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Ultra-high-energy collisions of particles in the field of near-extreme Kehagias-Sfetsos naked singularities and their appearance to distant observers Zdeněk Stuchlı́k, Jan Schee and Ahmadjon Abdujabbarov May 12, 2014 Abstract We demonstrate that ultra-high-energy collisions of particles falling freely from rest at infinity can occur in the field of near-extreme KehagiasSfetsos naked singularities related to the Hořava gravity. However, the efficiency of escaping of created ultra-relativistic particles and the energy efficiency of the collisional process relative to distant observers are significantly lowered due to large gravitational redshift, being substantially lower in comparison to those related to the collisions occurring close to the equatorial plane of near-extreme Kerr naked-singularity spacetimes. In the Kehagias-Sfetsos naked singularity spacetimes, the energy efficiency relative to distant observers corresponds to the covariant energy of the colliding particles only. Finally we demonstrate how the ultra-high energy collisions are modified for charged particles, if the Kehagias-Sfetsos naked singularities are immersed in an uniform magnetic field. 1 Introduction Collisions of particles freely falling from rest at infinity can give extremely high Centre-of-Mass (CM) energy, if they occur in a very close vicinity of the black hole horizon of extreme Kerr black holes with dimensionless spin a = 1, as shown in [10]. The so called Banados-Silk-White (BSW) processes with extremely large CM energy has been shown to be potentially realizable very close to the event horizon of near-extreme Kerr black holes, in the field of black holes considered in some alternative theories of gravity (braneworld black holes, Hořava-gravity black holes, etc.). Nevertheless, such processes need in all the considered cases a very fine tuning of the motion constants of the colliding particles and imply some doubts on reality of such processes [18, 52, 103, 104]. Moreover, it has been shown that due to the gravitational redshift effect, the energy of the ultra-relativistic particles created in the BSW processes has to be comparable to the rest energy of the colliding particles, if observed by distant observers. The energy efficiency of the BSW process is thus substantially reduced by the gravitational redshift and is close to the rest energy of the colliding particles [18, 51]. An exceptional situation is possible if the electromagnetic interaction can be relevant in the collisional process [77, 4, 97] 1 In the field of near-extreme superspinning Kerr geometries, i.e., Kerr naked singularities or primordial Kerr superspinars [35, 93, 84], in the final stages of their conversion to near-extreme black holes due to accretion processes [92], the extremely high CM energy can be obtained with no fine tuning of the motion constants. It has been demonstrated for particles freely falling from infinity in the equatorial plane [70, 71] or along ”radial” trajectories with arbitrary latitude [92, 19, 87, 82], if they collide at (or near) r = M , and for collisions of particles moving in the equatorial plane along the stable circular orbits located at r = M with any particle freely falling from infinity [85]. It can also be obtained in generic collisions of particles freely falling from infinity with covariant energy E = m assuming that the collisions occur just at the radius r = M [86]. The efficiency of escaping of the created highly-energetic particles and the energy-conversion efficiency relative to distant observers due to the frequency shift of the high-energy photons (ultra-relativistic particles) produced in the collisions are the crucial phenomena related to the observational relevance of the ultra-high-energy collisions. In the field of black holes, both these efficiencies are restricted by the gravitational redshift effect as demonstrated in [18]. On the other hand, both these efficiencies can remain large in the field of nearextreme superspinning Kerr geometry, if the collisions occur at r = M and close enough to the equatorial plane of the geometry [86]. For such an effect, both the non-existence of the black hole horizon and the strong rotation of the superspinning near-extreme Kerr geometry are probably relevant, as the efficiencies decrease substantially, if the particles collide near the symmetry axis of the Kerr geometry [86]. To clear up the situation, a study of the collisions of the same kind occurring in a spherically symmetric naked singularity spacetime has to be realized. Here we demonstrate the relevance of this statement by considering the phenomenon of the ultra-high-energy collisions in the field of near-extreme Kehagias-Sfetsos naked singularities that represent an interesting spherically symmetric solution of the modified Hořava gravity [55]. The present study of the acceleration process in the KS naked singularity spacetime is complementary to the recent works related to the particle acceleration mechanism in the field of 5-dimensional Kerr black holes [3] and black strings [97]. Moreover, the combined influence of the brane tension and the cosmological constant on the acceleration process has been considered in [77], while the acceleration of charged particle near the black holes with non vanishing gravitomagnetic charge has been studied in [4]. The Hořava (or Hořava-Lifshitz) gravity [42, 41, 43, 8, 36, 37, 99, 61] is recently considered as one of the promising approaches to the quantum gravity, being inspired by solid-body physics, namely the Lifshitz theory. The Hořava gravity breaks the Lorentz invariance at the high-energy limit, while at the low-energy limit it reduces to the General Relativity and satisfies the Lorentz invariance. The solutions of the Hořava effective gravitational equations have been