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Hindawi Advances in High Physics Volume 2017, Article ID 4383617, 7 pages https://doi.org/10.1155/2017/4383617

Research Article Particle Collision Near 1 + 1-Dimensional Horava-Lifshitz and Naked Singularity

M. Halilsoy and A. Ovgun

Physics Department, Eastern Mediterranean University, Gazimagusa, Northern Cyprus, Mersin 10, Turkey

Correspondence should be addressed to A. Ovgun; [email protected]

Received 5 October 2016; Revised 29 November 2016; Accepted 27 December 2016; Published 16 January 2017

Academic Editor: Tiberiu Harko

Copyright © 2017 M. Halilsoy and A. Ovgun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

The unbounded center-of-mass (CM) energy of oppositely moving colliding particles near horizon emerges alsoin 1+1- dimensional Horava-Lifshitz gravity. This theory has imprints of renormalizable characteristics in accordance with the method of simple power counting. Surprisingly the result obtained is not valid for a 1-dimensional Compton-like process between an outgoing and an infalling massless/massive particle. It is possible to achieve unbounded CM energy due to collision between infalling and particles. The source of outgoing particles may be attributed to an explosive process just outside the horizon for a black hole and the naturally repulsive character for the case of a naked singularity. It is found that absence of angular momenta in 1+1-dimension does not yield unbounded energy for collisions in the vicinity of naked singularities.

1. Introduction with infalling particles. Thus collision of two particles can only be argued if both are infalling toward the horizon It is known that in dimensions less than four of a black hole. Such a process, however, yields no BSW gravity has no life of its own unless supplemented by external effect in the nonrotating metrics, which is our main interest sources. With that addition we can have lower dimensional in this study. In order to have an unbounded CM energy gravity and we can talk of black holes, , , in a collision process both particles must be taken in the lensing effect, and so on in analogy with the higher dimen- same coordinate frame and in opposite directions. This is sions. One effect that attracted much interest in recent times possible in the vicinity of a naked singularity whose repulsive is the process of particle collisions near the horizon of black effect compels particles/photons to make collisions with an holes due to Banadosetal.[1]whichcametobeknownasthe˜ infalling particle/photon. From the outset we state that such BSWeffect.Thisproblemaroseasaresultofimitatingthe a collision taking place near the naked singularity in the rather expensive venture of high energy particle collisions in absence of angular momenta does not yield an unbounded laboratory. From curiosity the natural question arises: is there CM energy. To extend our study to cover also collisions a natural laboratory (a particle accelerator) in our cosmos that near black holes we assume that some unspecified process, we may extract information/energy in a cheaper way? This such as disintegration decay process of some particles, yields automatically drew attention to the strong gravity regions outgoing particles photons while the partners fall into the such as near horizon of black holes. Rotating black holes host hole. For a thorough analysis of all these problems covering greater energy reservoir due to their angular momenta and the ergosphere region of a Kerr black hole, Penrose process, attention naturally focused therein first [2, 3]. In case the particle collisions, and so on one must consult [4]. metric is static and diagonal, there are reasons to consider the In general one considers the radial geodesics and upon collision process in the vicinity of a naked singularity as well. energy-momentum conservation in the center-of-mass (CM) We note from physical grounds that outgoing particles frame the near horizon limit is checked whether the energy from the of a black hole cannot occur. Hawking is bounded/unbounded. Our aim in this study is to consider radiation particles/photons emerge too weak to compare black hole solutions in 1+1-dimensional Horava-Lifshitz 2 Advances in High Energy Physics

