Research ArticleParticle Collision Near 1 + 1-Dimensional Horava-LifshitzBlack Hole 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 aliovgunemuedutr
Received 5 October 2016 Revised 29 November 2016 Accepted 27 December 2016 Published 16 January 2017
Academic Editor Tiberiu Harko
Copyright copy 2017 M Halilsoy and A Ovgun This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3
The unbounded center-of-mass (CM) energy of oppositely moving colliding particles near horizon emerges also in 1 + 1-dimensional Horava-Lifshitz gravity This theory has imprints of renormalizable quantum gravity characteristics in accordancewith the method of simple power counting Surprisingly the result obtained is not valid for a 1-dimensional Compton-like processbetween an outgoing photon and an infalling masslessmassive particle It is possible to achieve unbounded CM energy due tocollision between infalling photons and particles The source of outgoing particles may be attributed to an explosive process justoutside the horizon for a black hole and the naturally repulsive character for the case of a naked singularity It is found that absenceof angular momenta in 1 + 1-dimension does not yield unbounded energy for collisions in the vicinity of naked singularities
1 Introduction
It is known that in spacetime dimensions less than fourgravity has no life of its own unless supplemented by externalsources With that addition we can have lower dimensionalgravity and we can talk of black holes wormholes geodesicslensing effect and so on in analogy with the higher dimen-sions One effect that attracted much interest in recent timesis the process of particle collisions near the horizon of blackholes due to Banados et al [1] which came to be known as theBSW effect This problem arose as a result of imitating therather expensive venture of high energy particle collisions inlaboratory From curiosity the natural question arises is therea natural laboratory (a particle accelerator) in our cosmos thatwe may extract informationenergy in a cheaper way Thisautomatically drew attention to the strong gravity regionssuch as near horizon of black holes Rotating black holes hostgreater energy reservoir due to their angular momenta andattention naturally focused therein first [2 3] In case themetric is static and diagonal there are reasons to consider thecollision process in the vicinity of a naked singularity as well
We note from physical grounds that outgoing particlesfrom the event horizon of a black hole cannot occur Hawkingradiation particlesphotons emerge too weak to compare
with infalling particles Thus collision of two particles canonly be argued if both are infalling toward the horizonof a black hole Such a process however yields no BSWeffect in the nonrotating metrics which is our main interestin this study In order to have an unbounded CM energyin a collision process both particles must be taken in thesame coordinate frame and in opposite directions This ispossible in the vicinity of a naked singularity whose repulsiveeffect compels particlesphotons to make collisions with aninfalling particlephoton From the outset we state that sucha collision taking place near the naked singularity in theabsence of angular momenta does not yield an unboundedCM energy To extend our study to cover also collisionsnear black holes we assume that some unspecified processsuch as disintegration decay process of some particles yieldsoutgoing particles photons while the partners fall into thehole For a thorough analysis of all these problems coveringthe ergosphere region of a Kerr black hole Penrose processparticle collisions and so on one must consult [4]
In general one considers the radial geodesics and uponenergy-momentum conservation in the center-of-mass (CM)frame the near horizon limit is checked whether the energyis boundedunbounded Our aim in this study is to considerblack hole solutions in 1 + 1-dimensional Horava-Lifshitz
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2017 Article ID 4383617 7 pageshttpsdoiorg10115520174383617
2 Advances in High Energy Physics
(HL) gravity [5] and check the BSW effect in such reduceddimensional theory Let us remark that at the Planck scalein higher dimensions the spherical part 1199032119889Ω2119863minus2 of the lineelement is less effective compared to the time and radialcomponents For this reason 1+1-dimension becomes signif-icant at the Planck scale For a number of reasons HL gravityis promising as a candidate for a renormalizable quantumgravity physics which has been yearned for a long time [6]The key idea in HL- gravity is the inhomogeneous scalingproperties of time and space coordinates which violate theLorentz invariance Arnowitt-Deser-Misner (ADM) splittingof space and time [7] constitutes its geometrical backgroundBSW effect in lowerhigher dimensions has been workedout by many authors [8ndash44] Following the similar ideawe consider black hole solutions and naked singularitiesin 1 + 1-dimension and search for the same effect in thislower dimension It should be added that with 1 + 1-dimensional HL theory the simplest nontrivial solution isthe solution describing an accelerated particle in the flatspace of Rindler frame This justifies also the meaning of thevector field (119886119894) as the acceleration in the HL gravity Therole of Rindler acceleration in 3 + 1-dimension as a possiblesource of flat rotation curves and geodesics motion has beendiscussed recently [45] It is our belief that the results in lowerdimensions are informative for higher dimensions and as atoy model can play the role as precursors in this regard Evena Compton-like process can be considered at the toy levelbetween amassless photon outgoing from a naked singularityand a particle falling into the naked singularityThe divergingCM energy results in the case of photon-particle collision in1 + 1-dimension under specific conditions
Organization of the paper is as follows In Section 2 wereview in brief the 1 + 1-D HL theory with a large classof black hole and naked singularity solutions CM energyof colliding particles near horizon and naked singularity isconsidered in Section 3 Section 4 proceeds with applicationsto particular examples The case of particle-photon collisionis studied separately in Section 5 The paper ends with ourconclusion in Section 6
2 1 + 1-D HL Black HoleNaked Singularity
HL formalism in 3 + 1-D makes use of the ADM splitting oftime and space components as follows
where 119870119894119895 is the extrinsic curvature tensor with trace 119870 andPlanck mass119872119875119897 119881(120601) stands for the potential function of ascalar field 120601 and 120582 is a constant (120582 gt 1) Reduction from3 + 1-D to 1 + 1-D results in the action [5]
where 120578= constant and120572= constantwill be chosen to be unityand 1198861 = (ln119873)1015840 Let us comment that a ldquoprimerdquo denotes119889119889119909 We note that also the first term in 119878 is inherited fromthe geometric part of the action while the other two terms arefrom the scalar field source For simplicity we have set also119872119875119897 = 1
It has been shown in [5] that by variational principle ageneral class of solutions is obtained as follows
This is a typical relation between the Hawking temperatureand the mass of black holes in 1 + 1-dimension [46]
In the case of 1198622 = 12 119861 = minus2119872 120578 = 1 and 119860 = 119862 =1198621 = 0 it gives a Schwarzschild-like solution119873(119909)2 = 1 minus 2119872119909 (15)
On the other hand the choice of the parameters for1198622 =12 119861 = minus2119872 119862 = 31198762 120578 = 1 and 119860 = 1198621 = 0 gives aReissnerndashNordstrom-like solution
119873(119909)2 = 1 minus 2119872119909 + 11987621199092 (16)
As in the general relativity we can make particular choiceof the parameters so that we end up with a naked singularityinstead of a black hole The choice 1198762 gt 1198722 in (16)for instance transforms the HL- black hole into a nakedsingularity at 119909 = 0 Similarly119872 lt 0 turns (15) into a nakedsingular metric at 119909 = 03 CM Energy of Particle Collision near
the Horizon of the 1 + 1-D HL Black Hole
Here we will derive the equations of motion of an˜unchargedmassive test particle by using the method of geodesic Lag-rangian Such equations can be derived from the Lagrangianequation
L = 12 [minus119873 (119909)2 ( 119889119905119889120591)2 + 1
The 1 + 1-D HL black hole has only one killing vector 120597119905The associated conserved quantity will be labeled by 119864 From(18) 119864 is related to119873(119909)2 as
minus119873 (119909)2 119905 = minus119864 (20)
Hence
119905 = 119864119873 (119909)2 (21)
The two velocities of the particles are given by 119906120583 =119889119909120583119889120591 We have already obtained 119906119905 in the above derivationTo find 119906119909 = the normalization condition for time-likeparticles 119906120583119906120583 = minus1 [1 47] can be used as
By substituting 119906119905 to (22) one obtains 119906119909 as(119906119909)2 = 1198642 minus 119873 (119909)2 (23)
for which an effective potential 119881eff can be defined by
(119906119909)2 + 119881eff = 1198642 (24)
Now the two velocities can be written as
119906119905 = 119905 = 119864119873 (119909)2
119906119909 = = radic1198642 minus 119873 (119909)2(25)
We proceed now to present the CM energy of twoparticles with two velocities 1199061205831 and 1199061205832 We will assume thatboth have rest mass1198980 = 1 The CM energy is given by
119864cm = radic2radic(1 minus 119892120583]1199061205831119906]2) (26)
So
1198642cm2 = 1 + 11986411198642119873(119909)2 minus120581radic11986421 minus 119873 (119909)2radic11986422 minus 119873 (119909)2
119873(119909)2 (27)
where 120581 = plusmn1 corresponds to particles moving in thesameopposite direction with respect to each other We wishto stress that our concern is for the case 120581 = plusmn1 since nophysical particle is ejected from the black hole Note that1198641 and 1198642 are the energy constants corresponding to eachparticle In case the second term under the square root is toosmall than the first one
radic1198642 minus 119873 (119909)2 asymp (119864 minus 119873 (119909)221198642 + sdot sdot sdot) (28)
so that the higher order terms can be neglected and CMenergy of two particles can be written as [23]
in which it gives unbounded CM energy near the horizon ofthe HL black holes provided an outgoing particle mechanismfrom the horizon is established Otherwise the yield of twoingoing particles collision remains finite
4 Advances in High Energy Physics
4 Some Examples
41 Schwarzschild-Like Solution In the case of 1198622 = 12 119861 =minus2119872 120578 = 1 and 119860 = 119862 = 1198621 = 0 it gives Schwarzschild-likesolution where
For the CM energy on the horizon we have to computethe limiting value of (27) as 119909 rarr 119909ℎ = 2119872 where the horizonof the black hole is Setting 120581 = minus1 as it is the CM energy nearthe event horizon for 1 + 1 D Schwarzschild BH is
This result for 4-D Schwarzschild Black hole is alreadycalculated by Baushev [24] Hence the condition of 120581 = minus1when the location of particle 1 approaches the horizon and onthe other hand the particle 2 runs outward from the horizondue to some unspecified physical process yet yields 1198642cm rarrinfin so there is BSW effect for 1+1 Schwarzschild-like solutionwhen the condition 120581 = minus1 is satisfied42 Reissner-Nordstrom-Like Solution On the other handthe choice of the parameters for 1198622 = 12 119861 = minus2119872119862 = 31198762 120578 = 1 and 119860 = 1198621 = 0 gives the ReissnerndashNordstrom-like solution
119873(119909)2 = 1 minus 2119872119909 + 11987621199092 (34)
45 Near Horizon Coordinates We have explored the regionnear the horizon by replacing 119903 by a coordinate 120588 The properdistance from the horizon 120588 [48] is given as follows
Thefirst example is the Schwarzschild-like solutionwhichis
119873(119909)2 = 1 minus 2119872119909 (45)
so that proper distance is calculated as
120588 = int119909119909ℎ
(1 minus 2119872119909 )minus12 1198891199091015840
= radic119909 (119909 minus 2119872) + 2119872119866 sinhminus1 (radic 1199092119872 minus 1) (46)
The new metric is
1198891199042 = minus(1 minus 2119872119909 ()) 1198891199052 + 1198892 (47)
where ≃ 2radic2119872(119909 minus 2119872) so that it gives approximately
1198891199042 ≃ minus 1205882(4119872)2 1198891199052 + 1198891205882 (48)
which is oncemore the Rindler-type line element Let us notethat this Rindler-type line element is valid within the nearhorizon limit approximation For practical purposes thereare advantages in adapting such an approximation whichconformswith the equivalence principle [48]TheCMenergyof two colliding particles is given by
1198642cm211989820 = 1