found in [12, 13, 14]. The spherically symmetric solution having asymptotically the Schwarzschild character has been found in the framework of the modified Hořava model - the solution is described by the so called KehagiasSfetsos (KS) metric [55, 69, 57], which allows for existence of both black hole and naked singularity spacetimes. Slowly rotating black hole solutions of the modified Hořava gravity has been found in [60, 6]. In connection to the accretion phenomena, the KS metric describing black holes has been extensively studied in a series of works related both to the particle 2 motion [39, 6, 2, 50, 32, 45, 40] and optical phenomena [5, 31, 9] that can be relevant for tests of validity of the Hořava gravity. The modified Hořava model has been also tested for the properties of the magnetic field near spherical stars [40]. The KS naked-singularity spacetimes have been discussed for embeddings of the direct and optical geometry in [38] and for the circular geodetical motion in [101]. The properties of the circular geodesics of the KS naked singularity spacetimes have been shown to be similar to those related to the well known spherically symmetric naked singularity spacetimes, namely the Reissner-Nordstrom [79, 73] and braneworld naked singularity spacetimes [58, 81, 75, 74, 80, 7], allowing for occurrence of principally new astrophysical phenomena. Here we concentrate on the particles freely falling from rest at infinity and colliding in the deep gravitational field of the KS naked singularities, searching for conditions allowing for occurrence of the ultra-high CM energy collisions. In analogy with our studies of these phenomena in the superspinning Kerr geometries, we expect the ultra-high-energy collisions to occur near the surface r = M , in the field of near-extreme KS spacetimes. Then we test the efficiency of the acceleration process for charged particles following circular orbits, assuming the KS naked singularities immersed in an asymptotically uniform magnetic field. We would like to stress that we consider high-energy collisions that still allow us to apply the Hořava gravity in its General Relativistic limit, i.e., we consider motion of the particles and photons along geodesics of the spacetime. 2 Kehagias-Sfetsos spacetimes 2.1 Geometry The spherically symmetric solution of the so called modified Hořava gravity, allowing for the Schwarzschild spacetime as an appropriate limit, is the KehagiasSfetsos (KS) spacetime [55], described in the standard Schwarzschild coordinates and the geometric units by the line element where ds2 = −f (r)dt2 + f −1 (r)dr2 + r2 (dθ2 + sin2 θdϕ2 ) (1) 4M 1/2 . f (r) = 1 + r ω 1 − 1 + 3 ωr (2) 4 1/2 . f (x) = 1 + x (ωM ) 1 − 1 + ωM 2 x3 (3) 2 In terms of x = r/M , 2 2 The pseudosingularities of the geometry, the horizons, are located at r 1 r± = M ± M 2 − . 2ω (4) Two horizons of the KS black hole spacetimes exist, if ω ≥ ωh = 3 1 . 2M 2 (5) The limit of the Schwarzschild black hole is obtained when ω → ∞. The horizon coincide when ω = ωh , giving an extreme KS black hole spacetime. The KS naked singularity spacetimes occur, if ω < ωh . (6) Recall that KS spacetimes are not Ricci flat [38] – in fact, in the Schwarzschild limit, as the parameter ω → ∞, the Ricci scalar R∼ 1 . ω (7) In the following, we shall use for simplicity the dimensionless radial coordinate and the dimensionless metric parameter introduced by r →r M and ωM 2 → ω, (8) as the dimensionality is governed by the mass parameter M of the KS solution of the modified Hořava gravity. The parameter ω (”Hořava parameter”) then governs modification of the gravitational law in the modified Hořava gravitational theory. The basic properties of the KS naked singularity spacetimes were described for the embedding diagrams in [38] and for circular geodesics in [101]. Here, we have to consider also general trajectories. 2.2 Geodesic equations The motion of test particles and photons is assumed to be governed by geodesics of the spacetime. The geodesic equation for the 4-momentum pµ of test particles (photons) Dpµ =0 (9) dλ can be in the spherically symmetric KS spacetimes separated and integrated easily. Due to the axial symmetry and stationarity of the KS spacetimes two constants of motion arise: pφ = L, pt = −E (10) that are identified with the axial angular momentum L and covariant energy E related to the distant static observers. In spherically symmetric spacetimes, the motion occurs in the central planes; for a single particle motion, the plane can be chosen to be the equatorial plane. Considering motion in a general central plane, an additional motion constant, L, corresponding to the total angular momentum of the test particle (photon), can be introduced. The geodesic equations in the integrated and separated form take then the form (see, e.g., [90]) pt = g tt pt = E , f (r; ω) L , sin2 θ θ 2 1 L2 , p = 4 L2 − r sin2 θ pφ = g φφ pφ = 4 r2 (11) (12) (13) L2 [p ] = E − f (r; ω) m + 2 r r 2 2 2 , (14) where m is the rest energy (mass) of the particle that is also a motion constant; for photons there is m = 0. In the analysis of massive-particle or masslessphoton motion, it is useful to use the notion of the effective potential that can be expressed in the form L2 (15) Vef f = f (r; ω) m2 + 2 . r For the motion in the equatorial plane (θ = π/2), there is L = L and pθ = 0. 