(HL) gravity [5] and check the BSW effect in such reduced where 𝜂 =constantand𝛼 =constantwillbechosentobeunity 󸀠 dimensional theory. Let us remark that at the Planck scale and 𝑎1 =(ln 𝑁) . Let us comment that a “prime” denotes 2 2 𝑑/𝑑𝑥. 𝑆 in higher dimensions the spherical part 𝑟 𝑑Ω𝐷−2 of the line We note that also the first term in is inherited from element is less effective compared to the time and radial the geometric part of the while the other two terms are components. For this reason 1+1-dimension becomes signif- from the scalar field source. For simplicity we have set also icantatthePlanckscale.ForanumberofreasonsHLgravity 𝑀𝑃𝑙 =1. is promising as a candidate for a renormalizable quantum It has been shown in [5] that by variational principle a gravity physics which has been yearned for a long time [6]. general class of solutions is obtained as follows: The key idea in HL- gravity is the inhomogeneous scaling 𝐴 𝐵 𝐶 properties of time and space coordinates which violate the 2 2 𝑁 (𝑥) =2𝐶2 + 𝑥 −2𝐶1𝑥+ + (4) Lorentz invariance. Arnowitt-Deser-Misner (ADM) splitting 𝜂 𝜂𝑥 3𝜂𝑥2 of space and time [7] constitutes its geometrical background. BSW effect in lower/higher dimensions has been worked in which 𝐶2, 𝐴, 𝐶1, 𝐵,and𝐶 are integration constants. out by many authors [8–44]. Following the similar idea Reference [5] must be consulted for the physical content of we consider black hole solutions and naked singularities these constants. in 1+1-dimensionandsearchforthesameeffectinthis The line element is 1+1 lower dimension. It should be added that with - 𝑑𝑥2 dimensional HL theory the simplest nontrivial solution is 𝑑𝑠2 =−𝑁(𝑥)2 𝑑𝑡2 + 2 (5) the solution describing an accelerated particle in the flat 𝑁 (𝑥) space of Rindler frame. This justifies also the meaning of the vector field (𝑎𝑖) as the in the HL gravity. The with the scalar field role of Rindler acceleration in 3+1-dimension as a possible 𝐴 𝐵 𝐶 source of flat rotation curves and geodesics motion has been 2 𝜙 (𝑥) = ln √2𝐶2 + 𝑥 −2𝐶1𝑥+ + . (6) discussed recently [45]. It is our belief that the results in lower 𝜂 𝜂𝑥 3𝜂𝑥2 dimensions are informative for higher dimensions and as a toy model can play the role as precursors in this regard. Even Note that the associated potential is a Compton-like process can be considered at the toy level 𝐵 𝐶 between a massless photon outgoing from a naked singularity 𝑉(𝜙(𝑥))=𝐴+ + (7) and a particle falling into the naked singularity. The diverging 𝑥3 𝑥4 CM energy results in the case of photon-particle collision in 1+1-dimension under specific conditions. and the Ricci scalar is calculated as Organization of the paper is as follows. In Section 2, we 2 𝐵 𝐶 1+1 𝑅=− (𝐴 + + ). review in brief the -D HL theory with a large class 𝜂 𝑥3 𝑥4 (8) ofblackholeandnakedsingularitysolutions.CMenergy of colliding particles near horizon and naked singularity is There is naked singularity when 𝐴=𝐶1 =0and 𝐶2 =𝐵= considered in Section 3. Section 4 proceeds with applications 𝐶=𝜂=1, so that there is no horizon for to particular examples. The case of particle-photon collision is studied separately in Section 5. The paper ends with our 1 1 𝑁 (𝑥)2 =2+ + . conclusion in Section 6. 𝑥 3𝑥2 (9)

2. 1 + 1-D HL Black Hole/Naked Singularity Another black hole solution reported by Bazeia et al. [5] is found by taking 𝐶1 =0̸ , 𝐶2 =0̸ , 𝐵 =0̸ ,and𝐴=𝐶=0. HL formalism in 3+1-D makes use of the ADM splitting of 𝐵 time and space components as follows: 𝑁 (𝑥)2 =2𝐶 −2𝐶 𝑥+ . 2 1 𝜂𝑥 (10) 2 2 2 𝑖 𝑖 𝑗 𝑗 𝑑𝑠 =−𝑁𝑑𝑡 +𝑔𝑖𝑗 (𝑑𝑥 +𝑁𝑑𝑡) (𝑑𝑥 +𝑁𝑑𝑡) , (1)