+ (4119872)2 (11986411198642 minus 120581radic11986421 minus 1205884 (4119872)4radic11986422 minus 1205884 (4119872)4)1205882
(49)
so that there is BSW effect for 120581 = minus1 when 120588 rarr 0
Advances in High Energy Physics 5
5 Particle Collision nearthe Naked Singularity
There is a naked singularity for our 1 + 1-D HL model at thelocation of 119909 = 0 with 1198762 gt 1198722 in (16) In addition 119872 lt 0turns (15) into a naked singular metric at 119909 = 0 There is alsonaked singularity whenwe choosemetric function as follows
For the case 120581 = plusmn1 when 119909 goes to zero the CM energyremains finite for radially moving particles
1198642cm2100381610038161003816100381610038161003816100381610038161003816119909=0 997888rarr 1 minus 120581 (52)
This suggests that although one of the particle is boostedby the naked singularity there is not any unlimited collisionalenergy near such singularity Note that Compton-like pro-cesses were considered first in [4] where rotational effect ofKerr black hole played a significant role Our case here isentirely free of rotational effects
6 Photon versus an Infalling Particle
A massless photon can naturally scatter an infalling particleor vice versa This phenomenon is analogous to a Comptonscattering taking place in 1+1-dimension Null-geodesics fora photon can be described simply by
119889119905119889120582 = 11986411198732119889119909119889120582 = plusmnradic11986421 minus 1198732
(53)
where 120582 is an affine parameter and 1198641 stands for the photonenergy Defining 1198641 = ℎ1205960 where 1205960 is the frequency (withthe choice ℎ = 1)we can parametrize energy of the photon by1205960 aloneTheCMenergy of a photon and the infalling particlecan be taken now as
1198642cm = minus (119901120583 + 119896120583)2 (54)
in which 119901120583 = 119898119906120583 and 119896120583 refer to the particle and photon 2momenta respectively This amounts to
1198642cm = 1198982 minus 2119898119892120583]119906120583119896] (55)
where we have for the particle
119901120583 = 119898( 11986421198732 radic11986422 minus 1198732) (56)
Note that for 120581 = minus1 we have 1198642cm given by
1198642cm = 1198982 (1 minus 11986411198981198642) (60)
which is finite between the collision of a photon and aninfalling particle and therefore is not of interest As a matterof fact the occurrence of outgoing photon from the eventhorizon cannot be justified unless an explosivedecay processis assumed to take place As a result for 120581 = +1 from(59) we obtain an unbounded 1198642cm between the collisionof infalling photon and particle Let us add that ldquoinverserdquoCompton process in the ergosphere of Kerr black holewas considered in [4] where the photonrsquos energy showedincrement due to rotational and curvature effectsThe energyhowever attained an upper bound which was finite Ourresult obtained here being entirely radial on the other handcan hardly be compared with those of [4]
7 Conclusion
Our aim was to investigate whether the BSW type effectwhich arises in higher dimensional black holes applies alsoto the 1 + 1-D naked singularityblack hole The theorywe adapted is not general relativity but instead the recentlypopular HL gravity We employed the class of 5-parameterblack holenaked singularity solutions found recently [5]Theclass has particular limits of flat Rindler Schwarzschild andReissner-Nordstrom-like solutions For each case we havecalculated the center-of-mass (CM) energy of the particlesand shown that the energy can grow unbounded for somecases In other words the strong gravity near the eventhorizon affects the collision process with unlimited sourceto turn it into a natural accelerator The model we useapplies also to the case of a photonparticle collision withdifferent characteristics It is observed that the CM energyof the infalling particles from the rest at infinity will remainfinite in the CM frame at the event horizon of a blackhole Contrariwise unlimited CM energy will be attainedbetween the collision of the outgoing particles from the eventhorizon region and infalling particles It is also possible toachieve the infinite energy between an infalling photon andan infalling massive particle However we found finite CMenergy between an outgoing photon and infalling particleFinally we must admit that absence of rotational effects in1 + 1-D restricts the problem to the level of a toy model inwhich particles move on pure radial geodesics yielding finiteCM energy in the vicinity of a naked singularity
6 Advances in High Energy Physics
Disclosure
This work was presented as a poster at Karl SchwarzschildMeeting 20ndash24 July 2015 Frankfurt Institute for AdvancedStudies
Competing Interests
The authors declare that they have no competing interests
References
[1] M Banados J Silk and S MWest ldquoKerr black holes as particleaccelerators to arbitrarily high energyrdquo Physical Review Lettersvol 103 no 11 Article ID 111102 2009
[2] T Jacobson and J P Sotiriou ldquoSpinning black holes as particleacceleratorsrdquo Physical Review Letters vol 104 no 2 Article ID021101 3 pages 2010
[3] K Lake ldquoParticle accelerators inside spinning black holesrdquoPhysical Review Letters vol 104 Article ID 211102 2010
[4] T Piran and J Shaham ldquoUpper bounds on collisional penroseprocesses near rotating black-hole horizonsrdquo Physical ReviewDvol 16 no 6 pp 1615ndash1635 1977
[5] D Bazeia F A Brito and F G Costa ldquoTwo-dimensionalHorava-Lifshitz black hole solutionsrdquo Physical Review D Par-ticles Fields Gravitation and Cosmology vol 91 no 4 ArticleID 044026 2015
[6] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 2009
[7] R Arnowitt S Deser and C W Misner ldquoRepublication ofthe dynamics of general relativityrdquo General Relativity andGravitation vol 40 no 9 pp 1997ndash2027 2008
[8] E Berti V Cardoso L Gualtieri F Pretorius and U SperhakeldquoComment on ldquokerr black holes as particle accelerators toarbitrarily high energyrdquordquo Physical Review Letters vol 103 no23 Article ID 239001 2009
[9] M Banados B Hassanain J Silk and S M West ldquoEmergentflux from particle collisions near a Kerr black holerdquo PhysicalReview D vol 83 no 2 Article ID 023004 2011
[10] T Jacobson and T P Sotiriou ldquoSpinning black holes as particleacceleratorsrdquo Physical Review Letters vol 104 no 2 Article ID021101 2010
[11] O B Zaslavskii ldquoAcceleration of particles by nonrotatingcharged black holesrdquo JETP Letters vol 92 no 9 pp 571ndash5742011
[12] S WWei Y X Liu H T Li and FW Chen ldquoParticle collisionson stringy black hole backgroundrdquo Journal of High EnergyPhysics vol 2010 no 12 article 066 2010
[13] O B Zaslavskii ldquoEnergy extraction from extremal chargedblack holes due to the Banados-Silk-West effectrdquo PhysicalReview D vol 86 Article ID 124039 2012
[14] H Saadat ldquoThe centre-of-mass energy of two colliding particlesin STU black holesrdquo Canadian Journal of Physics vol 92 no 12pp 1562ndash1564 2014
[15] N Tsukamoto and C Bambi ldquoHigh energy collision of twoparticles in wormhole spacetimesrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 91 no 8 Article ID084013 2015
[16] S G Ghosh P Sheoran and M Amir ldquoRotating Ayon-Beato-Garcıa black hole as a particle acceleratorrdquo Physical Review Dvol 90 no 10 Article ID 103006 2014
[17] A Galajinsky ldquoParticle collisions on near horizon extremalKerr backgroundrdquo Physical Review D vol 88 no 2 Article ID027505 2013
[18] V P Frolov ldquoWeakly magnetized black holes as particle acceler-atorsrdquo Physical Review D vol 85 no 2 Article ID 024020 2012
[19] A M Al Zahrani V P Frolov and A A Shoom ldquoCriticalescape velocity for a charged particle moving around a weaklymagnetized Schwarzschild black holerdquo Physical Review D vol87 no 8 Article ID 084043 2013
[20] J Sadeghi and B Pourhassan ldquoParticle acceleration in Horava-Lifshitz black holesrdquo The European Physical Journal C vol 72no 4 article 1984 2012
[21] J Sadeghi B Pourhassan and H Farahani ldquoRotating chargedhairy black hole in (2+1) dimensions and particle accelerationrdquoCommunications in Theoretical Physics vol 62 no 3 pp 358ndash362 2014
[22] C Liu S Chen CDing and J Jing ldquoParticle acceleration on thebackground of the KerrndashTaubndashNUT spacetimerdquo Physics LettersB vol 701 no 3 pp 285ndash290 2011
[23] M Patil and P S Joshi ldquoUltrahigh energy particle collisions ina regular spacetime without black holes or naked singularitiesrdquoPhysical Review D vol 86 no 4 Article ID 044040 2012
[24] A N Baushev ldquoDark matter annihilation in the gravitationalfield of a black holerdquo International Journal of Modern Physics Dvol 18 no 8 pp 1195ndash1203 2009
[25] M Patil and P S Joshi ldquoParticle acceleration by MajumdarndashPapapetrou di-holerdquoGeneral Relativity and Gravitation vol 46no 10 2014
[26] J D Schnittman ldquoRevised upper limit to energy extractionfrom a kerr black holerdquo Physical Review Letters vol 113 no 26Article ID 261102 2014
[27] M Patil and P S Joshi ldquoNaked singularities as particle accel-eratorsrdquo Physical Review D vol 82 no 10 Article ID 1040492010
[28] M Patil P S Joshi and D Malafarina ldquoNaked singularities asparticle accelerators IIrdquoPhysical ReviewD vol 83 no 6 ArticleID 064007 2011
[29] A Grib andY Pavlov ldquoOn particle collisions in the gravitationalfield of the Kerr black holerdquo Astroparticle Physics vol 34 no 7pp 581ndash586 2011
[30] M Sharif and N Haider ldquoStudy of center of mass energy byparticles collision in some black holesrdquo Astrophysics and SpaceScience vol 346 no 1 pp 111ndash117 2013
[31] I Hussain M Jamil and B Majeed ldquoA slowly rotating blackhole in horava-lifshitz gravity and a 3+1 dimensional topo-logical black hole motion of particles and BSW mechanismrdquoInternational Journal of Theoretical Physics vol 54 no 5 pp1567ndash1577 2015
[32] S Hussain I Hussain and M Jamil ldquoDynamics of a chargedparticle around a slowly rotating Kerr black hole immersed inmagnetic fieldrdquoThe European Physical Journal C vol 74 no 122014
[33] M Amir and S G Ghosh ldquoRotating Haywardrsquos regular blackhole as particle acceleratorrdquo Journal of High Energy Physics vol2015 no 7 article 015 2015
[34] B Pourhassan andUDebnath ldquoParticle acceleration in rotatingmodified hayward and bardeen black holesrdquo httpsarxivorgabs150603443
[35] A A Grib and Y V Pavlov ldquoAre black holes totally blackrdquoGravitation and Cosmology vol 21 no 1 pp 13ndash18 2015
Advances in High Energy Physics 7
[36] A A Grib and Y V Pavlov ldquoHigh energy physics in the vicinityof rotating black holesrdquo Theoretical and Mathematical Physicsvol 185 no 1 pp 1425ndash1432 2015
[37] C Ding C Liu andQQuo ldquoSpacetime noncommutative effecton black hole as particle acceleratorsrdquo International Journal ofModern Physics D vol 22 no 04 Article ID 1350013 2013
[38] J Yang Y-L Li Y Li S-W Wei and Y-X Liu ldquoParticlecollisions in the lower dimensional rotating black hole space-time with the cosmological constantrdquo Advances in High EnergyPhysics vol 2014 Article ID 204016 7 pages 2014
[39] H Nemoto UMiyamoto T Harada and T Kokubu ldquoEscape ofsuperheavy and highly energetic particles produced by particlecollisions near maximally charged black holesrdquo Physical ReviewD vol 87 no 12 Article ID 127502 2013
[40] C Zhong and S Gao ldquoParticle collisions near the cosmologicalhorizon of a Reissner-Nordstrom-de Sitter black holerdquo JETPLetters vol 94 no 8 pp 589ndash592 2011
[41] C Liu S Chen and J Jing ldquoCollision of two general geodesicparticles around a kerrmdashnewman black holerdquo Chinese PhysicsLetters vol 30 no 10 Article ID 100401 2013
[42] Y Zhu S Wu Y Liu and Y Jiang ldquoGeneral stationary chargedblack holes as charged particle acceleratorsrdquo Physical Review Dvol 84 no 4 Article ID 043006 2011
[43] U Miyamoto H Nemoto and M Shimano ldquoParticle creationby naked singularities in higher dimensionsrdquo Physical ReviewD vol 83 no 8 Article ID 084054 2011
[44] Y Li J Yang Y-L Li S-W Wei and Y-X Liu ldquoParticleacceleration in Kerr-(anti-)de Sitter black hole backgroundsrdquoClassical and Quantum Gravity vol 28 no 22 Article ID225006 2011
[45] M Halilsoy O Gurtug and S H Mazharimousavi ldquoRindlermodified Schwarzschild geodesicsrdquo General Relativity andGravitation vol 45 no 11 pp 2363ndash2381 2013
[46] S W Hawking ldquoBlack holes and thermodynamicsrdquo PhysicalReview D vol 13 no 2 pp 191ndash197 1976
[47] CWMisner K SThorne and J AWheelerGravitation WHFreeman amp Co San Francisco Calif USA 1972
[48] L Susskind and J Lindesay An Introduction to Black HolesInformation and the String Theory Revolution The HolographicUniverse World Scientific Hackensack NJ USA 2005