3 Photon motion The radial component of the photon four-momentum reads 2 L 2 [pr ] = E 2 − f (r; ω) . r2 (16) We consider the equatorial motion with L = L. The trajectories of photons are independent of energy, therefore, it is convenient to relate the effective potential of photons to the impact parameter l= L ; E (17) then the turning points of the radial photon motion are given in terms of the effective potential by the relation l2 = Vef f /ph ≡ r2 . f (r) (18) We illustrate the radial profile of the effective potential Vef f /ph (r; ω) for representative choices of the parameter ω in Figure 1. The photon circular orbits, i.e., their radii rph and their impact parameters lph can be found from the condition dVef f /ph 3M 2 = 0. r− q = (19) dr f (r)2 1 + 4M r3 ω In the dimensionless form, we find out that the photon circular orbits are located at radii satisfying the cubic equation r3 − 9r + 4 = 0. ω (20) The character of the solutions of the equation (20) is determined by the value of the parameter ω. The critical value of this parameter related to the photon circular orbits reads 4 2 ωph = . (21) 27 We can distinguish four cases for the occurrence of the photon circular orbits. 5 1. ω ≥ ωh - in the black hole spacetimes, there is only one photon orbit at radius √ 2π 2 1 . (22) − cos−1 − √ rph1 = 2 3 cos 3 3 3 3ω 2. ωh > ω > ωph - there are two photon circular orbits, the outer one at the radius given by the formula (22) is unstable relative to radial perturbations, and the inner one, stable relative to radial perturbations, located at the radius radius √ 1 2 rph2 = 2 3 cos . (23) cos−1 − √ 3 3 3ω 3. ω = ωph - the two photon circular orbits coincide √ and there remains only one photon circular orbit at the radius rph = 3. 4. ω < ωph - there is no photon circular orbit in this case. The corresponding value of impact parameter l of the circular photon orbits follows from the equation (18), i.e., there is 2 lph = r2 1 + r2 ω 1 − q 1+ 4 r3 ω , (24) where r = rph1 or r = rph2 . The values of the impact parameter of the photon circular orbits are given as a function of the Hořava parameter ω in Figure 2. For the extreme KS black holes with ω = ωh , the photon circular orbit is located at the radius rph(h) = 2.37M , and the related impact parameter takes the value . lph(h) = 4.69. For ω = ωph , the photon circular orbit is located at rph(c) = and the value of corresponding impact parameter reads . lph(c) = 4.40. 4 (25) √ . 3M = 1.73M (26) Keplerian circular orbits The equatorial circular orbits of test particles (Keplerian orbits) are determined by the condition " # r dVef f 4 2 2 4 2 2 5 r L −r ω 1 + 3 + (r − 3L + r ω) = 0 =− q dr r ω r4 1 + 4 r3 ω (27) that implies the radial profile of the specific angular momentum (related to the unit rest mass of the particle) of the Keplerian orbits in the form r2 3 + ωr3 A(r; ω)(1 − A(r; ω)) L2K (r; ω) = (28) m2 rA(r; ω) − 3 6 . The radial profile of the specific energy (related to the unit rest mass of the particle) of the Keplerian orbits then takes the form 2 rA(r; ω)f 2 (r; ω) EK (r; ω) = m2 rA(r; ω) − 3 (29) where we have introduced the function A(r; ω) = r 1+ 4 r3 ω . (30) Using the components of the test particle 4-momentum given by Equations (10) and (11), the radial profile of the angular frequency on the Keplerian circular orbits, related to distant static observers, is then given by the formula s r3 ω[A(r; ω) − 1] − 1 f (r; ω) LK pφ = . (31) ΩK = t = p r 2 EK r3 A(r; ω) 2.5 40 2.0 0.4 30 1.5 E2K L2K 20 0.36 1.0 0.4 0.36 10 0.5 0 0.0 0 1 2 3 4 5 0 r 1 2 3 4 5 r 2 Figure 1: Plot of the radial profiles of the specific energy EK (specific angular 2 momentum LK ) of the Keplerian orbits on the left (right) panel for two representative values of parameter ω = 0.36 and 0.4 when both the outer and inner discs are allowed. The inner disc is constituted only by stable Keplerian orbits. In the outer disc, the internal part corresponds to the unstable Keplerian orbits that are represented by the dotted part of the curve. Properties of the radial profiles of the specific energy, specific angular momentum and angular frequency of the Keplerian orbits strongly depend on the values of the parameter ω and were studied in detail in [101]. We shortly summarize that the divergence of both the specific energy and specific angular momentum is given by the condition rA(r; ω) − 3 = 0 (32) that is identical with the condition for the circular photon orbits. The zero points of the radial profiles of the specific angular momentum (L2K (r; ω) = 0) give the so called static radii corresponding to equilibrium points where the particle is at rest relative to distant static observers. The static radius is given by rstat (ω) = (2ω)−1/3 . (33) 7 The static radii give the lower limit for existence of Keplerian orbits of test particles in the field of KS naked singularity spacetimes (ω < 1/2), in analogy with the Reissner-Nordstrom naked singularity spacetimes [79, 73]. The angular velocity of the Keplerian orbits in the field of KS naked singularities vanishes at the static radius. The local maximum of the radial profile of the angular velocity of the Keplerian orbits is then located at the radius 1/3 2 . rΩ = ω (34) There are no zero points of the specific energy Keplerian radial profile, since L2 2 EK = f (r, ω)(m2 + rK 2 ). At the static radius, the specific energy of the Keplerian circular orbits approaches its minimum EK (r = rstat , ω) ≡ Estat = 1 − (2ω)1/3 . (35) There is Estat (ω → 1/2) → 0, and Estat (ω → 0) → 1. In the near-extreme KS naked singularity spacetimes, there are two regions of circular orbits: the inner region extends between the static radius and the stable inner circular photon orbit, contains only stable orbits, and the energy of these Keplerian orbits diverges as their radius approaches the stable inner circular photon orbit; the outer region of the circular orbits is the standard one. We illustrate the situation in Figure 3. 