𝑖 This solution develops the following horizons: where 𝑁(𝑡) and 𝑁 are the lapse and shift functions, respec- 2 tively. The action of this theory is ± 𝐶 𝐶 𝐵 𝑥 = 2 ± √Δ, Δ = 2 + . (11) 2 ℎ 2𝐶 4𝐶2 2𝜂𝐶 𝑀𝑃𝑙 3 𝑖𝑗 2 1 1 1 𝑆= ∫ 𝑑 𝑥𝑑𝑡√𝑔(𝐾𝑖𝑗 𝐾 +𝜆𝐾 + 𝑉 (𝜙)) , (2) 2 + − As Δ=0they degenerate; that is, 𝑥ℎ =𝑥ℎ . where 𝐾𝑖𝑗 is the extrinsic curvature tensor with trace 𝐾 and The Hawking temperature is given in terms of the outer 𝑀 𝑉(𝜙) + Planck mass 𝑃𝑙. standsforthepotentialfunctionofa (𝑥ℎ ) horizon as follows: scalar field 𝜙,and𝜆 is a constant (𝜆 > 1). Reduction from 3+1 1+1 󸀠 󵄨 -D to -D results in the action [5]: (𝑁 (𝑥)2) 󵄨 󵄨 1 𝑇𝐻 = 󵄨 . (12) 𝑆=∫ 𝑑𝑡 𝑑𝑥 (− 𝜂𝑁2𝑎2 +𝛼𝑁2𝜙󸀠2 −𝑉(𝜙)) 4𝜋 󵄨 1 (3) 󵄨𝑥=𝑥+ 2 ℎ Advances in High Energy Physics 3

𝜇 For the special case 𝐶2 =0, 𝐶1 =−𝑀,and𝐵=−2𝑀the The two of the particles are given by 𝑢 = 𝜇 𝑡 horizons are independent of the mass 𝑀: 𝑑𝑥 /𝑑𝜏.Wehavealreadyobtained𝑢 in the above derivation. 𝑥 To find 𝑢 = 𝑥̇, the normalization condition for time-like ± 1 𝜇 𝑥 =± (𝜂>0). particles, 𝑢 𝑢𝜇 =−1[1,47],canbeusedas ℎ √𝜂 (13) 2 2 𝑔 (𝑢𝑡) +𝑔 (𝑢𝑥) =−1. (22) The temperature is then given simply by 𝑡𝑡 𝑥𝑥 𝑢𝑡 𝑢𝑥 𝑀 By substituting to (22), one obtains as 𝑇𝐻 = . (14) 𝑥 2 2 2 𝜋 (𝑢 ) =𝐸 −𝑁(𝑥) (23)

This is a typical relation between the Hawking temperature for which an effective potential 𝑉eff canbedefinedby and the mass of black holes in 1+1-dimension [46]. 𝑥 2 2 (𝑢 ) +𝑉eff =𝐸. (24) In the case of 𝐶2 =1/2, 𝐵=−2𝑀, 𝜂=1,and𝐴=𝐶= 𝐶1 =0it gives a Schwarzschild-like solution; Now,thetwovelocitiescanbewrittenas 𝐸 2 2𝑀 𝑢𝑡 = 𝑡=̇ , 𝑁 (𝑥) =1− . 2 𝑥 (15) 𝑁 (𝑥) (25) On the other hand, the choice of the parameters, for 𝐶2 = 𝑥 √ 2 2 2 𝑢 = 𝑥=̇ 𝐸 −𝑁(𝑥) . 