(HL) gravity [5] and check the BSW effect in such reduceddimensional theory Let us remark that at the Planck scalein higher dimensions the spherical part 1199032119889Ω2119863minus2 of the lineelement is less effective compared to the time and radialcomponents For this reason 1+1-dimension becomes signif-icant at the Planck scale For a number of reasons HL gravityis promising as a candidate for a renormalizable quantumgravity physics which has been yearned for a long time [6]The key idea in HL- gravity is the inhomogeneous scalingproperties of time and space coordinates which violate theLorentz invariance Arnowitt-Deser-Misner (ADM) splittingof space and time [7] constitutes its geometrical backgroundBSW effect in lowerhigher dimensions has been workedout by many authors [8ndash44] Following the similar ideawe consider black hole solutions and naked singularitiesin 1 + 1-dimension and search for the same effect in thislower dimension It should be added that with 1 + 1-dimensional HL theory the simplest nontrivial solution isthe solution describing an accelerated particle in the flatspace of Rindler frame This justifies also the meaning of thevector field (119886119894) as the acceleration in the HL gravity Therole of Rindler acceleration in 3 + 1-dimension as a possiblesource of flat rotation curves and geodesics motion has beendiscussed recently [45] It is our belief that the results in lowerdimensions are informative for higher dimensions and as atoy model can play the role as precursors in this regard Evena Compton-like process can be considered at the toy levelbetween amassless photon outgoing from a naked singularityand a particle falling into the naked singularityThe divergingCM energy results in the case of photon-particle collision in1 + 1-dimension under specific conditions
Organization of the paper is as follows In Section 2 wereview in brief the 1 + 1-D HL theory with a large classof black hole and naked singularity solutions CM energyof colliding particles near horizon and naked singularity isconsidered in Section 3 Section 4 proceeds with applicationsto particular examples The case of particle-photon collisionis studied separately in Section 5 The paper ends with ourconclusion in Section 6
2 1 + 1-D HL Black HoleNaked Singularity
HL formalism in 3 + 1-D makes use of the ADM splitting oftime and space components as follows
where 119870119894119895 is the extrinsic curvature tensor with trace 119870 andPlanck mass119872119875119897 119881(120601) stands for the potential function of ascalar field 120601 and 120582 is a constant (120582 gt 1) Reduction from3 + 1-D to 1 + 1-D results in the action [5]
where 120578= constant and120572= constantwill be chosen to be unityand 1198861 = (ln119873)1015840 Let us comment that a ldquoprimerdquo denotes119889119889119909 We note that also the first term in 119878 is inherited fromthe geometric part of the action while the other two terms arefrom the scalar field source For simplicity we have set also119872119875119897 = 1
It has been shown in [5] that by variational principle ageneral class of solutions is obtained as follows
This is a typical relation between the Hawking temperatureand the mass of black holes in 1 + 1-dimension [46]
In the case of 1198622 = 12 119861 = minus2119872 120578 = 1 and 119860 = 119862 =1198621 = 0 it gives a Schwarzschild-like solution119873(119909)2 = 1 minus 2119872119909 (15)
On the other hand the choice of the parameters for1198622 =12 119861 = minus2119872 119862 = 31198762 120578 = 1 and 119860 = 1198621 = 0 gives aReissnerndashNordstrom-like solution
119873(119909)2 = 1 minus 2119872119909 + 11987621199092 (16)
As in the general relativity we can make particular choiceof the parameters so that we end up with a naked singularityinstead of a black hole The choice 1198762 gt 1198722 in (16)for instance transforms the HL- black hole into a nakedsingularity at 119909 = 0 Similarly119872 lt 0 turns (15) into a nakedsingular metric at 119909 = 03 CM Energy of Particle Collision near
the Horizon of the 1 + 1-D HL Black Hole
Here we will derive the equations of motion of an˜unchargedmassive test particle by using the method of geodesic Lag-rangian Such equations can be derived from the Lagrangianequation
L = 12 [minus119873 (119909)2 ( 119889119905119889120591)2 + 1
The 1 + 1-D HL black hole has only one killing vector 120597119905The associated conserved quantity will be labeled by 119864 From(18) 119864 is related to119873(119909)2 as
minus119873 (119909)2 119905 = minus119864 (20)
Hence
119905 = 119864119873 (119909)2 (21)
The two velocities of the particles are given by 119906120583 =119889119909120583119889120591 We have already obtained 119906119905 in the above derivationTo find 119906119909 = the normalization condition for time-likeparticles 119906120583119906120583 = minus1 [1 47] can be used as
By substituting 119906119905 to (22) one obtains 119906119909 as(119906119909)2 = 1198642 minus 119873 (119909)2 (23)
for which an effective potential 119881eff can be defined by
(119906119909)2 + 119881eff = 1198642 (24)
Now the two velocities can be written as
119906119905 = 119905 = 119864119873 (119909)2
119906119909 = = radic1198642 minus 119873 (119909)2(25)
We proceed now to present the CM energy of twoparticles with two velocities 1199061205831 and 1199061205832 We will assume thatboth have rest mass1198980 = 1 The CM energy is given by
119864cm = radic2radic(1 minus 119892120583]1199061205831119906]2) (26)
So
1198642cm2 = 1 + 11986411198642119873(119909)2 minus120581radic11986421 minus 119873 (119909)2radic11986422 minus 119873 (119909)2
119873(119909)2 (27)
where 120581 = plusmn1 corresponds to particles moving in thesameopposite direction with respect to each other We wishto stress that our concern is for the case 120581 = plusmn1 since nophysical particle is ejected from the black hole Note that1198641 and 1198642 are the energy constants corresponding to eachparticle In case the second term under the square root is toosmall than the first one
radic1198642 minus 119873 (119909)2 asymp (119864 minus 119873 (119909)221198642 + sdot sdot sdot) (28)
so that the higher order terms can be neglected and CMenergy of two particles can be written as [23]
in which it gives unbounded CM energy near the horizon ofthe HL black holes provided an outgoing particle mechanismfrom the horizon is established Otherwise the yield of twoingoing particles collision remains finite
4 Advances in High Energy Physics
4 Some Examples
41 Schwarzschild-Like Solution In the case of 1198622 = 12 119861 =minus2119872 120578 = 1 and 119860 = 119862 = 1198621 = 0 it gives Schwarzschild-likesolution where
For the CM energy on the horizon we have to computethe limiting value of (27) as 119909 rarr 119909ℎ = 2119872 where the horizonof the black hole is Setting 120581 = minus1 as it is the CM energy nearthe event horizon for 1 + 1 D Schwarzschild BH is
This result for 4-D Schwarzschild Black hole is alreadycalculated by Baushev [24] Hence the condition of 120581 = minus1when the location of particle 1 approaches the horizon and onthe other hand the particle 2 runs outward from the horizondue to some unspecified physical process yet yields 1198642cm rarrinfin so there is BSW effect for 1+1 Schwarzschild-like solutionwhen the condition 120581 = minus1 is satisfied42 Reissner-Nordstrom-Like Solution On the other handthe choice of the parameters for 1198622 = 12 119861 = minus2119872119862 = 31198762 120578 = 1 and 119860 = 1198621 = 0 gives the ReissnerndashNordstrom-like solution
119873(119909)2 = 1 minus 2119872119909 + 11987621199092 (34)
45 Near Horizon Coordinates We have explored the regionnear the horizon by replacing 119903 by a coordinate 120588 The properdistance from the horizon 120588 [48] is given as follows
Thefirst example is the Schwarzschild-like solutionwhichis
119873(119909)2 = 1 minus 2119872119909 (45)
so that proper distance is calculated as
120588 = int119909119909ℎ
(1 minus 2119872119909 )minus12 1198891199091015840
= radic119909 (119909 minus 2119872) + 2119872119866 sinhminus1 (radic 1199092119872 minus 1) (46)
The new metric is
1198891199042 = minus(1 minus 2119872119909 ()) 1198891199052 + 1198892 (47)
where ≃ 2radic2119872(119909 minus 2119872) so that it gives approximately
1198891199042 ≃ minus 1205882(4119872)2 1198891199052 + 1198891205882 (48)
which is oncemore the Rindler-type line element Let us notethat this Rindler-type line element is valid within the nearhorizon limit approximation For practical purposes thereare advantages in adapting such an approximation whichconformswith the equivalence principle [48]TheCMenergyof two colliding particles is given by
1198642cm211989820 = 1
+ (4119872)2 (11986411198642 minus 120581radic11986421 minus 1205884 (4119872)4radic11986422 minus 1205884 (4119872)4)1205882
(49)
so that there is BSW effect for 120581 = minus1 when 120588 rarr 0
Advances in High Energy Physics 5
5 Particle Collision nearthe Naked Singularity
There is a naked singularity for our 1 + 1-D HL model at thelocation of 119909 = 0 with 1198762 gt 1198722 in (16) In addition 119872 lt 0turns (15) into a naked singular metric at 119909 = 0 There is alsonaked singularity whenwe choosemetric function as follows
For the case 120581 = plusmn1 when 119909 goes to zero the CM energyremains finite for radially moving particles
1198642cm2100381610038161003816100381610038161003816100381610038161003816119909=0 997888rarr 1 minus 120581 (52)
This suggests that although one of the particle is boostedby the naked singularity there is not any unlimited collisionalenergy near such singularity Note that Compton-like pro-cesses were considered first in [4] where rotational effect ofKerr black hole played a significant role Our case here isentirely free of rotational effects
6 Photon versus an Infalling Particle
A massless photon can naturally scatter an infalling particleor vice versa This phenomenon is analogous to a Comptonscattering taking place in 1+1-dimension Null-geodesics fora photon can be described simply by
119889119905119889120582 = 11986411198732119889119909119889120582 = plusmnradic11986421 minus 1198732
(53)
where 120582 is an affine parameter and 1198641 stands for the photonenergy Defining 1198641 = ℎ1205960 where 1205960 is the frequency (withthe choice ℎ = 1)we can parametrize energy of the photon by1205960 aloneTheCMenergy of a photon and the infalling particlecan be taken now as
1198642cm = minus (119901120583 + 119896120583)2 (54)
in which 119901120583 = 119898119906120583 and 119896120583 refer to the particle and photon 2momenta respectively This amounts to
1198642cm = 1198982 minus 2119898119892120583]119906120583119896] (55)
where we have for the particle
119901120583 = 119898( 11986421198732 radic11986422 minus 1198732) (56)
Note that for 120581 = minus1 we have 1198642cm given by
1198642cm = 1198982 (1 minus 11986411198981198642) (60)
which is finite between the collision of a photon and aninfalling particle and therefore is not of interest As a matterof fact the occurrence of outgoing photon from the eventhorizon cannot be justified unless an explosivedecay processis assumed to take place As a result for 120581 = +1 from(59) we obtain an unbounded 1198642cm between the collisionof infalling photon and particle Let us add that ldquoinverserdquoCompton process in the ergosphere of Kerr black holewas considered in [4] where the photonrsquos energy showedincrement due to rotational and curvature effectsThe energyhowever attained an upper bound which was finite Ourresult obtained here being entirely radial on the other handcan hardly be compared with those of [4]
7 Conclusion
Our aim was to investigate whether the BSW type effectwhich arises in higher dimensional black holes applies alsoto the 1 + 1-D naked singularityblack hole The theorywe adapted is not general relativity but instead the recentlypopular HL gravity We employed the class of 5-parameterblack holenaked singularity solutions found recently [5]Theclass has particular limits of flat Rindler Schwarzschild andReissner-Nordstrom-like solutions For each case we havecalculated the center-of-mass (CM) energy of the particlesand shown that the energy can grow unbounded for somecases In other words the strong gravity near the eventhorizon affects the collision process with unlimited sourceto turn it into a natural accelerator The model we useapplies also to the case of a photonparticle collision withdifferent characteristics It is observed that the CM energyof the infalling particles from the rest at infinity will remainfinite in the CM frame at the event horizon of a blackhole Contrariwise unlimited CM energy will be attainedbetween the collision of the outgoing particles from the eventhorizon region and infalling particles It is also possible toachieve the infinite energy between an infalling photon andan infalling massive particle However we found finite CMenergy between an outgoing photon and infalling particleFinally we must admit that absence of rotational effects in1 + 1-D restricts the problem to the level of a toy model inwhich particles move on pure radial geodesics yielding finiteCM energy in the vicinity of a naked singularity