5 Ultra-high CM energy of colliding particles We calculate the CM energy of colliding particles in three characteristic cases. Two of them correspond to the situations when the ultra-high-energy collisions were treated in the field of near-extreme superspinning Kerr geometry [92, 84, 86] - namely the collisions of particles in radial free fall from infinity (E = m and L = 0) with particles following geodesic circular orbits, or with radially escaping particles having E = m and L = 0 that inverted their motion at r = 0. As the third case we consider collisions of particles moving in the equatorial plane with opposite angular momentum at the turning point of their radial motion. For simplicity, we consider collisions of identical particles, having the same rest energy. Let the 4-momenta of the colliding particles are pµ1 and pν2 with total 4momentum pµtot = pµ1 + pµ2 . The corresponding CM energy then reads β 2 (36) ECM = −ptot · ptot = 2m2 (1 − u1 · u2 ) = 2m2 1 − gαβ uα 1 u2 . 5.1 Collisions of particles on circular geodesics with radially falling particles The circular orbits are assumed to be located at the equatorial plane. We assume for simplicity the particle falling from infinity also in the equatorial plane. The 4-velocity of the particle on the circular geodesic is then given by EK /m LK /m α u1 = , (37) , 0, 0, f r2 8 where the constants of the motion on the circular orbits are the specific energy EK (r; ω)/m and specific angular momentum LK (r; ω)/m. For a particle radially falling from infinity with covariant energy E/m = 1 and zero angular momentum, the 4-velocity reads p 1 , − uα = 1 − f (r, ω), 0, 0 . (38) 1 f (r, ω) The CM energy of the colliding particles reads 2 EK 1 ECM 1+ . = 4m2 2 f (39) The CM energy is normed to the rest energy of the colliding particles that equals 2m. The energy of the test particle moving along a circular orbit at a radius r is given by the relation 29. The behaviour of the CM energy occurring at this kind of collisions is presented in Fig. 2 . One clearly sees that the maximum of the collisional CM energy decreases with decreasing value of parameter ω. The location of the maximum shifts to the larger values of the radius r with decreasing value of the parameter ω. 1.8 2 ECM H4m2 L 1.6 0.281 1.4 0.2 1.2 0.15 0.1 1.0 2 3 4 5 6 r Figure 2: The plots of the CM energy of the collisions between radially falling particles and orbiting particles on Keplerian orbits are given for five representative values of the parameter ω = 0.1, 0.15, 0.2, 0.281, and 0.3. The dashed line represents ECM at the static radius rstat = (2ω)11/3 . The dotted line indicates the ECM for collisions that appear in the region of unstable Keplerian orbits. For the stable circular orbits corresponding to matter in accretion discs extended to large distance from the KS naked singularity, the centre-of-mass energy remains finite for all values of the Hořava parameter ω corresponding to KS naked singularities. However, in the field of KS naked singularities with 1/2 > ω > 0.3849, an additional, inner accretion disc can occur that can extend from the inner edge at the static radius rs = (2/ω)1/3 where particles with L = 0 are located, to the outer edge corresponding to the inner photon circular orbit at rph2 that is stable relative to the radial perturbations. As the circular orbit at the inner disc is approaching the outer edge, both the specific energy and the specific angular momentum of the particle diverge (Figure 1). Then also the CM energy of the colliding particles diverges; moreover, it is also influenced 9 by the metric factor 1/f (r; ω) that is very large in the near-extreme KS naked singularity spacetimes in vicinity of the specific radius r = M . Unfortunately, it is unclear, how to obtain by natural physical processes the ultra-relativistic particles following the stable circular orbits. Some ultra-relativistic particles following the stable circular orbits could be created during preceding collisional processes, however, their covariant energy corresponds to the rest energy of the freely infalling particles. 5.2 Collisions of particles with opposite angular momentum Let two particles fall from infinity with covariant energy E1,2 = m with opposite angular momenta L1 = −L2 = L. Assuming an opposite angular momentum of colliding particles, both particles have to move in the same central plane, and it is enough to consider only the equatorial motion to describe the problem in full generality. The components of the 4-velocities for both particles read s ! L2 /m2 L/m 1 µ u1 = , 0, − 2 , (40) , 1 − f (r) 1 + f r2 r s ! L2 /m2 L/m 1 µ , 0, + 2 . (41) , 1 − f (r) 1 + u2 = f r2 r The square of the CM energy then reads 2 ECM L2 /m2 = 1 + . 4m2 r2 (42) When we let such particles to collide at the turning point rt = rt (ω, |L|) of their radial motion, which is given by the equation 1 − f (r)(1 − L2 /r2 ) = 0, the specific angular momentum of the colliding particles at a given rt is then determined by 1 L2t (r; ω) = rt2 ( − 1), (43) f (rt ; ω) giving the square of the CM energy in the form 2 1 ECM = . 4m2 f (rt , ω) (44) 2 The behaviour of the CM energy ECM /4m2 as the function of the radius r and the parameter ω is illustrated in Figs 3 . 10 14 5 0.28 3 10 0.2 2 0.4 12 0.25 2 ECM H4m2 L 2 ECM H4m2 L 4 0.1 0.37 8 0.35 6 0.3 4 1 2 0 0 0 1 2 3 4 5 6 0 r 1 2 3 4 5 6 L Figure 3: The plots of the radial profile of the CM energy of collisions of particles with angular momenta L1 = −L2 = L and covariant energy E1 /m = E2 /m = 1. The plots are constructed for eight representative values of the parameter ω = 0.1, 0.2, 0.25, 0.28 (left-handside figure), 0.3, 0.35, 0.37, and 0.4 (right-handside figure). The curves constructed for ω = 0.35, 0.37 and 0.4 have discontinuity at the point Ld that satisfies the conditions V (rd , Ld , ω) = 1, dV (rd , Ld , ω)/dr = 0 and d2 V (rd , Ld , ω)/dr2 = 0. The discontinuity in the curves plotted for parameter ω = 0.35, 0.37, and 0.4 is present due to the existence of the circular Keplerian orbits with covariant energy E = m. This kind of circular orbits exists for ω ≥ ωC1 = 0.327764. The 2 maxima of ECM radial profiles for this kind of collisions increase with increasing value of the parameter ω. The corresponding angular momentum