1/2, 𝐵=−2𝑀, 𝐶=3𝑄, 𝜂=1,and𝐴=𝐶1 =0,givesa Reissner–Nordstrom-like solution. We proceed now to present the CM energy of two 𝜇 𝜇 2 particles with two velocities 𝑢1 and 𝑢2 . We will assume that 2 2𝑀 𝑄 𝑁 (𝑥) =1− + . (16) both have rest mass 𝑚0 =1.TheCMenergyisgivenby 𝑥 𝑥2 𝜇 𝐸 = √2√(1 − 𝑔 𝑢 𝑢]). (26) As in the we can make particular choice cm 𝜇] 1 2 of the parameters so that we end up with a naked singularity So 𝑄2 >𝑀2 instead of a black hole. The choice in (16), 2 2 𝐸2 𝐸 𝐸 𝜅√𝐸2 −𝑁(𝑥) √𝐸2 −𝑁(𝑥) for instance, transforms the HL- black hole into a naked cm =1+ 1 2 − 1 2 , (27) singularity at 𝑥=0.Similarly𝑀<0turns (15) into a naked 2 𝑁 (𝑥)2 𝑁 (𝑥)2 singular metric at 𝑥=0. where 𝜅=±1corresponds to particles moving in the same/opposite direction with respect to each other. We wish 3. CM Energy of Particle Collision near to stress that our concern is for the case 𝜅=±1since no the Horizon of the 1 + 1-D HL Black Hole physical particle is ejected from the black hole. Note that 𝐸1 and 𝐸2 are the energy constants corresponding to each Herewewillderivetheequationsofmotionofan˜uncharged particle.Incasethesecondtermunderthesquarerootistoo massive test particle by using the method of Lag- small than the first one, rangian. Such equations can be derived from the Lagrangian 𝑁 (𝑥)2 equation, √𝐸2 −𝑁(𝑥)2 ≈(𝐸− +⋅⋅⋅) 2𝐸2 (28) 2 2 1 2 𝑑𝑡 1 𝑑𝑥 L = [−𝑁 (𝑥) ( ) + ( ) ] , (17) so that the higher order terms can be neglected and CM 2 𝑑𝜏 𝑁 (𝑥)2 𝑑𝜏 energy of two particles can be written as [23] in which 𝜏 isthepropertimefortime-likegeodesics(or 𝐸2 𝐸 𝐸 𝜅 𝐸 𝐸 cm ≈1+(1−𝜅) 1 2 + ( 2 + 1 ) . massive particles). The canonical momenta are 2 (29) 2 𝑁 (𝑥) 2 𝐸1 𝐸2 𝑑L 𝑝 = =−𝑁(𝑥)2 𝑡,̇ The case with 𝜅=+1is obvious, in which the CM energy 𝑡 ̇ (18) 𝑑𝑡 becomes 𝑑L 𝑥̇ 2 (𝐸2 +𝐸2) 𝑝 = = 𝐸cm 2 1 𝑥 𝑑𝑥̇ 𝑁 (𝑥)2 (19) ≈1+ , (30) 2 2𝐸1𝐸2