6 Advances in High Energy Physics
Disclosure
This work was presented as a poster at Karl SchwarzschildMeeting 20ndash24 July 2015 Frankfurt Institute for AdvancedStudies
Competing Interests
The authors declare that they have no competing interests
References
[1] M Banados J Silk and S MWest ldquoKerr black holes as particleaccelerators to arbitrarily high energyrdquo Physical Review Lettersvol 103 no 11 Article ID 111102 2009
[2] T Jacobson and J P Sotiriou ldquoSpinning black holes as particleacceleratorsrdquo Physical Review Letters vol 104 no 2 Article ID021101 3 pages 2010
[3] K Lake ldquoParticle accelerators inside spinning black holesrdquoPhysical Review Letters vol 104 Article ID 211102 2010
[4] T Piran and J Shaham ldquoUpper bounds on collisional penroseprocesses near rotating black-hole horizonsrdquo Physical ReviewDvol 16 no 6 pp 1615ndash1635 1977
[5] D Bazeia F A Brito and F G Costa ldquoTwo-dimensionalHorava-Lifshitz black hole solutionsrdquo Physical Review D Par-ticles Fields Gravitation and Cosmology vol 91 no 4 ArticleID 044026 2015
[6] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 2009
[7] R Arnowitt S Deser and C W Misner ldquoRepublication ofthe dynamics of general relativityrdquo General Relativity andGravitation vol 40 no 9 pp 1997ndash2027 2008
[8] E Berti V Cardoso L Gualtieri F Pretorius and U SperhakeldquoComment on ldquokerr black holes as particle accelerators toarbitrarily high energyrdquordquo Physical Review Letters vol 103 no23 Article ID 239001 2009
[9] M Banados B Hassanain J Silk and S M West ldquoEmergentflux from particle collisions near a Kerr black holerdquo PhysicalReview D vol 83 no 2 Article ID 023004 2011
[10] T Jacobson and T P Sotiriou ldquoSpinning black holes as particleacceleratorsrdquo Physical Review Letters vol 104 no 2 Article ID021101 2010
[11] O B Zaslavskii ldquoAcceleration of particles by nonrotatingcharged black holesrdquo JETP Letters vol 92 no 9 pp 571ndash5742011
[12] S WWei Y X Liu H T Li and FW Chen ldquoParticle collisionson stringy black hole backgroundrdquo Journal of High EnergyPhysics vol 2010 no 12 article 066 2010
[13] O B Zaslavskii ldquoEnergy extraction from extremal chargedblack holes due to the Banados-Silk-West effectrdquo PhysicalReview D vol 86 Article ID 124039 2012
[14] H Saadat ldquoThe centre-of-mass energy of two colliding particlesin STU black holesrdquo Canadian Journal of Physics vol 92 no 12pp 1562ndash1564 2014
[15] N Tsukamoto and C Bambi ldquoHigh energy collision of twoparticles in wormhole spacetimesrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 91 no 8 Article ID084013 2015
[16] S G Ghosh P Sheoran and M Amir ldquoRotating Ayon-Beato-Garcıa black hole as a particle acceleratorrdquo Physical Review Dvol 90 no 10 Article ID 103006 2014
[17] A Galajinsky ldquoParticle collisions on near horizon extremalKerr backgroundrdquo Physical Review D vol 88 no 2 Article ID027505 2013
[18] V P Frolov ldquoWeakly magnetized black holes as particle acceler-atorsrdquo Physical Review D vol 85 no 2 Article ID 024020 2012
[19] A M Al Zahrani V P Frolov and A A Shoom ldquoCriticalescape velocity for a charged particle moving around a weaklymagnetized Schwarzschild black holerdquo Physical Review D vol87 no 8 Article ID 084043 2013
[20] J Sadeghi and B Pourhassan ldquoParticle acceleration in Horava-Lifshitz black holesrdquo The European Physical Journal C vol 72no 4 article 1984 2012
[21] J Sadeghi B Pourhassan and H Farahani ldquoRotating chargedhairy black hole in (2+1) dimensions and particle accelerationrdquoCommunications in Theoretical Physics vol 62 no 3 pp 358ndash362 2014
[22] C Liu S Chen CDing and J Jing ldquoParticle acceleration on thebackground of the KerrndashTaubndashNUT spacetimerdquo Physics LettersB vol 701 no 3 pp 285ndash290 2011
[23] M Patil and P S Joshi ldquoUltrahigh energy particle collisions ina regular spacetime without black holes or naked singularitiesrdquoPhysical Review D vol 86 no 4 Article ID 044040 2012
[24] A N Baushev ldquoDark matter annihilation in the gravitationalfield of a black holerdquo International Journal of Modern Physics Dvol 18 no 8 pp 1195ndash1203 2009
[25] M Patil and P S Joshi ldquoParticle acceleration by MajumdarndashPapapetrou di-holerdquoGeneral Relativity and Gravitation vol 46no 10 2014
[26] J D Schnittman ldquoRevised upper limit to energy extractionfrom a kerr black holerdquo Physical Review Letters vol 113 no 26Article ID 261102 2014
[27] M Patil and P S Joshi ldquoNaked singularities as particle accel-eratorsrdquo Physical Review D vol 82 no 10 Article ID 1040492010
[28] M Patil P S Joshi and D Malafarina ldquoNaked singularities asparticle accelerators IIrdquoPhysical ReviewD vol 83 no 6 ArticleID 064007 2011
[29] A Grib andY Pavlov ldquoOn particle collisions in the gravitationalfield of the Kerr black holerdquo Astroparticle Physics vol 34 no 7pp 581ndash586 2011
[30] M Sharif and N Haider ldquoStudy of center of mass energy byparticles collision in some black holesrdquo Astrophysics and SpaceScience vol 346 no 1 pp 111ndash117 2013
[31] I Hussain M Jamil and B Majeed ldquoA slowly rotating blackhole in horava-lifshitz gravity and a 3+1 dimensional topo-logical black hole motion of particles and BSW mechanismrdquoInternational Journal of Theoretical Physics vol 54 no 5 pp1567ndash1577 2015
[32] S Hussain I Hussain and M Jamil ldquoDynamics of a chargedparticle around a slowly rotating Kerr black hole immersed inmagnetic fieldrdquoThe European Physical Journal C vol 74 no 122014
[33] M Amir and S G Ghosh ldquoRotating Haywardrsquos regular blackhole as particle acceleratorrdquo Journal of High Energy Physics vol2015 no 7 article 015 2015
[34] B Pourhassan andUDebnath ldquoParticle acceleration in rotatingmodified hayward and bardeen black holesrdquo httpsarxivorgabs150603443
[35] A A Grib and Y V Pavlov ldquoAre black holes totally blackrdquoGravitation and Cosmology vol 21 no 1 pp 13ndash18 2015
Advances in High Energy Physics 7
[36] A A Grib and Y V Pavlov ldquoHigh energy physics in the vicinityof rotating black holesrdquo Theoretical and Mathematical Physicsvol 185 no 1 pp 1425ndash1432 2015
[37] C Ding C Liu andQQuo ldquoSpacetime noncommutative effecton black hole as particle acceleratorsrdquo International Journal ofModern Physics D vol 22 no 04 Article ID 1350013 2013
[38] J Yang Y-L Li Y Li S-W Wei and Y-X Liu ldquoParticlecollisions in the lower dimensional rotating black hole space-time with the cosmological constantrdquo Advances in High EnergyPhysics vol 2014 Article ID 204016 7 pages 2014
[39] H Nemoto UMiyamoto T Harada and T Kokubu ldquoEscape ofsuperheavy and highly energetic particles produced by particlecollisions near maximally charged black holesrdquo Physical ReviewD vol 87 no 12 Article ID 127502 2013
[40] C Zhong and S Gao ldquoParticle collisions near the cosmologicalhorizon of a Reissner-Nordstrom-de Sitter black holerdquo JETPLetters vol 94 no 8 pp 589ndash592 2011
[41] C Liu S Chen and J Jing ldquoCollision of two general geodesicparticles around a kerrmdashnewman black holerdquo Chinese PhysicsLetters vol 30 no 10 Article ID 100401 2013
[42] Y Zhu S Wu Y Liu and Y Jiang ldquoGeneral stationary chargedblack holes as charged particle acceleratorsrdquo Physical Review Dvol 84 no 4 Article ID 043006 2011
[43] U Miyamoto H Nemoto and M Shimano ldquoParticle creationby naked singularities in higher dimensionsrdquo Physical ReviewD vol 83 no 8 Article ID 084054 2011
[44] Y Li J Yang Y-L Li S-W Wei and Y-X Liu ldquoParticleacceleration in Kerr-(anti-)de Sitter black hole backgroundsrdquoClassical and Quantum Gravity vol 28 no 22 Article ID225006 2011
[45] M Halilsoy O Gurtug and S H Mazharimousavi ldquoRindlermodified Schwarzschild geodesicsrdquo General Relativity andGravitation vol 45 no 11 pp 2363ndash2381 2013
[46] S W Hawking ldquoBlack holes and thermodynamicsrdquo PhysicalReview D vol 13 no 2 pp 191ndash197 1976
[47] CWMisner K SThorne and J AWheelerGravitation WHFreeman amp Co San Francisco Calif USA 1972
[48] L Susskind and J Lindesay An Introduction to Black HolesInformation and the String Theory Revolution The HolographicUniverse World Scientific Hackensack NJ USA 2005
This is a typical relation between the Hawking temperatureand the mass of black holes in 1 + 1-dimension [46]
In the case of 1198622 = 12 119861 = minus2119872 120578 = 1 and 119860 = 119862 =1198621 = 0 it gives a Schwarzschild-like solution119873(119909)2 = 1 minus 2119872119909 (15)
On the other hand the choice of the parameters for1198622 =12 119861 = minus2119872 119862 = 31198762 120578 = 1 and 119860 = 1198621 = 0 gives aReissnerndashNordstrom-like solution
119873(119909)2 = 1 minus 2119872119909 + 11987621199092 (16)
As in the general relativity we can make particular choiceof the parameters so that we end up with a naked singularityinstead of a black hole The choice 1198762 gt 1198722 in (16)for instance transforms the HL- black hole into a nakedsingularity at 119909 = 0 Similarly119872 lt 0 turns (15) into a nakedsingular metric at 119909 = 03 CM Energy of Particle Collision near
the Horizon of the 1 + 1-D HL Black Hole
Here we will derive the equations of motion of an˜unchargedmassive test particle by using the method of geodesic Lag-rangian Such equations can be derived from the Lagrangianequation
L = 12 [minus119873 (119909)2 ( 119889119905119889120591)2 + 1
The 1 + 1-D HL black hole has only one killing vector 120597119905The associated conserved quantity will be labeled by 119864 From(18) 119864 is related to119873(119909)2 as
minus119873 (119909)2 119905 = minus119864 (20)
Hence
119905 = 119864119873 (119909)2 (21)
The two velocities of the particles are given by 119906120583 =119889119909120583119889120591 We have already obtained 119906119905 in the above derivationTo find 119906119909 = the normalization condition for time-likeparticles 119906120583119906120583 = minus1 [1 47] can be used as
By substituting 119906119905 to (22) one obtains 119906119909 as(119906119909)2 = 1198642 minus 119873 (119909)2 (23)
for which an effective potential 119881eff can be defined by
(119906119909)2 + 119881eff = 1198642 (24)
Now the two velocities can be written as
119906119905 = 119905 = 119864119873 (119909)2
119906119909 = = radic1198642 minus 119873 (119909)2(25)
We proceed now to present the CM energy of twoparticles with two velocities 1199061205831 and 1199061205832 We will assume thatboth have rest mass1198980 = 1 The CM energy is given by
119864cm = radic2radic(1 minus 119892120583]1199061205831119906]2) (26)
So
1198642cm2 = 1 + 11986411198642119873(119909)2 minus120581radic11986421 minus 119873 (119909)2radic11986422 minus 119873 (119909)2
119873(119909)2 (27)
where 120581 = plusmn1 corresponds to particles moving in thesameopposite direction with respect to each other We wishto stress that our concern is for the case 120581 = plusmn1 since nophysical particle is ejected from the black hole Note that1198641 and 1198642 are the energy constants corresponding to eachparticle In case the second term under the square root is toosmall than the first one
radic1198642 minus 119873 (119909)2 asymp (119864 minus 119873 (119909)221198642 + sdot sdot sdot) (28)
so that the higher order terms can be neglected and CMenergy of two particles can be written as [23]
in which it gives unbounded CM energy near the horizon ofthe HL black holes provided an outgoing particle mechanismfrom the horizon is established Otherwise the yield of twoingoing particles collision remains finite
4 Advances in High Energy Physics
4 Some Examples
41 Schwarzschild-Like Solution In the case of 1198622 = 12 119861 =minus2119872 120578 = 1 and 119860 = 119862 = 1198621 = 0 it gives Schwarzschild-likesolution where
For the CM energy on the horizon we have to computethe limiting value of (27) as 119909 rarr 119909ℎ = 2119872 where the horizonof the black hole is Setting 120581 = minus1 as it is the CM energy nearthe event horizon for 1 + 1 D Schwarzschild BH is
This result for 4-D Schwarzschild Black hole is alreadycalculated by Baushev [24] Hence the condition of 120581 = minus1when the location of particle 1 approaches the horizon and onthe other hand the particle 2 runs outward from the horizondue to some unspecified physical process yet yields 1198642cm rarrinfin so there is BSW effect for 1+1 Schwarzschild-like solutionwhen the condition 120581 = minus1 is satisfied42 Reissner-Nordstrom-Like Solution On the other handthe choice of the parameters for 1198622 = 12 119861 = minus2119872119862 = 31198762 120578 = 1 and 119860 = 1198621 = 0 gives the ReissnerndashNordstrom-like solution
119873(119909)2 = 1 minus 2119872119909 + 11987621199092 (34)
