistic as-trophysics. Volume 2 – The Structure and Evo-lution of the Universe (University of ChicagoPress, Chicago, IL, 1983).[9] E. Fermi,
Sopra i fenomeni che avvengono invicinanza di una linea oraria , Rend. Lincei (1), 21—23, 51—52, 101—103 (1922).[10] P. K. Rashevsky, Riemannian Geometry andTensor Analysis (Nauka, Moscow, 1967) (InRussian).[11] L. D. Landau, E. M. Lifshitz,
The Classical The-ory of Fields (Pergamon Press, Oxford, 1983).[12] N. Bocharova, K. Bronnikov, V. Melnikov,
Onan exact solution of the Einstein and masslessscalar field equations , Vestn. Mosk. Univ., Ser.3: Fiz. Astron. , 706 (1970) (Moscow Univ.Phys. Bull. , 80 (1970)).[13] K. A. Bronnikov, Scalar-tensor theoryand scalar charge , Acta Phys. Polon. B , 251 (1973).[14] J. D. Bekenstein, Exact solutions of Einstein-conformal scalar equations , Ann. Phys. (N.Y.) , 535 (1974).[15] J. D. Bekenstein, Black holes with a scalarcharge , Ann. Phys. (N.Y.) , 75 (1975).[16] Yu. G. Ignatyev, R. F. Miftakhov, Statisticalsystems of particles with scalar interactionin cosmology , Grav. Cosmol. , 1 (2006);arXiv:1101.1655.[17] Yu. G. Ignatyev, D. Yu. Ignatyev, Statisticalsystem with fantom scalar interaction in grav-itation theory. I. Microscopic dynamics , Grav.Cosmol. , 299 (2014).[18] T. Piran, J. Shaham, Upper bounds on colli-sional Penrose process near rotating black-holehorizons , Phys. Rev. D , 1615 (1977). KINEMATIC CENSORSHIP AS A CONSTRAINT[19] O. B. Zaslavskii, Is the super-Penrose processpossible near black holes?
Phys. Rev. D , 024056 (2016).[20] A. A. Grib, Yu. V. Pavlov, Are black holes to-tally black?
Grav. Cosmol. , 13 (2015).[21] O. B. Zaslavskii, White holes as particle accel-erators , Grav. Cosmol. , 92 (2018).[22] A. V. Toporensky, O. B. Zaslavskii, Redshift ofa photon emitted along the black hole horizon ,Eur. Phys. J. C , 179 (2017).[23] O. B. Zaslavskii, Acceleration of particles asuniversal property of rotating black holes , Phys.Rev. D , 083004 (2010).[24] T. Jacobson, T. P. Sotiriou, Spinning blackholes as particle accelerators , Phys. Rev. Lett. , 021101 (2010).[25] O. B. Zaslavskii,
Acceleration of particlesby nonrotating charged black holes , Pis’mav ZhETF , 635 (2010), (JETP Letters , 571 (2010)).[26] A. A. Grib, Yu. V. Pavlov, On particle colli-sions near rotating black holes , Grav. Cosmol. , 42 (2011).[27] K. Lake, Particle accelerators insidespinning black holes , Phys. Rev. Lett. , 211102 (2010).[28] K. Lake,
Erratum: Particle accelerators in-side spinning black holes , Phys. Rev. Lett. , 259903 (2010).[29] A. A. Grib, Yu. V. Pavlov,
On particle collisionsin the gravitational field of the Kerr black hole ,Astropart. Phys. , 581 (2011).[30] O. B. Zaslavskii, Acceleration of particles nearthe inner black hole horizon , Phys. Rev. D , 024029 (2012).[31] S. Gao, C. Zhong, Non-extremal Kerr blackholes as particle accelerators , Phys. Rev. D , 044006 (2011).[32] C. Zhong, S. Gao, Particle Collisions nearthe cosmological horizon of a Reissner–Nordstr¨om–de Sitter black hole , JETP Letters, , 589 (2011).[33] O. B. Zaslavskii, Black hole with a scalar fieldas a particle accelerator , Int. J. Mod. Phys. D , 1750108 (2017). [34] M. Patil, P. Joshi, Kerr naked singularitiesas particle accelerators , Class. Quantum Grav. , 235012 (2011).[35] R. B. Mann, I. Nagle, D. R. Terno, Transitionto light-like trajectories in thin shell dynamics ,Nucl. Phys. B936