The 1+1-D HL black hole has only one killing vector 𝜕𝑡. where the CM energy is independent of metric function, and The associated conserved quantity will be labeled by 𝐸.From it gives always a finite energy. On the other hand 𝜅=−1gives 𝐸 𝑁(𝑥)2 (18), is related to as 2 (𝐸2 +𝐸2) 𝐸cm 2𝐸1𝐸2 2 1 2 ≈1+ − −𝑁 (𝑥) 𝑡=−𝐸.̇ 2 (31) (20) 2 𝑁 (𝑥) 2𝐸1𝐸2 Hence, in which it gives unbounded CM energy near the horizon of 𝐸 theHLblackholesprovidedanoutgoingparticlemechanism 𝑡=̇ . 2 (21) from the horizon is established. Otherwise the yield of two 𝑁 (𝑥) ingoing particles collision remains finite. 4 Advances in High Energy Physics

4. Some Examples with the potential 𝐵 4.1. Schwarzschild-Like Solution. In the case of 𝐶2 =1/2, 𝐵= 𝑉 (𝜙 (𝑥)) = . 𝑥3 (40) −2𝑀, 𝜂=1,and𝐴=𝐶=𝐶1 =0it gives Schwarzschild-like solution, where For the special case 𝐶2 =0, 𝐶1 =−𝑀,and𝐵=−2𝑀we 2𝑀 have 𝑉(𝜙(𝑥))=− , 2 2𝑀 𝑥3 𝑁 (𝑥) =2𝑀𝑥− (41) (32) 𝜂𝑥 2𝑀 𝑁 (𝑥)2 =1− . 𝑥 with suitable potential which is 2𝑀 𝑉(𝜙(𝑥))=− . For the CM energy on the horizon, we have to compute 𝑥3 (42) the limiting value of (27) as 𝑥→𝑥ℎ =2𝑀, where the horizon oftheblackholeis.Setting𝜅=−1as it is, the CM energy near The CM energy of two colliding particles is calculated by the event horizon for 1+1D Schwarzschild BH is taking the limiting values of (31) 2 2 𝐸cm (𝑥 󳨀→ℎ 𝑥 )=∞. (43) 𝐸cm (𝑥 󳨀→ℎ 𝑥 )=∞. (33) Hence the BSW effect arises here as well. This result for 4-D Schwarzschild Black hole is already calculated by Baushev [24]. Hence, the condition of 𝜅=−1, 4.5. Near Horizon Coordinates. We have explored the region whenthelocationofparticle1approachesthehorizonandon near the horizon by replacing 𝑟 by a coordinate 𝜌.Theproper the other hand the particle 2 runs outward from the horizon distance from the horizon 𝜌 [48] is given as follows: 𝐸2 → due to some unspecified physical process, yet yields cm 𝑥 1 ∞ so there is BSW effect for 1+1Schwarzschild-like solution 𝜌=∫ √𝑔 (𝑥󸀠)𝑑𝑥󸀠 = ∫ 𝑑𝑥󸀠. 𝑥𝑥 󸀠 (44) 𝑥 𝑁(𝑥 ) when the condition 𝜅=−1is satisfied. ℎ The first example is the Schwarzschild-like solution which 4.2. Reissner-Nordstrom-Like Solution. On the other hand, is 𝐶 =1/2𝐵=−2𝑀 thechoiceoftheparameters,for 2 , , 2 2𝑀 𝐶=3𝑄2 𝜂=1 𝐴=𝐶 =0 𝑁 (𝑥) =1− (45) , ,and 1 gives the Reissner– 𝑥 Nordstrom-like solution. so that proper distance is calculated as 2 2 2𝑀 𝑄 𝑥 −1/2 𝑁 (𝑥) =1− + , (34) 2𝑀 󸀠 𝑥 𝑥2 𝜌=∫ (1 − ) 𝑑𝑥 𝑥 𝑥ℎ 2𝑀 3𝑄2 (46) 𝑉(𝜙(𝑥))=− + (35) 𝑥 𝑥3 𝑥4 = √𝑥 (𝑥−2𝑀) +2𝑀𝐺 −1 (√ −1). sinh 2𝑀 So the CM energy is calculated by using the limiting value The new metric is of (31) 2 2𝑀 2 2 2 𝑑𝑠 =−(1− )𝑑𝑡 +𝑑𝜌̃ , (47) 𝐸cm (𝑥󳨀→𝑥ℎ=𝑀+√(𝑀2−𝑄2)) =∞. (36) 𝑥(𝜌)̃ 𝜌≃2̃ √2𝑀(𝑥 − 2𝑀) So there is a BSW effect. where so that it gives approximately 2 2 𝜌 2 2 𝑑𝑠 ≃− 𝑑𝑡 +𝑑𝜌 (48) 4.3.TheExtremalCaseoftheReissner-Nordstrom-LikeBlack (4𝑀)2 Hole. Fortheextremalcasewehavewith𝑀=𝑄,from(34), which is once more the Rindler-type line element. Let us note 𝑀 2 that this Rindler-type line element is valid within the near 𝑁 (𝑥)2 = (1− ) (37) 𝑥 horizon limit approximation. For practical purposes there are advantages in adapting such an approximation which so that it also gives the same answer from (31) as conforms with the [48]. The CM energy of two colliding particles is given by 𝐸2 (𝑥 󳨀→ 𝑥 )=∞. cm ℎ (38) 𝐸2 cm =1 2 2𝑚0 4.4. Specific New Black Hole Case. The new 3-parameter black (49) hole solution given by Bazeia et al. [5] is chosen as 2 √ 2 4 4√ 2 4 4 (4𝑀) (𝐸1𝐸2 −𝜅 𝐸1 −𝜌 / (4𝑀) 𝐸2 −𝜌 / (4𝑀) ) + 𝐵 𝜌2 𝑁 (𝑥)2 =2𝐶 −2𝐶 𝑥+ 2 1 𝜂𝑥 (39) so that there is BSW effect for 𝜅=−1when 𝜌→0. Advances in High Energy Physics 5