45 Near Horizon Coordinates We have explored the regionnear the horizon by replacing 119903 by a coordinate 120588 The properdistance from the horizon 120588 [48] is given as follows
Thefirst example is the Schwarzschild-like solutionwhichis
119873(119909)2 = 1 minus 2119872119909 (45)
so that proper distance is calculated as
120588 = int119909119909ℎ
(1 minus 2119872119909 )minus12 1198891199091015840
= radic119909 (119909 minus 2119872) + 2119872119866 sinhminus1 (radic 1199092119872 minus 1) (46)
The new metric is
1198891199042 = minus(1 minus 2119872119909 ()) 1198891199052 + 1198892 (47)
where ≃ 2radic2119872(119909 minus 2119872) so that it gives approximately
1198891199042 ≃ minus 1205882(4119872)2 1198891199052 + 1198891205882 (48)
which is oncemore the Rindler-type line element Let us notethat this Rindler-type line element is valid within the nearhorizon limit approximation For practical purposes thereare advantages in adapting such an approximation whichconformswith the equivalence principle [48]TheCMenergyof two colliding particles is given by
1198642cm211989820 = 1
+ (4119872)2 (11986411198642 minus 120581radic11986421 minus 1205884 (4119872)4radic11986422 minus 1205884 (4119872)4)1205882
(49)
so that there is BSW effect for 120581 = minus1 when 120588 rarr 0
Advances in High Energy Physics 5
5 Particle Collision nearthe Naked Singularity
There is a naked singularity for our 1 + 1-D HL model at thelocation of 119909 = 0 with 1198762 gt 1198722 in (16) In addition 119872 lt 0turns (15) into a naked singular metric at 119909 = 0 There is alsonaked singularity whenwe choosemetric function as follows
For the case 120581 = plusmn1 when 119909 goes to zero the CM energyremains finite for radially moving particles
1198642cm2100381610038161003816100381610038161003816100381610038161003816119909=0 997888rarr 1 minus 120581 (52)
This suggests that although one of the particle is boostedby the naked singularity there is not any unlimited collisionalenergy near such singularity Note that Compton-like pro-cesses were considered first in [4] where rotational effect ofKerr black hole played a significant role Our case here isentirely free of rotational effects
6 Photon versus an Infalling Particle
A massless photon can naturally scatter an infalling particleor vice versa This phenomenon is analogous to a Comptonscattering taking place in 1+1-dimension Null-geodesics fora photon can be described simply by
119889119905119889120582 = 11986411198732119889119909119889120582 = plusmnradic11986421 minus 1198732
(53)
where 120582 is an affine parameter and 1198641 stands for the photonenergy Defining 1198641 = ℎ1205960 where 1205960 is the frequency (withthe choice ℎ = 1)we can parametrize energy of the photon by1205960 aloneTheCMenergy of a photon and the infalling particlecan be taken now as
1198642cm = minus (119901120583 + 119896120583)2 (54)
in which 119901120583 = 119898119906120583 and 119896120583 refer to the particle and photon 2momenta respectively This amounts to
1198642cm = 1198982 minus 2119898119892120583]119906120583119896] (55)
where we have for the particle
119901120583 = 119898( 11986421198732 radic11986422 minus 1198732) (56)
Note that for 120581 = minus1 we have 1198642cm given by
1198642cm = 1198982 (1 minus 11986411198981198642) (60)
which is finite between the collision of a photon and aninfalling particle and therefore is not of interest As a matterof fact the occurrence of outgoing photon from the eventhorizon cannot be justified unless an explosivedecay processis assumed to take place As a result for 120581 = +1 from(59) we obtain an unbounded 1198642cm between the collisionof infalling photon and particle Let us add that ldquoinverserdquoCompton process in the ergosphere of Kerr black holewas considered in [4] where the photonrsquos energy showedincrement due to rotational and curvature effectsThe energyhowever attained an upper bound which was finite Ourresult obtained here being entirely radial on the other handcan hardly be compared with those of [4]
7 Conclusion
Our aim was to investigate whether the BSW type effectwhich arises in higher dimensional black holes applies alsoto the 1 + 1-D naked singularityblack hole The theorywe adapted is not general relativity but instead the recentlypopular HL gravity We employed the class of 5-parameterblack holenaked singularity solutions found recently [5]Theclass has particular limits of flat Rindler Schwarzschild andReissner-Nordstrom-like solutions For each case we havecalculated the center-of-mass (CM) energy of the particlesand shown that the energy can grow unbounded for somecases In other words the strong gravity near the eventhorizon affects the collision process with unlimited sourceto turn it into a natural accelerator The model we useapplies also to the case of a photonparticle collision withdifferent characteristics It is observed that the CM energyof the infalling particles from the rest at infinity will remainfinite in the CM frame at the event horizon of a blackhole Contrariwise unlimited CM energy will be attainedbetween the collision of the outgoing particles from the eventhorizon region and infalling particles It is also possible toachieve the infinite energy between an infalling photon andan infalling massive particle However we found finite CMenergy between an outgoing photon and infalling particleFinally we must admit that absence of rotational effects in1 + 1-D restricts the problem to the level of a toy model inwhich particles move on pure radial geodesics yielding finiteCM energy in the vicinity of a naked singularity
6 Advances in High Energy Physics
Disclosure
This work was presented as a poster at Karl SchwarzschildMeeting 20ndash24 July 2015 Frankfurt Institute for AdvancedStudies
Competing Interests
The authors declare that they have no competing interests
References
[1] M Banados J Silk and S MWest ldquoKerr black holes as particleaccelerators to arbitrarily high energyrdquo Physical Review Lettersvol 103 no 11 Article ID 111102 2009
[2] T Jacobson and J P Sotiriou ldquoSpinning black holes as particleacceleratorsrdquo Physical Review Letters vol 104 no 2 Article ID021101 3 pages 2010
[3] K Lake ldquoParticle accelerators inside spinning black holesrdquoPhysical Review Letters vol 104 Article ID 211102 2010
[4] T Piran and J Shaham ldquoUpper bounds on collisional penroseprocesses near rotating black-hole horizonsrdquo Physical ReviewDvol 16 no 6 pp 1615ndash1635 1977
[5] D Bazeia F A Brito and F G Costa ldquoTwo-dimensionalHorava-Lifshitz black hole solutionsrdquo Physical Review D Par-ticles Fields Gravitation and Cosmology vol 91 no 4 ArticleID 044026 2015
[6] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 2009
[7] R Arnowitt S Deser and C W Misner ldquoRepublication ofthe dynamics of general relativityrdquo General Relativity andGravitation vol 40 no 9 pp 1997ndash2027 2008
[8] E Berti V Cardoso L Gualtieri F Pretorius and U SperhakeldquoComment on ldquokerr black holes as particle accelerators toarbitrarily high energyrdquordquo Physical Review Letters vol 103 no23 Article ID 239001 2009
[9] M Banados B Hassanain J Silk and S M West ldquoEmergentflux from particle collisions near a Kerr black holerdquo PhysicalReview D vol 83 no 2 Article ID 023004 2011
[10] T Jacobson and T P Sotiriou ldquoSpinning black holes as particleacceleratorsrdquo Physical Review Letters vol 104 no 2 Article ID021101 2010
[11] O B Zaslavskii ldquoAcceleration of particles by nonrotatingcharged black holesrdquo JETP Letters vol 92 no 9 pp 571ndash5742011
[12] S WWei Y X Liu H T Li and FW Chen ldquoParticle collisionson stringy black hole backgroundrdquo Journal of High EnergyPhysics vol 2010 no 12 article 066 2010
[13] O B Zaslavskii ldquoEnergy extraction from extremal chargedblack holes due to the Banados-Silk-West effectrdquo PhysicalReview D vol 86 Article ID 124039 2012
[14] H Saadat ldquoThe centre-of-mass energy of two colliding particlesin STU black holesrdquo Canadian Journal of Physics vol 92 no 12pp 1562ndash1564 2014
[15] N Tsukamoto and C Bambi ldquoHigh energy collision of twoparticles in wormhole spacetimesrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 91 no 8 Article ID084013 2015
[16] S G Ghosh P Sheoran and M Amir ldquoRotating Ayon-Beato-Garcıa black hole as a particle acceleratorrdquo Physical Review Dvol 90 no 10 Article ID 103006 2014
[17] A Galajinsky ldquoParticle collisions on near horizon extremalKerr backgroundrdquo Physical Review D vol 88 no 2 Article ID027505 2013
[18] V P Frolov ldquoWeakly magnetized black holes as particle acceler-atorsrdquo Physical Review D vol 85 no 2 Article ID 024020 2012
[19] A M Al Zahrani V P Frolov and A A Shoom ldquoCriticalescape velocity for a charged particle moving around a weaklymagnetized Schwarzschild black holerdquo Physical Review D vol87 no 8 Article ID 084043 2013
[20] J Sadeghi and B Pourhassan ldquoParticle acceleration in Horava-Lifshitz black holesrdquo The European Physical Journal C vol 72no 4 article 1984 2012
[21] J Sadeghi B Pourhassan and H Farahani ldquoRotating chargedhairy black hole in (2+1) dimensions and particle accelerationrdquoCommunications in Theoretical Physics vol 62 no 3 pp 358ndash362 2014
[22] C Liu S Chen CDing and J Jing ldquoParticle acceleration on thebackground of the KerrndashTaubndashNUT spacetimerdquo Physics LettersB vol 701 no 3 pp 285ndash290 2011
[23] M Patil and P S Joshi ldquoUltrahigh energy particle collisions ina regular spacetime without black holes or naked singularitiesrdquoPhysical Review D vol 86 no 4 Article ID 044040 2012
[24] A N Baushev ldquoDark matter annihilation in the gravitationalfield of a black holerdquo International Journal of Modern Physics Dvol 18 no 8 pp 1195ndash1203 2009
[25] M Patil and P S Joshi ldquoParticle acceleration by MajumdarndashPapapetrou di-holerdquoGeneral Relativity and Gravitation vol 46no 10 2014
[26] J D Schnittman ldquoRevised upper limit to energy extractionfrom a kerr black holerdquo Physical Review Letters vol 113 no 26Article ID 261102 2014
[27] M Patil and P S Joshi ldquoNaked singularities as particle accel-eratorsrdquo Physical Review D vol 82 no 10 Article ID 1040492010
[28] M Patil P S Joshi and D Malafarina ldquoNaked singularities asparticle accelerators IIrdquoPhysical ReviewD vol 83 no 6 ArticleID 064007 2011
[29] A Grib andY Pavlov ldquoOn particle collisions in the gravitationalfield of the Kerr black holerdquo Astroparticle Physics vol 34 no 7pp 581ndash586 2011
[30] M Sharif and N Haider ldquoStudy of center of mass energy byparticles collision in some black holesrdquo Astrophysics and SpaceScience vol 346 no 1 pp 111ndash117 2013
[31] I Hussain M Jamil and B Majeed ldquoA slowly rotating blackhole in horava-lifshitz gravity and a 3+1 dimensional topo-logical black hole motion of particles and BSW mechanismrdquoInternational Journal of Theoretical Physics vol 54 no 5 pp1567ndash1577 2015
[32] S Hussain I Hussain and M Jamil ldquoDynamics of a chargedparticle around a slowly rotating Kerr black hole immersed inmagnetic fieldrdquoThe European Physical Journal C vol 74 no 122014
[33] M Amir and S G Ghosh ldquoRotating Haywardrsquos regular blackhole as particle acceleratorrdquo Journal of High Energy Physics vol2015 no 7 article 015 2015
[34] B Pourhassan andUDebnath ldquoParticle acceleration in rotatingmodified hayward and bardeen black holesrdquo httpsarxivorgabs150603443
[35] A A Grib and Y V Pavlov ldquoAre black holes totally blackrdquoGravitation and Cosmology vol 21 no 1 pp 13ndash18 2015
Advances in High Energy Physics 7
[36] A A Grib and Y V Pavlov ldquoHigh energy physics in the vicinityof rotating black holesrdquo Theoretical and Mathematical Physicsvol 185 no 1 pp 1425ndash1432 2015
[37] C Ding C Liu andQQuo ldquoSpacetime noncommutative effecton black hole as particle acceleratorsrdquo International Journal ofModern Physics D vol 22 no 04 Article ID 1350013 2013
[38] J Yang Y-L Li Y Li S-W Wei and Y-X Liu ldquoParticlecollisions in the lower dimensional rotating black hole space-time with the cosmological constantrdquo Advances in High EnergyPhysics vol 2014 Article ID 204016 7 pages 2014
[39] H Nemoto UMiyamoto T Harada and T Kokubu ldquoEscape ofsuperheavy and highly energetic particles produced by particlecollisions near maximally charged black holesrdquo Physical ReviewD vol 87 no 12 Article ID 127502 2013
[40] C Zhong and S Gao ldquoParticle collisions near the cosmologicalhorizon of a Reissner-Nordstrom-de Sitter black holerdquo JETPLetters vol 94 no 8 pp 589ndash592 2011
[41] C Liu S Chen and J Jing ldquoCollision of two general geodesicparticles around a kerrmdashnewman black holerdquo Chinese PhysicsLetters vol 30 no 10 Article ID 100401 2013
[42] Y Zhu S Wu Y Liu and Y Jiang ldquoGeneral stationary chargedblack holes as charged particle acceleratorsrdquo Physical Review Dvol 84 no 4 Article ID 043006 2011