shifts to larger values with increasing value of ω. Now, we turn our attention to the case of the collisions at the radius r = M , and in the near-extreme KS naked singularity spacetimes with the Hořava parameter 1 ω = − δ, δ << 1. (45) 2 Then the metric coefficient f (r = M, ω = 1/2 − δ) ∼ 2δ/3, and the CM energy takes the value of 2 3 ECM ∼ (46) 2 4m 2δ that diverges as ω → 1/2 - this a similar situation as those occurring in the Kerr naked singularity spacetimes [84, 86]. In this special case of the ultrahigh-energy collisions, the angular momentum of the colliding particles has to take very large value 3 L2HE ∼ . (47) 2δ 5.3 Purely radial collisions Since the KS naked singularity spacetimes demonstrate the ”antigravity” or ”repulsive gravity” effect that converts inward directed purely radial motion of uncharged test particles into an outward directed motion [101], similarly to an analogous effect occuring in the Kerr naked singularity spacetimes [87], we can consider a simple possibility to find the ultra-high-energy collisions by purely 11 radial collision of two particles with energy E = m. 1 The 4-velocities of the colliding particles read p 1 (48) uµ1 = , 1 − f (r, ω), 0, 0 f (r, ω) p 1 uµ2 = (49) , − 1 − f (r, ω), 0, 0 f (r, ω) The corresponding CM energy of the collision is given by formula 2 ECM 1 = . 2 4m f (r, ω) (50) The radial coordinate where we obtain the maximal CM energy of the collision in a given naked singularity spacetime is obtained when f (r, ω) has a minimum, i.e. 6 df (r, ω) =0= 2 + 2r(1 − A(r, ω))ω. (51) dr r A(r, ω) 2 2 2 The behavior of the ECM /4m2 radial function, and of its maxima ECM max /4m , is illustrated in Figure 4. As magnitude of the parameter ω increases towards ω = 0.5, the radius of the maximum of CM energy approaches to r = 1. For ω → 2 ECM 3 0.5 the maximal value of CM energy diverges as 4m 2 ∼ 2δ . This is identical result as those obtained above for the collisions of particles with oppositely oriented angular momentum at the turning point of their radial motion that occur in the direction perpendicular to the radial direction. 1 In the Kerr naked singularity spacetimes the ”antigravity” effect is related to the distribution of the mass-energy of the spacetime that is attractive everywhere, but its specific configuration in the spacetime results in the apparently repulsive gravitational phenomena, as demonstrated by [28]. On the other hand, there is also the well known gravitational repulsion effect related to the vacuum energy, with the corresponding relevant effects on the test particle motion discussed, e.g., in [90, 83]. It is not quite clear what is the cause of the ”antigravity” effects in the KS spacetimes. However, we could expect some role of repulsive quasi-vacuum effects in the Hořava gravity, because of the asymptotic behavior of the f (r; ω) function near the physical singularity resembling behavior of the quintessential fields causing accelerating expansion of the Universe. 12 2 ECM H4m2 L 25 0.45 20 15 0.4 10 0.3 5 0.1 0 1 2 3 4 r Figure 4: The plots of the radial profiles of the CM energy (solid lines) and of its maximal value (dashed line) occurring due to the head-on collision of the radially moving particles. Each solid line is marked with the value of the corresponding parameter ω. Therefore, in both cases considered above, the ultra-high-energy collisions can occur at the radius r = 1 in the field of near-extreme KS naked singularities with ω → 1/2, demonstrating the same dependence on the parameter δ. In both of these kinds of particle collisions, the centre-of-mass reference system coincides with the static reference system at the position of the collision, as can be easily demonstrated following the arguments presented in [86]. Therefore, these cases can be used in a very simple way to estimate the efficiency of the ultra-highenergy collisions in relation to the static distant observers. We can construct the escape cones of the isotropically generated radiation, and calculate the frequency shift of the radiated high-energy photons at infinity, in an analogous way to the calculations in the Locally Non-Rotating Frames in the Kerr naked singularity spacetimes [86] that correspond to the static frames in the spherically symmetric spacetimes. 6 Escaping cones and frequency shift of photons generated at the static reference systems We can properly estimate the efficiency of the ultra-high-energy collisions relative to the distant observers by treating the escaping cones and frequency shift of photons generated in the CM reference system as properties of these photons well represent properties of highly ultra-relativistic particles. We focus our attention to the phenomena related to the static reference systems as those systems correspond to the CM systems for the purely radial collissions of particles having identical rest energy, and to the collisions occuring at the turning point of particles of identical rest energy and opposite angular momentum. 13 r=1.3M × er × er × er × er × er r=1.05M r=5M r=1.8M r=1.45M r=5M r=1.4M r=1.16M × er × er × er Figure 5: The top (bottom ) plots of the central sections of the escape cones constructed for the KS naked singularity spacetimes having value of the parameter ω = 0.4 (0.45) are given for the four representative values of the emitter location at the radii r = 1.3M (1.05M ), r = 1.45M (1.16M ), r = 1.8M (1.4M ), and r = 5M . The two circular photon orbits of the KS spacetimes are located at the rph1 ≈ 1.45M (1.16M ) (stable inner orbit) and rph2 = 2M (2.25M ) (unstable outer orbit). The shaded region corresponds to the trapped photons. The arrow gives the outward radial direction. r=1.M × er r=1.M r=1.M × er × er r=1.M × er Figure 6: Plots of the central sections of the escape cones are constructed for the four representative values of the parameter ω = 0.433, 0.45, 0.49, 0.499 and for the emitter located at the radius r = 1M where maximal CM energy of collisional processes can be obtained. The shaded region corresponds to the trapped photons. 