5. Particle Collision near andforthephoton the Naked Singularity 𝐸 𝑘𝜇 =( 1 ,−𝐸 ). 𝑁2 1 (57) There is a naked singularity for our 1+1-D HL model at the 2 2 location of 𝑥=0,with𝑄 >𝑀 in (16). In addition 𝑀<0 One obtains 𝑥=0 2𝑚𝐸 turns(15)intoanakedsingularmetricat .Thereisalso 𝐸2 =𝑚2 + 1 (𝐸 +𝜅√𝐸2 −𝑁2). naked singularity when we choose metric function as follows: cm 𝑁2 2 2 (58) 2 In the near horizon limit this reduces to 2 1 1 6𝑥 +3𝑥+1 𝑁 (𝑥) =2+ + = . 2 2 2 (50) 2𝑚𝐸 𝑁 𝑥 3𝑥 3𝑥 𝐸2 =𝑚2 + 1 (𝐸 +𝜅𝐸 − ). cm 2 2 2 (59) 𝑁 2𝐸2 As it is given in (27), CM energy of the collision of two 2 particles generally is (for 𝑁(𝑥) → ∞). Note that for 𝜅=−1we have 𝐸cm given by 𝐸 𝐸2 2 2 1 cm 1 2 2 𝐸cm =𝑚 (1 − ) (60) ≈1−𝜅+ [2𝐸 𝐸 +𝜅(𝐸 +𝐸 )] . (51) 𝑚𝐸 2 2𝑁 (𝑥)2 1 2 1 2 2 which is finite between the collision of a photon and an For the case 𝜅=±1,when𝑥 goes to zero, the CM energy infalling particle and therefore is not of interest. As a matter remains finite for radially moving particles. of fact the occurrence of outgoing photon from the event 󵄨 horizon cannot be justified unless an explosive/decay process 𝐸2 󵄨 𝜅=+1 c.m. 󵄨 󳨀→ 1 − 𝜅. is assumed to take place. As a result for from 󵄨 (52) 𝐸2 2 󵄨𝑥=0 (59) we obtain an unbounded cm between the collision of infalling photon and particle. Let us add that “inverse” This suggests that although one of the particle is boosted ComptonprocessintheergosphereofKerrblackhole by the naked singularity, there is not any unlimited collisional was considered in [4] where the photon’s energy showed energy near such singularity. Note that Compton-like pro- increment due to rotational and curvature effects. The energy, cesses were considered first in [4], where rotational effect of however, attained an upper bound which was finite. Our Kerr black hole played a significant role. Our case here is result obtained here being entirely radial on the other hand entirely free of rotational effects. can hardly be compared with those of [4].

6. Photon versus an Infalling Particle 7. Conclusion A massless photon can naturally scatter an infalling particle Our aim was to investigate whether the BSW type effect or vice versa. This phenomenon is analogous to a Compton which arises in higher dimensional black holes applies also scattering taking place in 1+1-dimension. Null-geodesics for to the 1+1-D naked singularity/black hole. The theory a photon can be described simply by we adapted is not general relativity but instead the recently popular HL gravity. We employed the class of 5-parameter 𝑑𝑡 𝐸 = 1 black hole/naked singularity solutions found recently [5]. The 𝑑𝜆 𝑁2 class has particular limits of flat Rindler, Schwarzschild, and (53) 𝑑𝑥 Reissner-Nordstrom-like solutions. For each case we have =±√𝐸2 −𝑁2, calculated the center-of-mass (CM) energy of the particles 𝑑𝜆 1 andshownthattheenergycangrowunboundedforsome where 𝜆 is an affine parameter and 𝐸1 stands for the photon cases. In other words the strong gravity near the event energy. Defining 𝐸1 =ℎ𝜔0,where𝜔0 is the frequency (with horizon affects the collision process with unlimited source the choice ℎ=1)we can parametrize energy of the photon by to turn it into a natural accelerator. The model we use 𝜔0 alone. The CM energy of a photon and the infalling particle applies also to the case of a photon/particle collision with canbetakennowas different characteristics. It is observed that the CM energy of the infalling particles from the rest at infinity will remain 2 𝜇 𝜇 2 𝐸cm =−(𝑝 +𝑘 ) (54) finite in the CM frame at the event horizon of a black hole. Contrariwise, unlimited CM energy will be attained 𝜇 𝜇 𝜇 in which 𝑝 =𝑚𝑢 and 𝑘 refer to the particle and photon, 2 between the collision of the outgoing particles from the event momenta, respectively. This amounts to horizon region and infalling particles. It is also possible to achieve the infinite energy between an infalling photon and 2 2 𝜇 ] 𝐸cm =𝑚 −2𝑚𝑔𝜇]𝑢 𝑘 , (55) an infalling massive particle. However, we found finite CM energy between an outgoing photon and infalling particle. wherewehavefortheparticle Finally, we must admit that absence of rotational effects in 1+1-Drestrictstheproblemtothelevelofatoymodelin 𝐸 𝜇 2 √ 2 2 which particles move on pure radial geodesics yielding finite 𝑝 =𝑚( , 𝐸2 −𝑁 ) (56) 𝑁2 CM energy in the vicinity of a naked singularity. 6 Advances in High Energy Physics

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