[43] U Miyamoto H Nemoto and M Shimano ldquoParticle creationby naked singularities in higher dimensionsrdquo Physical ReviewD vol 83 no 8 Article ID 084054 2011
[44] Y Li J Yang Y-L Li S-W Wei and Y-X Liu ldquoParticleacceleration in Kerr-(anti-)de Sitter black hole backgroundsrdquoClassical and Quantum Gravity vol 28 no 22 Article ID225006 2011
[45] M Halilsoy O Gurtug and S H Mazharimousavi ldquoRindlermodified Schwarzschild geodesicsrdquo General Relativity andGravitation vol 45 no 11 pp 2363ndash2381 2013
[46] S W Hawking ldquoBlack holes and thermodynamicsrdquo PhysicalReview D vol 13 no 2 pp 191ndash197 1976
[47] CWMisner K SThorne and J AWheelerGravitation WHFreeman amp Co San Francisco Calif USA 1972
[48] L Susskind and J Lindesay An Introduction to Black HolesInformation and the String Theory Revolution The HolographicUniverse World Scientific Hackensack NJ USA 2005
41 Schwarzschild-Like Solution In the case of 1198622 = 12 119861 =minus2119872 120578 = 1 and 119860 = 119862 = 1198621 = 0 it gives Schwarzschild-likesolution where
For the CM energy on the horizon we have to computethe limiting value of (27) as 119909 rarr 119909ℎ = 2119872 where the horizonof the black hole is Setting 120581 = minus1 as it is the CM energy nearthe event horizon for 1 + 1 D Schwarzschild BH is
This result for 4-D Schwarzschild Black hole is alreadycalculated by Baushev [24] Hence the condition of 120581 = minus1when the location of particle 1 approaches the horizon and onthe other hand the particle 2 runs outward from the horizondue to some unspecified physical process yet yields 1198642cm rarrinfin so there is BSW effect for 1+1 Schwarzschild-like solutionwhen the condition 120581 = minus1 is satisfied42 Reissner-Nordstrom-Like Solution On the other handthe choice of the parameters for 1198622 = 12 119861 = minus2119872119862 = 31198762 120578 = 1 and 119860 = 1198621 = 0 gives the ReissnerndashNordstrom-like solution
119873(119909)2 = 1 minus 2119872119909 + 11987621199092 (34)
45 Near Horizon Coordinates We have explored the regionnear the horizon by replacing 119903 by a coordinate 120588 The properdistance from the horizon 120588 [48] is given as follows
Thefirst example is the Schwarzschild-like solutionwhichis
119873(119909)2 = 1 minus 2119872119909 (45)
so that proper distance is calculated as
120588 = int119909119909ℎ
(1 minus 2119872119909 )minus12 1198891199091015840
= radic119909 (119909 minus 2119872) + 2119872119866 sinhminus1 (radic 1199092119872 minus 1) (46)
The new metric is
1198891199042 = minus(1 minus 2119872119909 ()) 1198891199052 + 1198892 (47)
where ≃ 2radic2119872(119909 minus 2119872) so that it gives approximately
1198891199042 ≃ minus 1205882(4119872)2 1198891199052 + 1198891205882 (48)
which is oncemore the Rindler-type line element Let us notethat this Rindler-type line element is valid within the nearhorizon limit approximation For practical purposes thereare advantages in adapting such an approximation whichconformswith the equivalence principle [48]TheCMenergyof two colliding particles is given by
1198642cm211989820 = 1
+ (4119872)2 (11986411198642 minus 120581radic11986421 minus 1205884 (4119872)4radic11986422 minus 1205884 (4119872)4)1205882
(49)
so that there is BSW effect for 120581 = minus1 when 120588 rarr 0
Advances in High Energy Physics 5
5 Particle Collision nearthe Naked Singularity
There is a naked singularity for our 1 + 1-D HL model at thelocation of 119909 = 0 with 1198762 gt 1198722 in (16) In addition 119872 lt 0turns (15) into a naked singular metric at 119909 = 0 There is alsonaked singularity whenwe choosemetric function as follows
For the case 120581 = plusmn1 when 119909 goes to zero the CM energyremains finite for radially moving particles
1198642cm2100381610038161003816100381610038161003816100381610038161003816119909=0 997888rarr 1 minus 120581 (52)
This suggests that although one of the particle is boostedby the naked singularity there is not any unlimited collisionalenergy near such singularity Note that Compton-like pro-cesses were considered first in [4] where rotational effect ofKerr black hole played a significant role Our case here isentirely free of rotational effects
6 Photon versus an Infalling Particle
A massless photon can naturally scatter an infalling particleor vice versa This phenomenon is analogous to a Comptonscattering taking place in 1+1-dimension Null-geodesics fora photon can be described simply by
119889119905119889120582 = 11986411198732119889119909119889120582 = plusmnradic11986421 minus 1198732
(53)
where 120582 is an affine parameter and 1198641 stands for the photonenergy Defining 1198641 = ℎ1205960 where 1205960 is the frequency (withthe choice ℎ = 1)we can parametrize energy of the photon by1205960 aloneTheCMenergy of a photon and the infalling particlecan be taken now as
1198642cm = minus (119901120583 + 119896120583)2 (54)
in which 119901120583 = 119898119906120583 and 119896120583 refer to the particle and photon 2momenta respectively This amounts to
1198642cm = 1198982 minus 2119898119892120583]119906120583119896] (55)
where we have for the particle
119901120583 = 119898( 11986421198732 radic11986422 minus 1198732) (56)
Note that for 120581 = minus1 we have 1198642cm given by
1198642cm = 1198982 (1 minus 11986411198981198642) (60)
which is finite between the collision of a photon and aninfalling particle and therefore is not of interest As a matterof fact the occurrence of outgoing photon from the eventhorizon cannot be justified unless an explosivedecay processis assumed to take place As a result for 120581 = +1 from(59) we obtain an unbounded 1198642cm between the collisionof infalling photon and particle Let us add that ldquoinverserdquoCompton process in the ergosphere of Kerr black holewas considered in [4] where the photonrsquos energy showedincrement due to rotational and curvature effectsThe energyhowever attained an upper bound which was finite Ourresult obtained here being entirely radial on the other handcan hardly be compared with those of [4]
7 Conclusion
Our aim was to investigate whether the BSW type effectwhich arises in higher dimensional black holes applies alsoto the 1 + 1-D naked singularityblack hole The theorywe adapted is not general relativity but instead the recentlypopular HL gravity We employed the class of 5-parameterblack holenaked singularity solutions found recently [5]Theclass has particular limits of flat Rindler Schwarzschild andReissner-Nordstrom-like solutions For each case we havecalculated the center-of-mass (CM) energy of the particlesand shown that the energy can grow unbounded for somecases In other words the strong gravity near the eventhorizon affects the collision process with unlimited sourceto turn it into a natural accelerator The model we useapplies also to the case of a photonparticle collision withdifferent characteristics It is observed that the CM energyof the infalling particles from the rest at infinity will remainfinite in the CM frame at the event horizon of a blackhole Contrariwise unlimited CM energy will be attainedbetween the collision of the outgoing particles from the eventhorizon region and infalling particles It is also possible toachieve the infinite energy between an infalling photon andan infalling massive particle However we found finite CMenergy between an outgoing photon and infalling particleFinally we must admit that absence of rotational effects in1 + 1-D restricts the problem to the level of a toy model inwhich particles move on pure radial geodesics yielding finiteCM energy in the vicinity of a naked singularity
6 Advances in High Energy Physics
Disclosure
This work was presented as a poster at Karl SchwarzschildMeeting 20ndash24 July 2015 Frankfurt Institute for AdvancedStudies
Competing Interests
The authors declare that they have no competing interests
References
[1] M Banados J Silk and S MWest ldquoKerr black holes as particleaccelerators to arbitrarily high energyrdquo Physical Review Lettersvol 103 no 11 Article ID 111102 2009
[2] T Jacobson and J P Sotiriou ldquoSpinning black holes as particleacceleratorsrdquo Physical Review Letters vol 104 no 2 Article ID021101 3 pages 2010
[3] K Lake ldquoParticle accelerators inside spinning black holesrdquoPhysical Review Letters vol 104 Article ID 211102 2010
[4] T Piran and J Shaham ldquoUpper bounds on collisional penroseprocesses near rotating black-hole horizonsrdquo Physical ReviewDvol 16 no 6 pp 1615ndash1635 1977
[5] D Bazeia F A Brito and F G Costa ldquoTwo-dimensionalHorava-Lifshitz black hole solutionsrdquo Physical Review D Par-ticles Fields Gravitation and Cosmology vol 91 no 4 ArticleID 044026 2015
[6] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 2009
[7] R Arnowitt S Deser and C W Misner ldquoRepublication ofthe dynamics of general relativityrdquo General Relativity andGravitation vol 40 no 9 pp 1997ndash2027 2008
[8] E Berti V Cardoso L Gualtieri F Pretorius and U SperhakeldquoComment on ldquokerr black holes as particle accelerators toarbitrarily high energyrdquordquo Physical Review Letters vol 103 no23 Article ID 239001 2009
[9] M Banados B Hassanain J Silk and S M West ldquoEmergentflux from particle collisions near a Kerr black holerdquo PhysicalReview D vol 83 no 2 Article ID 023004 2011
[10] T Jacobson and T P Sotiriou ldquoSpinning black holes as particleacceleratorsrdquo Physical Review Letters vol 104 no 2 Article ID021101 2010
[11] O B Zaslavskii ldquoAcceleration of particles by nonrotatingcharged black holesrdquo JETP Letters vol 92 no 9 pp 571ndash5742011
[12] S WWei Y X Liu H T Li and FW Chen ldquoParticle collisionson stringy black hole backgroundrdquo Journal of High EnergyPhysics vol 2010 no 12 article 066 2010
[13] O B Zaslavskii ldquoEnergy extraction from extremal chargedblack holes due to the Banados-Silk-West effectrdquo PhysicalReview D vol 86 Article ID 124039 2012
[14] H Saadat ldquoThe centre-of-mass energy of two colliding particlesin STU black holesrdquo Canadian Journal of Physics vol 92 no 12pp 1562ndash1564 2014
[15] N Tsukamoto and C Bambi ldquoHigh energy collision of twoparticles in wormhole spacetimesrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 91 no 8 Article ID084013 2015
[16] S G Ghosh P Sheoran and M Amir ldquoRotating Ayon-Beato-Garcıa black hole as a particle acceleratorrdquo Physical Review Dvol 90 no 10 Article ID 103006 2014
[17] A Galajinsky ldquoParticle collisions on near horizon extremalKerr backgroundrdquo Physical Review D vol 88 no 2 Article ID027505 2013
[18] V P Frolov ldquoWeakly magnetized black holes as particle acceler-atorsrdquo Physical Review D vol 85 no 2 Article ID 024020 2012
[19] A M Al Zahrani V P Frolov and A A Shoom ldquoCriticalescape velocity for a charged particle moving around a weaklymagnetized Schwarzschild black holerdquo Physical Review D vol87 no 8 Article ID 084043 2013
[20] J Sadeghi and B Pourhassan ldquoParticle acceleration in Horava-Lifshitz black holesrdquo The European Physical Journal C vol 72no 4 article 1984 2012
[21] J Sadeghi B Pourhassan and H Farahani ldquoRotating chargedhairy black hole in (2+1) dimensions and particle accelerationrdquoCommunications in Theoretical Physics vol 62 no 3 pp 358ndash362 2014
[22] C Liu S Chen CDing and J Jing ldquoParticle acceleration on thebackground of the KerrndashTaubndashNUT spacetimerdquo Physics LettersB vol 701 no 3 pp 285ndash290 2011
[23] M Patil and P S Joshi ldquoUltrahigh energy particle collisions ina regular spacetime without black holes or naked singularitiesrdquoPhysical Review D vol 86 no 4 Article ID 044040 2012
[24] A N Baushev ldquoDark matter annihilation in the gravitationalfield of a black holerdquo International Journal of Modern Physics Dvol 18 no 8 pp 1195ndash1203 2009
[25] M Patil and P S Joshi ldquoParticle acceleration by MajumdarndashPapapetrou di-holerdquoGeneral Relativity and Gravitation vol 46no 10 2014
[26] J D Schnittman ldquoRevised upper limit to energy extractionfrom a kerr black holerdquo Physical Review Letters vol 113 no 26Article ID 261102 2014
[27] M Patil and P S Joshi ldquoNaked singularities as particle accel-eratorsrdquo Physical Review D vol 82 no 10 Article ID 1040492010
[28] M Patil P S Joshi and D Malafarina ldquoNaked singularities asparticle accelerators IIrdquoPhysical ReviewD vol 83 no 6 ArticleID 064007 2011
[29] A Grib andY Pavlov ldquoOn particle collisions in the gravitationalfield of the Kerr black holerdquo Astroparticle Physics vol 34 no 7pp 581ndash586 2011
[30] M Sharif and N Haider ldquoStudy of center of mass energy byparticles collision in some black holesrdquo Astrophysics and SpaceScience vol 346 no 1 pp 111ndash117 2013
[31] I Hussain M Jamil and B Majeed ldquoA slowly rotating blackhole in horava-lifshitz gravity and a 3+1 dimensional topo-logical black hole motion of particles and BSW mechanismrdquoInternational Journal of Theoretical Physics vol 54 no 5 pp1567ndash1577 2015