14 6.1 Escape cones and the KS naked singularity silhouette The opening angle of the escaping cones related to the static reference frames is in the spherically symmetric spacetimes determined by the simple formula [90, 78, 91] # "p f (r; ω) l (52) αesc = arcsin r The dependence of the opening angle on the location of the static frames is given for some typical values of the Hořava parameter ω corresponding to the naked singularity spacetimes allowing for existence of the stable and unstable photon circular geodesics in Figure 5. We can see that the captured photons exist only if radiated by sources located under the outer unstable photon circular orbit. All photons radiated above the radius of the outer photon circular orbit escape to infinity. In Figure 6, we construct the escaping cones at the special radius r = M for a properly chosen sequence of the parameter ω, and we demonstrate how the opening angle shrinks to zero as ω → 1/2. It should be stressed that in the KS naked singularity spacetimes having no photon circular orbits, all photons can escape to infinity from any position in the spacetime. 1.6 1.4 VEff 1.2 1.0 0.8 0.6 1 2 3 4 5 r@MD Figure 7: A representative plot of the effective potential of the photon motion in the field of a KS naked singularity spacetimes allowing for existence of photon circular orbits. The effective potential illustrates formation of the KS naked singularity silhouette - the dot represents the unstable photon circular orbit which captures photons having the impact parameter with value lph2 corresponding to the orbit. The distant-observer sky represented by the parameters [α, β] related to the impact parameters of the outcoming photons then reflects a dark circle of the radius governed by the value of the impact parameter lph2 of photons trapped on the photon orbit that determines the KS naked singularity silhouette. No silhouette appears for the KS naked singularity spacetimes not allowing for existence of photon circular orbits. The notion of the escape cones enables us to determine in a simple way [74, 93] the shape and extension of the silhouette of the KS naked singularities in dependence on their parameter ω. The shape of the silhouette is spherical 15 and its extension is given by the impact parameter related to the outer photon circular orbit - see Figure 7. The silhouette of KS naked singularity spacetimes with parameter ω > 0.3849 are shown for some typical values of the parameter ω in Figure 8. Of course, for KS naked singularities with ω < 0.3849, no silhouette exists. 4.7 4 4.6 2 Β 0 Β 4.5 -2 4.4 -4 -4 -2 0 2 4.3 0.0 4 0.2 0.4 Α 0.6 0.8 1.0 Α Figure 8: Silhouettes of the KS naked singularity spacetimes are generated for five representative values of the parameter ω = 0.39 (black), 0.4 (dashed), 0.45 (dotted), 0.49 (gray), and 0.499 (blue). Image on the right represents an enlarged region of the silhouette. 6.2 Frequency shift The frequency shift of the radiated photons is given by the general formula (gµν k µ U ν )o νobs = . νemit (gµν k µ U ν )e (53) For the static observers at infinity, and the static observers (emitters) near the KS naked singularity, the 4-velocities read Uoµ Ueµ = = (1, 0, 0, 0) , 1 p f (r, ω) (54) ! , 0, 0, 0 . (55) For simplicity, we shall consider the frequency shift of the purely radially emitted photons as these are the most relevant for considering the escaping phenomena of the collisional processes at the radius r = M in the field of the near-extreme KS naked singularities. At the position of observation and emission, the photon 4-momentum of the radially directed photons in the near-extreme KS naked singularity spacetimes reads koµ = keµ = (E, E, 0, 0) , E , E, 0, 0 , f (r, ω) (56) (57) where E is the covariant energy of the photon related to its frequency at infinity. The frequency shift of the radially directed escaping photons in the KS geometry 16 then reads p νobs (58) = f (r; ω). νemit For photons radiated at the radius r = M in the field of near-extreme KS naked singularities, we arrive to the formula r 2 νobs ∼ . (59) νemit 3δ We see that the redshift diverges as ω → 1/2. 6.3 Efficiency of the collisional process In order to understand the efficiency of the collisional process, we have to relate the energy of the photon at the point of its emission, to its energy at infinity, as measured by static observers there. Notice that the energy generated per particle at the collision reads 1 ECM =p , 2m f (r; ω) (60) while the redshift factor of the radially emitted photons is just inverse of this expression. Therefore, the energy of the radially emitted photon measured by static observers at infinity is given by Einf = g ECM = m. 2 (61) Clearly, the gravitational redshift simply cancels the energy excess locally generated at the ultra-high energy collisions. Therefore, we can see that in the field of KS naked singularities, the energy observed at infinity due to the ultra-high energy collisions in the strong gravity is essentially governed by the conversion of the rest energy of the colliding particles. A similar argument can be clearly applied in the case of any spherically symmetric naked singularity spacetime allowing for the existence of the high-energy BSW processes, e.g., in the case of the Reissner-Nordstrom naked singularity spacetime. Notice that the same reduction factor works in the case of the collisions of the radially falling particles with those moving along the ultra-relativistic stable circular orbits belonging to the inner discs in the field of the near-extreme KS naked singularities. The efficiency of such collisions relative to the distant √ observers is governed by the squared