[32] S Hussain I Hussain and M Jamil ldquoDynamics of a chargedparticle around a slowly rotating Kerr black hole immersed inmagnetic fieldrdquoThe European Physical Journal C vol 74 no 122014
[33] M Amir and S G Ghosh ldquoRotating Haywardrsquos regular blackhole as particle acceleratorrdquo Journal of High Energy Physics vol2015 no 7 article 015 2015
[34] B Pourhassan andUDebnath ldquoParticle acceleration in rotatingmodified hayward and bardeen black holesrdquo httpsarxivorgabs150603443
[35] A A Grib and Y V Pavlov ldquoAre black holes totally blackrdquoGravitation and Cosmology vol 21 no 1 pp 13ndash18 2015
Advances in High Energy Physics 7
[36] A A Grib and Y V Pavlov ldquoHigh energy physics in the vicinityof rotating black holesrdquo Theoretical and Mathematical Physicsvol 185 no 1 pp 1425ndash1432 2015
[37] C Ding C Liu andQQuo ldquoSpacetime noncommutative effecton black hole as particle acceleratorsrdquo International Journal ofModern Physics D vol 22 no 04 Article ID 1350013 2013
[38] J Yang Y-L Li Y Li S-W Wei and Y-X Liu ldquoParticlecollisions in the lower dimensional rotating black hole space-time with the cosmological constantrdquo Advances in High EnergyPhysics vol 2014 Article ID 204016 7 pages 2014
[39] H Nemoto UMiyamoto T Harada and T Kokubu ldquoEscape ofsuperheavy and highly energetic particles produced by particlecollisions near maximally charged black holesrdquo Physical ReviewD vol 87 no 12 Article ID 127502 2013
[40] C Zhong and S Gao ldquoParticle collisions near the cosmologicalhorizon of a Reissner-Nordstrom-de Sitter black holerdquo JETPLetters vol 94 no 8 pp 589ndash592 2011
[41] C Liu S Chen and J Jing ldquoCollision of two general geodesicparticles around a kerrmdashnewman black holerdquo Chinese PhysicsLetters vol 30 no 10 Article ID 100401 2013
[42] Y Zhu S Wu Y Liu and Y Jiang ldquoGeneral stationary chargedblack holes as charged particle acceleratorsrdquo Physical Review Dvol 84 no 4 Article ID 043006 2011
[43] U Miyamoto H Nemoto and M Shimano ldquoParticle creationby naked singularities in higher dimensionsrdquo Physical ReviewD vol 83 no 8 Article ID 084054 2011
[44] Y Li J Yang Y-L Li S-W Wei and Y-X Liu ldquoParticleacceleration in Kerr-(anti-)de Sitter black hole backgroundsrdquoClassical and Quantum Gravity vol 28 no 22 Article ID225006 2011
[45] M Halilsoy O Gurtug and S H Mazharimousavi ldquoRindlermodified Schwarzschild geodesicsrdquo General Relativity andGravitation vol 45 no 11 pp 2363ndash2381 2013
[46] S W Hawking ldquoBlack holes and thermodynamicsrdquo PhysicalReview D vol 13 no 2 pp 191ndash197 1976
[47] CWMisner K SThorne and J AWheelerGravitation WHFreeman amp Co San Francisco Calif USA 1972
[48] L Susskind and J Lindesay An Introduction to Black HolesInformation and the String Theory Revolution The HolographicUniverse World Scientific Hackensack NJ USA 2005
There is a naked singularity for our 1 + 1-D HL model at thelocation of 119909 = 0 with 1198762 gt 1198722 in (16) In addition 119872 lt 0turns (15) into a naked singular metric at 119909 = 0 There is alsonaked singularity whenwe choosemetric function as follows
For the case 120581 = plusmn1 when 119909 goes to zero the CM energyremains finite for radially moving particles
1198642cm2100381610038161003816100381610038161003816100381610038161003816119909=0 997888rarr 1 minus 120581 (52)
This suggests that although one of the particle is boostedby the naked singularity there is not any unlimited collisionalenergy near such singularity Note that Compton-like pro-cesses were considered first in [4] where rotational effect ofKerr black hole played a significant role Our case here isentirely free of rotational effects
6 Photon versus an Infalling Particle
A massless photon can naturally scatter an infalling particleor vice versa This phenomenon is analogous to a Comptonscattering taking place in 1+1-dimension Null-geodesics fora photon can be described simply by
119889119905119889120582 = 11986411198732119889119909119889120582 = plusmnradic11986421 minus 1198732
(53)
where 120582 is an affine parameter and 1198641 stands for the photonenergy Defining 1198641 = ℎ1205960 where 1205960 is the frequency (withthe choice ℎ = 1)we can parametrize energy of the photon by1205960 aloneTheCMenergy of a photon and the infalling particlecan be taken now as
1198642cm = minus (119901120583 + 119896120583)2 (54)
in which 119901120583 = 119898119906120583 and 119896120583 refer to the particle and photon 2momenta respectively This amounts to
1198642cm = 1198982 minus 2119898119892120583]119906120583119896] (55)
where we have for the particle
119901120583 = 119898( 11986421198732 radic11986422 minus 1198732) (56)
Note that for 120581 = minus1 we have 1198642cm given by
1198642cm = 1198982 (1 minus 11986411198981198642) (60)
which is finite between the collision of a photon and aninfalling particle and therefore is not of interest As a matterof fact the occurrence of outgoing photon from the eventhorizon cannot be justified unless an explosivedecay processis assumed to take place As a result for 120581 = +1 from(59) we obtain an unbounded 1198642cm between the collisionof infalling photon and particle Let us add that ldquoinverserdquoCompton process in the ergosphere of Kerr black holewas considered in [4] where the photonrsquos energy showedincrement due to rotational and curvature effectsThe energyhowever attained an upper bound which was finite Ourresult obtained here being entirely radial on the other handcan hardly be compared with those of [4]
7 Conclusion
Our aim was to investigate whether the BSW type effectwhich arises in higher dimensional black holes applies alsoto the 1 + 1-D naked singularityblack hole The theorywe adapted is not general relativity but instead the recentlypopular HL gravity We employed the class of 5-parameterblack holenaked singularity solutions found recently [5]Theclass has particular limits of flat Rindler Schwarzschild andReissner-Nordstrom-like solutions For each case we havecalculated the center-of-mass (CM) energy of the particlesand shown that the energy can grow unbounded for somecases In other words the strong gravity near the eventhorizon affects the collision process with unlimited sourceto turn it into a natural accelerator The model we useapplies also to the case of a photonparticle collision withdifferent characteristics It is observed that the CM energyof the infalling particles from the rest at infinity will remainfinite in the CM frame at the event horizon of a blackhole Contrariwise unlimited CM energy will be attainedbetween the collision of the outgoing particles from the eventhorizon region and infalling particles It is also possible toachieve the infinite energy between an infalling photon andan infalling massive particle However we found finite CMenergy between an outgoing photon and infalling particleFinally we must admit that absence of rotational effects in1 + 1-D restricts the problem to the level of a toy model inwhich particles move on pure radial geodesics yielding finiteCM energy in the vicinity of a naked singularity
6 Advances in High Energy Physics
Disclosure
This work was presented as a poster at Karl SchwarzschildMeeting 20ndash24 July 2015 Frankfurt Institute for AdvancedStudies
Competing Interests
The authors declare that they have no competing interests
References
[1] M Banados J Silk and S MWest ldquoKerr black holes as particleaccelerators to arbitrarily high energyrdquo Physical Review Lettersvol 103 no 11 Article ID 111102 2009
[2] T Jacobson and J P Sotiriou ldquoSpinning black holes as particleacceleratorsrdquo Physical Review Letters vol 104 no 2 Article ID021101 3 pages 2010
[3] K Lake ldquoParticle accelerators inside spinning black holesrdquoPhysical Review Letters vol 104 Article ID 211102 2010
[4] T Piran and J Shaham ldquoUpper bounds on collisional penroseprocesses near rotating black-hole horizonsrdquo Physical ReviewDvol 16 no 6 pp 1615ndash1635 1977
[5] D Bazeia F A Brito and F G Costa ldquoTwo-dimensionalHorava-Lifshitz black hole solutionsrdquo Physical Review D Par-ticles Fields Gravitation and Cosmology vol 91 no 4 ArticleID 044026 2015
[6] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 2009
[7] R Arnowitt S Deser and C W Misner ldquoRepublication ofthe dynamics of general relativityrdquo General Relativity andGravitation vol 40 no 9 pp 1997ndash2027 2008
[8] E Berti V Cardoso L Gualtieri F Pretorius and U SperhakeldquoComment on ldquokerr black holes as particle accelerators toarbitrarily high energyrdquordquo Physical Review Letters vol 103 no23 Article ID 239001 2009
[9] M Banados B Hassanain J Silk and S M West ldquoEmergentflux from particle collisions near a Kerr black holerdquo PhysicalReview D vol 83 no 2 Article ID 023004 2011
[10] T Jacobson and T P Sotiriou ldquoSpinning black holes as particleacceleratorsrdquo Physical Review Letters vol 104 no 2 Article ID021101 2010
[11] O B Zaslavskii ldquoAcceleration of particles by nonrotatingcharged black holesrdquo JETP Letters vol 92 no 9 pp 571ndash5742011
[12] S WWei Y X Liu H T Li and FW Chen ldquoParticle collisionson stringy black hole backgroundrdquo Journal of High EnergyPhysics vol 2010 no 12 article 066 2010
[13] O B Zaslavskii ldquoEnergy extraction from extremal chargedblack holes due to the Banados-Silk-West effectrdquo PhysicalReview D vol 86 Article ID 124039 2012
[14] H Saadat ldquoThe centre-of-mass energy of two colliding particlesin STU black holesrdquo Canadian Journal of Physics vol 92 no 12pp 1562ndash1564 2014
[15] N Tsukamoto and C Bambi ldquoHigh energy collision of twoparticles in wormhole spacetimesrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 91 no 8 Article ID084013 2015
[16] S G Ghosh P Sheoran and M Amir ldquoRotating Ayon-Beato-Garcıa black hole as a particle acceleratorrdquo Physical Review Dvol 90 no 10 Article ID 103006 2014
[17] A Galajinsky ldquoParticle collisions on near horizon extremalKerr backgroundrdquo Physical Review D vol 88 no 2 Article ID027505 2013
[18] V P Frolov ldquoWeakly magnetized black holes as particle acceler-atorsrdquo Physical Review D vol 85 no 2 Article ID 024020 2012
[19] A M Al Zahrani V P Frolov and A A Shoom ldquoCriticalescape velocity for a charged particle moving around a weaklymagnetized Schwarzschild black holerdquo Physical Review D vol87 no 8 Article ID 084043 2013
[20] J Sadeghi and B Pourhassan ldquoParticle acceleration in Horava-Lifshitz black holesrdquo The European Physical Journal C vol 72no 4 article 1984 2012
[21] J Sadeghi B Pourhassan and H Farahani ldquoRotating chargedhairy black hole in (2+1) dimensions and particle accelerationrdquoCommunications in Theoretical Physics vol 62 no 3 pp 358ndash362 2014
[22] C Liu S Chen CDing and J Jing ldquoParticle acceleration on thebackground of the KerrndashTaubndashNUT spacetimerdquo Physics LettersB vol 701 no 3 pp 285ndash290 2011
[23] M Patil and P S Joshi ldquoUltrahigh energy particle collisions ina regular spacetime without black holes or naked singularitiesrdquoPhysical Review D vol 86 no 4 Article ID 044040 2012
[24] A N Baushev ldquoDark matter annihilation in the gravitationalfield of a black holerdquo International Journal of Modern Physics Dvol 18 no 8 pp 1195ndash1203 2009
[25] M Patil and P S Joshi ldquoParticle acceleration by MajumdarndashPapapetrou di-holerdquoGeneral Relativity and Gravitation vol 46no 10 2014
[26] J D Schnittman ldquoRevised upper limit to energy extractionfrom a kerr black holerdquo Physical Review Letters vol 113 no 26Article ID 261102 2014
[27] M Patil and P S Joshi ldquoNaked singularities as particle accel-eratorsrdquo Physical Review D vol 82 no 10 Article ID 1040492010
[28] M Patil P S Joshi and D Malafarina ldquoNaked singularities asparticle accelerators IIrdquoPhysical ReviewD vol 83 no 6 ArticleID 064007 2011
[29] A Grib andY Pavlov ldquoOn particle collisions in the gravitationalfield of the Kerr black holerdquo Astroparticle Physics vol 34 no 7pp 581ndash586 2011
[30] M Sharif and N Haider ldquoStudy of center of mass energy byparticles collision in some black holesrdquo Astrophysics and SpaceScience vol 346 no 1 pp 111ndash117 2013
[31] I Hussain M Jamil and B Majeed ldquoA slowly rotating blackhole in horava-lifshitz gravity and a 3+1 dimensional topo-logical black hole motion of particles and BSW mechanismrdquoInternational Journal of Theoretical Physics vol 54 no 5 pp1567ndash1577 2015
[32] S Hussain I Hussain and M Jamil ldquoDynamics of a chargedparticle around a slowly rotating Kerr black hole immersed inmagnetic fieldrdquoThe European Physical Journal C vol 74 no 122014
[33] M Amir and S G Ghosh ldquoRotating Haywardrsquos regular blackhole as particle acceleratorrdquo Journal of High Energy Physics vol2015 no 7 article 015 2015
[34] B Pourhassan andUDebnath ldquoParticle acceleration in rotatingmodified hayward and bardeen black holesrdquo httpsarxivorgabs150603443
[35] A A Grib and Y V Pavlov ldquoAre black holes totally blackrdquoGravitation and Cosmology vol 21 no 1 pp 13ndash18 2015
Advances in High Energy Physics 7
[36] A A Grib and Y V Pavlov ldquoHigh energy physics in the vicinityof rotating black holesrdquo Theoretical and Mathematical Physicsvol 185 no 1 pp 1425ndash1432 2015
[37] C Ding C Liu andQQuo ldquoSpacetime noncommutative effecton black hole as particle acceleratorsrdquo International Journal ofModern Physics D vol 22 no 04 Article ID 1350013 2013
[38] J Yang Y-L Li Y Li S-W Wei and Y-X Liu ldquoParticlecollisions in the lower dimensional rotating black hole space-time with the cosmological constantrdquo Advances in High EnergyPhysics vol 2014 Article ID 204016 7 pages 2014
[39] H Nemoto UMiyamoto T Harada and T Kokubu ldquoEscape ofsuperheavy and highly energetic particles produced by particlecollisions near maximally charged black holesrdquo Physical ReviewD vol 87 no 12 Article ID 127502 2013
[40] C Zhong and S Gao ldquoParticle collisions near the cosmologicalhorizon of a Reissner-Nordstrom-de Sitter black holerdquo JETPLetters vol 94 no 8 pp 589ndash592 2011
[41] C Liu S Chen and J Jing ldquoCollision of two general geodesicparticles around a kerrmdashnewman black holerdquo Chinese PhysicsLetters vol 30 no 10 Article ID 100401 2013
[42] Y Zhu S Wu Y Liu and Y Jiang ldquoGeneral stationary chargedblack holes as charged particle acceleratorsrdquo Physical Review Dvol 84 no 4 Article ID 043006 2011
[43] U Miyamoto H Nemoto and M Shimano ldquoParticle creationby naked singularities in higher dimensionsrdquo Physical ReviewD vol 83 no 8 Article ID 084054 2011
[44] Y Li J Yang Y-L Li S-W Wei and Y-X Liu ldquoParticleacceleration in Kerr-(anti-)de Sitter black hole backgroundsrdquoClassical and Quantum Gravity vol 28 no 22 Article ID225006 2011
[45] M Halilsoy O Gurtug and S H Mazharimousavi ldquoRindlermodified Schwarzschild geodesicsrdquo General Relativity andGravitation vol 45 no 11 pp 2363ndash2381 2013
[46] S W Hawking ldquoBlack holes and thermodynamicsrdquo PhysicalReview D vol 13 no 2 pp 191ndash197 1976
[47] CWMisner K SThorne and J AWheelerGravitation WHFreeman amp Co San Francisco Calif USA 1972
[48] L Susskind and J Lindesay An Introduction to Black HolesInformation and the String Theory Revolution The HolographicUniverse World Scientific Hackensack NJ USA 2005
This work was presented as a poster at Karl SchwarzschildMeeting 20ndash24 July 2015 Frankfurt Institute for AdvancedStudies
Competing Interests
The authors declare that they have no competing interests
References
[1] M Banados J Silk and S MWest ldquoKerr black holes as particleaccelerators to arbitrarily high energyrdquo Physical Review Lettersvol 103 no 11 Article ID 111102 2009
[2] T Jacobson and J P Sotiriou ldquoSpinning black holes as particleacceleratorsrdquo Physical Review Letters vol 104 no 2 Article ID021101 3 pages 2010
[3] K Lake ldquoParticle accelerators inside spinning black holesrdquoPhysical Review Letters vol 104 Article ID 211102 2010
[4] T Piran and J Shaham ldquoUpper bounds on collisional penroseprocesses near rotating black-hole horizonsrdquo Physical ReviewDvol 16 no 6 pp 1615ndash1635 1977
[5] D Bazeia F A Brito and F G Costa ldquoTwo-dimensionalHorava-Lifshitz black hole solutionsrdquo Physical Review D Par-ticles Fields Gravitation and Cosmology vol 91 no 4 ArticleID 044026 2015
[6] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 2009
[7] R Arnowitt S Deser and C W Misner ldquoRepublication ofthe dynamics of general relativityrdquo General Relativity andGravitation vol 40 no 9 pp 1997ndash2027 2008
[8] E Berti V Cardoso L Gualtieri F Pretorius and U SperhakeldquoComment on ldquokerr black holes as particle accelerators toarbitrarily high energyrdquordquo Physical Review Letters vol 103 no23 Article ID 239001 2009
[9] M Banados B Hassanain J Silk and S M West ldquoEmergentflux from particle collisions near a Kerr black holerdquo PhysicalReview D vol 83 no 2 Article ID 023004 2011
[10] T Jacobson and T P Sotiriou ldquoSpinning black holes as particleacceleratorsrdquo Physical Review Letters vol 104 no 2 Article ID021101 2010
[11] O B Zaslavskii ldquoAcceleration of particles by nonrotatingcharged black holesrdquo JETP Letters vol 92 no 9 pp 571ndash5742011
[12] S WWei Y X Liu H T Li and FW Chen ldquoParticle collisionson stringy black hole backgroundrdquo Journal of High EnergyPhysics vol 2010 no 12 article 066 2010
[13] O B Zaslavskii ldquoEnergy extraction from extremal chargedblack holes due to the Banados-Silk-West effectrdquo PhysicalReview D vol 86 Article ID 124039 2012
[14] H Saadat ldquoThe centre-of-mass energy of two colliding particlesin STU black holesrdquo Canadian Journal of Physics vol 92 no 12pp 1562ndash1564 2014
[15] N Tsukamoto and C Bambi ldquoHigh energy collision of twoparticles in wormhole spacetimesrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 91 no 8 Article ID084013 2015
[16] S G Ghosh P Sheoran and M Amir ldquoRotating Ayon-Beato-Garcıa black hole as a particle acceleratorrdquo Physical Review Dvol 90 no 10 Article ID 103006 2014
[17] A Galajinsky ldquoParticle collisions on near horizon extremalKerr backgroundrdquo Physical Review D vol 88 no 2 Article ID027505 2013
[18] V P Frolov ldquoWeakly magnetized black holes as particle acceler-atorsrdquo Physical Review D vol 85 no 2 Article ID 024020 2012
[19] A M Al Zahrani V P Frolov and A A Shoom ldquoCriticalescape velocity for a charged particle moving around a weaklymagnetized Schwarzschild black holerdquo Physical Review D vol87 no 8 Article ID 084043 2013
[20] J Sadeghi and B Pourhassan ldquoParticle acceleration in Horava-Lifshitz black holesrdquo The European Physical Journal C vol 72no 4 article 1984 2012
[21] J Sadeghi B Pourhassan and H Farahani ldquoRotating chargedhairy black hole in (2+1) dimensions and particle accelerationrdquoCommunications in Theoretical Physics vol 62 no 3 pp 358ndash362 2014
[22] C Liu S Chen CDing and J Jing ldquoParticle acceleration on thebackground of the KerrndashTaubndashNUT spacetimerdquo Physics LettersB vol 701 no 3 pp 285ndash290 2011
[23] M Patil and P S Joshi ldquoUltrahigh energy particle collisions ina regular spacetime without black holes or naked singularitiesrdquoPhysical Review D vol 86 no 4 Article ID 044040 2012
[24] A N Baushev ldquoDark matter annihilation in the gravitationalfield of a black holerdquo International Journal of Modern Physics Dvol 18 no 8 pp 1195ndash1203 2009
[25] M Patil and P S Joshi ldquoParticle acceleration by MajumdarndashPapapetrou di-holerdquoGeneral Relativity and Gravitation vol 46no 10 2014
[26] J D Schnittman ldquoRevised upper limit to energy extractionfrom a kerr black holerdquo Physical Review Letters vol 113 no 26Article ID 261102 2014
[27] M Patil and P S Joshi ldquoNaked singularities as particle accel-eratorsrdquo Physical Review D vol 82 no 10 Article ID 1040492010
[28] M Patil P S Joshi and D Malafarina ldquoNaked singularities asparticle accelerators IIrdquoPhysical ReviewD vol 83 no 6 ArticleID 064007 2011
[29] A Grib andY Pavlov ldquoOn particle collisions in the gravitationalfield of the Kerr black holerdquo Astroparticle Physics vol 34 no 7pp 581ndash586 2011
[30] M Sharif and N Haider ldquoStudy of center of mass energy byparticles collision in some black holesrdquo Astrophysics and SpaceScience vol 346 no 1 pp 111ndash117 2013
[31] I Hussain M Jamil and B Majeed ldquoA slowly rotating blackhole in horava-lifshitz gravity and a 3+1 dimensional topo-logical black hole motion of particles and BSW mechanismrdquoInternational Journal of Theoretical Physics vol 54 no 5 pp1567ndash1577 2015
[32] S Hussain I Hussain and M Jamil ldquoDynamics of a chargedparticle around a slowly rotating Kerr black hole immersed inmagnetic fieldrdquoThe European Physical Journal C vol 74 no 122014
[33] M Amir and S G Ghosh ldquoRotating Haywardrsquos regular blackhole as particle acceleratorrdquo Journal of High Energy Physics vol2015 no 7 article 015 2015
[34] B Pourhassan andUDebnath ldquoParticle acceleration in rotatingmodified hayward and bardeen black holesrdquo httpsarxivorgabs150603443
[35] A A Grib and Y V Pavlov ldquoAre black holes totally blackrdquoGravitation and Cosmology vol 21 no 1 pp 13ndash18 2015
Advances in High Energy Physics 7
[36] A A Grib and Y V Pavlov ldquoHigh energy physics in the vicinityof rotating black holesrdquo Theoretical and Mathematical Physicsvol 185 no 1 pp 1425ndash1432 2015
[37] C Ding C Liu andQQuo ldquoSpacetime noncommutative effecton black hole as particle acceleratorsrdquo International Journal ofModern Physics D vol 22 no 04 Article ID 1350013 2013
[38] J Yang Y-L Li Y Li S-W Wei and Y-X Liu ldquoParticlecollisions in the lower dimensional rotating black hole space-time with the cosmological constantrdquo Advances in High EnergyPhysics vol 2014 Article ID 204016 7 pages 2014
[39] H Nemoto UMiyamoto T Harada and T Kokubu ldquoEscape ofsuperheavy and highly energetic particles produced by particlecollisions near maximally charged black holesrdquo Physical ReviewD vol 87 no 12 Article ID 127502 2013
[40] C Zhong and S Gao ldquoParticle collisions near the cosmologicalhorizon of a Reissner-Nordstrom-de Sitter black holerdquo JETPLetters vol 94 no 8 pp 589ndash592 2011
[41] C Liu S Chen and J Jing ldquoCollision of two general geodesicparticles around a kerrmdashnewman black holerdquo Chinese PhysicsLetters vol 30 no 10 Article ID 100401 2013
[42] Y Zhu S Wu Y Liu and Y Jiang ldquoGeneral stationary chargedblack holes as charged particle acceleratorsrdquo Physical Review Dvol 84 no 4 Article ID 043006 2011
[43] U Miyamoto H Nemoto and M Shimano ldquoParticle creationby naked singularities in higher dimensionsrdquo Physical ReviewD vol 83 no 8 Article ID 084054 2011
[44] Y Li J Yang Y-L Li S-W Wei and Y-X Liu ldquoParticleacceleration in Kerr-(anti-)de Sitter black hole backgroundsrdquoClassical and Quantum Gravity vol 28 no 22 Article ID225006 2011
[45] M Halilsoy O Gurtug and S H Mazharimousavi ldquoRindlermodified Schwarzschild geodesicsrdquo General Relativity andGravitation vol 45 no 11 pp 2363ndash2381 2013
[46] S W Hawking ldquoBlack holes and thermodynamicsrdquo PhysicalReview D vol 13 no 2 pp 191ndash197 1976
[47] CWMisner K SThorne and J AWheelerGravitation WHFreeman amp Co San Francisco Calif USA 1972
[48] L Susskind and J Lindesay An Introduction to Black HolesInformation and the String Theory Revolution The HolographicUniverse World Scientific Hackensack NJ USA 2005
[36] A A Grib and Y V Pavlov ldquoHigh energy physics in the vicinityof rotating black holesrdquo Theoretical and Mathematical Physicsvol 185 no 1 pp 1425ndash1432 2015
[37] C Ding C Liu andQQuo ldquoSpacetime noncommutative effecton black hole as particle acceleratorsrdquo International Journal ofModern Physics D vol 22 no 04 Article ID 1350013 2013
[38] J Yang Y-L Li Y Li S-W Wei and Y-X Liu ldquoParticlecollisions in the lower dimensional rotating black hole space-time with the cosmological constantrdquo Advances in High EnergyPhysics vol 2014 Article ID 204016 7 pages 2014
[39] H Nemoto UMiyamoto T Harada and T Kokubu ldquoEscape ofsuperheavy and highly energetic particles produced by particlecollisions near maximally charged black holesrdquo Physical ReviewD vol 87 no 12 Article ID 127502 2013
[40] C Zhong and S Gao ldquoParticle collisions near the cosmologicalhorizon of a Reissner-Nordstrom-de Sitter black holerdquo JETPLetters vol 94 no 8 pp 589ndash592 2011
[41] C Liu S Chen and J Jing ldquoCollision of two general geodesicparticles around a kerrmdashnewman black holerdquo Chinese PhysicsLetters vol 30 no 10 Article ID 100401 2013
[42] Y Zhu S Wu Y Liu and Y Jiang ldquoGeneral stationary chargedblack holes as charged particle acceleratorsrdquo Physical Review Dvol 84 no 4 Article ID 043006 2011
[43] U Miyamoto H Nemoto and M Shimano ldquoParticle creationby naked singularities in higher dimensionsrdquo Physical ReviewD vol 83 no 8 Article ID 084054 2011
[44] Y Li J Yang Y-L Li S-W Wei and Y-X Liu ldquoParticleacceleration in Kerr-(anti-)de Sitter black hole backgroundsrdquoClassical and Quantum Gravity vol 28 no 22 Article ID225006 2011
[45] M Halilsoy O Gurtug and S H Mazharimousavi ldquoRindlermodified Schwarzschild geodesicsrdquo General Relativity andGravitation vol 45 no 11 pp 2363ndash2381 2013
[46] S W Hawking ldquoBlack holes and thermodynamicsrdquo PhysicalReview D vol 13 no 2 pp 191ndash197 1976
[47] CWMisner K SThorne and J AWheelerGravitation WHFreeman amp Co San Francisco Calif USA 1972
[48] L Susskind and J Lindesay An Introduction to Black HolesInformation and the String Theory Revolution The HolographicUniverse World Scientific Hackensack NJ USA 2005