covariant energy of the circular orbit EK , i.e., it is again given by the covariant energy of the colliding particles. Of course, for particles falling from infinity, E = m is again the covariant energy that is decisive. No extra factor in addition to the covariant energy can appear due to the considered collisions in the spherically symmetric naked singularity spacetimes and these can work in close similarity to the particle accelerators where the covariant energy is conserved during the collisional process. The extra energy-efficiency factors have to be related to the rotating spacetimes and occur because of an efficient Penrose process if the particle collisions are realized near the equatorial plane of the Kerr spacetime, as shown in [86]. Another possibility is related to some other, non-gravitational, phenomena [3]. We study here briefly the role of an external asymptotically uniform magnetic field combined with the spherically symmetric gravitational field of the KS naked singularity spacetimes. 17 7 Motion of charged test particles in the field of KS naked singularities immersed into asymptotically uniform magnetic field Finally we would like to study the collisions of charged particles moving in the combined gravitational and electromagnetic field assuming the gravitational field given by the KS naked singularity spacetimes, and the electromagnetic field to be an axially symmetric test magnetic field having asymptotically uniform form and strength B > 0 [102]. The magnetic field is thus influenced by the spacetime, but its influence on the geometry is considered to be negligible. Motion of charged test particles is then governed by the general relativistic Lorentz equation. We shall consider collisions of charged particles following circular orbits in the equatorial plane of the combined gravitational and magnetic field, and electrically neutral particles incoming from rest state at infinity. Alternatively we consider collisions of oppositely charged particles moving along the circular orbits in opposite directions. Such collisions can in an appropriate way demonstrate the role of the magnetic field in the acceleration of the charged particles and the ultra-high energy collisions as discussed in an analogous situation of a 5D black ring in [97]. 7.1 Uniform magnetic field in the KS spacetime First, we have to give the form of the electromagnetic field related to the asymptotically uniform magnetic field. The situation is complicated by the fact that the KS spacetimes are not Ricci flat [55]. The Ricci tensor of the KS spacetimes has the following nonzero components: Rtt = Rrr = Rθθ = Rφφ = 3(2 + 6r3 ω + r6 ω 2 − r6 ω 2 A3 ) r6 ωA3 3 3(2 + r ω − r3 ωA) r3 A (62) (63) where A(r, ω) is given by Eq. 30. Let us consider a KS naked singularity or black hole in an external asymptotically uniform magnetic field. In order to find the related vector potential of the electromagnetic field we apply the Wald method that uses the Killing vectors of the spacetime [102]. The Killing vector field ξ µ , being an infinitesimal generator of an isometry, fulfills the equation ξα;β + ξβ;α = 0. (64) The equation (64) implies the equation ξα;β;γ − ξα;γ;β = −ξ λ Rλαβγ (65) containing the Riemann curvature tensor Rλαβγ . Therefore, the Killing vectors reflecting the symmetries of the KS spacetimes have to satisfy the relation α;β ξ;β = Rµα ξ µ . (66) The electromagnetic potential satisfying the Lorentz calibration condition has to fulfill the relation [102] α µ Aα;β (67) ;β = −Rµ A . 18 Asymptotically (for r → ∞) these two equations reduce to the homogeneous form [39, 1] α;β ξ;β = Aα;β (68) ;β = 0. In order to find the solution of the Maxwell equations for the vector potential Aα , one may use the following anzatz: α α Aα = C1 ξ(t) + C2 ξ(φ) − aα (69) where aα is the correction due to the non-zero Ricci tensor of the KS spacetime, while the first two terms in this equation are related to the homogeneous solution that is directly related to the Killing vectors of the KS spacetimes. The constants related to the Killing vectors can be found easily – the constant C2 = B/2, if we have the gravitational source immersed in the uniform magnetic field B that is parallel to the axis of rotation corresponding to the axial symmetry of the spacetime. The remaining constant C1 has to vanish as can be easily shown due to the asymptotic properties of the KS spacetimes at infinity [2]. The third term is the partial solution related to the Ricci tensor of the spacetime. The partial solution is governed by the equations aα;β;β aα;β;β = γ C1 Rαγ ξ(t) = γ C2 Rαγ ξ(φ) (70) . (71) Solving the equations (70)–(71) by using the expressions (62)–(63), one can get the full expression for the electromagnetic potential. In our case it can be found only by numerical calculations in fully general situation, but we can find analytic approximative solutions. Considering the first order approximation in ω −1 of the KS-geometry Ricci tensor contribution, corresponding to the KS black holes close to the Schwarzschild limit ω → ∞, the vector potential of the electromagnetic field takes the form [2] 3M 2 B ). (72) Aα = (0, 0, 0, 1 + 2 10ωr4 There is only one non-zero covariant component of the vector potential Aφ = B 2 2 3M 2 ). r sin θ(1 + 2 10ωr4 (73) In the case of KS naked singularities, small values of the Hořava parameter, ω < 1/2, enter the play, and the approximate limit of ω ≪ 1 becomes relevant. The solution of the equation (71) then gets the following form r ω ω B 2 α 0, 0, 0, 1 + 3 + + O(ω ) . (74) A = 2 r3 r2 For the situations we consider in this paper, the contribution of the vector aα to the total electromagnetic field around the KS naked singularity, as well as to the particle motion and collisional processes, is small and can be considered as negligible. Really, the Fig. 9 compares the results of the collisional processes in terms of the CM energy obtained by numerical calculations in the general case 19 8 Ecm 6 4 2 0 1 2 3 4 5 4 5 rM 35 30 25 Ecm 20 15 10 5 0 1 2 3 rM Figure 9: . The radial profile of the CM energy ECM for the head-on collisions of oppositely charged and oppositely moving particles on the circular orbits is given for characterictic values of the parameter ω = 0.2 (left) and ω = 0.3 (left). The blue line correspond to the electromagnetic field potential including the term aα and black lines corresponds to the pure Wald solution related to the axial Killing vector. The magnetic parameter is taken to be B = 10 fully considering the influence of the Ricci tensor on the electromagnetic field, to the results of analytic calculations based on the approximation corresponding to the vector potential related to the axial Killing vector only. One can see that the influence of the part of the vector potential related to the Ricci tensor is not exceeding ∼ 5%. Therefore, it is enough to consider here the basic approximation where the vector potential is represented in the simple form proportional to the axial Killing vector. Such an approximation is precise asymptotically at large distances from the KS naked singularities, and its precision is high enough in arbitrary positions. In this approximation we thus investigate the role of the vector potential B (75) Aφ = r2 sin2 θ. 2 20 7.2 Motion of charged test particles and its effective potential The Hamiltonian of charged test particles moving in the combined gravitational and electromagnetic fields given by the metric gµν and potential of the electromagnetic field Aµ reads H= 1 µν g (πµ − eAµ ) (πν − eAν ) 2 (76) where πµ are the components of generalized momentum. From the first set of Hamilton equations dxµ ∂H = (77) dλ ∂πµ one immediately obtains the relation pµ ≡ dxµ = π µ − eAµ . dλ (78) The metric and electromagnetic field of interest does not depend on coordinate time t and on the azimuthal angle φ. From the second set of the Hamilton equations ∂H dπµ (79) = − µ, dλ ∂x we see two integrals of motion πφ = L and πt = −E. (80) Introducing effective quantities, i.e., quantities expressed in unit mass E= E , m L= L eB , ,B = m 2m2 (81) the equations for the temporal, azimuthal and radial components of the 4momentum of the charged test particles then read pφ = pt = r 2 (p ) = L − B, r2 E , f " 2 E −f 1+r (82) (83) 2 L −B r2 2 # . (84) The last equation determines the effective potential governing the radial motion, and the regions allowed for the particle motion " 2 # L −B . (85) Vef f = f (r) 1 + r2 r2 21 7.3 Circular orbits of charged particles For the KS naked singularities (black holes) immersed in an asymptotically uniform magnetic field, the circular orbits are allowed in the equatorial plane and are determined by the condition dVef f =0 dr (86) This equation implies a quadratic relation for the angular momentum of charged particles moving on a circular orbit at a given radius r. Therefore, we obtain two families of the circular orbits, with the angular momentum determined by the roots of the equation 2 ′ − L2 + 4Brω (−1 + A + 2A′ /2) L + r 2 B 2 + ω + 2B 2 r2 ω + ωA 3 r −2 ω + 2B 2 r2 ω A − r 1 + B 2 r2 ωA′ = 0 (87) The roots are Lc± √ −b ± D = 2a (88) with a b D = = = 1 + r3 ωA′ /2, 4 (89) 5 ′ 2Br ω(−1 + A) + Br ωA , (90) 2 4 2 4 2 2 ′ 2 3 ′ r 4 ω + B(1 + r ω) + ω 4B r ωA − rA (2 − 2r ω + r ωA ) −2A(2 + 4B 2 (r2 + r4 ω) + r3 ωA′ )) . (91) This equation implies directly the symmetry of the solutions related to the simultaneous transformations L → −L and B → −B. Therefore, it is enough to study the case B > 0 only. The corresponding specific energy of the particles following the circular orbits of the two families given by Eq. (88) is given by Ec± = V (r, Lc± ). (92) The angular velocity relative to distant observers of the charged particle on the circular orbits reads f Lc± uφ dφ = t = 2 , (93) Ω= dt u r Ec± with the specific angular momentum of the circular orbit L given by formula (88). The marginally stable circular orbits occur, if the condition d2 Vef f = 0, dr2 (94) governing the inflexion points of the effective potential of the particle radial dVef f = 0. We have motion, is satisfied simultaneously with the condition dr to look for the solution of the inflexion points for both ”plus” and ”minus” families of the circular orbits. The results are illustrated in Figure 10 for the plus family orbits, and in Figure 11 for the minus family orbits. We can see that increasing magnetic field (magnetic parameter B) has a tendency to stabilize the 22 circular motion, i.e., the region of the unstable motion, if it exists, shrinks with increasing B. This phenomenon is more efficient for the plus family orbits, in which case the unstable orbits in the KS naked singularity spacetimes are allowed for 1/2 > ω > ωcrit (B), with ωcrit (B) significantly increasing with increasing parameter B. Moreover, the influence of the magnetic field causes also a substantial shift of the static radius rstat , i.e., the inner boundary of the circular orbits, to vicinity of the naked singularity at r = 0 even for small values of the Hořava parameter ω, as demonstrated in Figures 11 and 12. 1.0 1.0 B=0 B=0.1 0.6 0.6 Ω 0.8 Ω 0.8 0.4 0.4 0.2 0.2 0.0 0.0 0 2 4 6 8 10 0 2 4 r 6 8 10 r 1.0 1.0 B=0.3 B=1 0.6 0.6 Ω 0.8 Ω 0.8 0.4 0.4 0.2 0.2 0.0 0.0 0 2 4 6 8 10 0 r 2 4 6 8 10 r Figure 10: The shaded gray (light gray) regions represent (r, ω) positions of the stable (unstable) circular orbits of charged particles of the family with specific angular momentum L = L+ in the field of KS naked singularities and black holes immersed in an axisymmetric, asymptotically uniform magnetic field. The figures are constructed for typical values of the parameter B characterizing the interaction of the charged particles with the magnetic field. 23 1.0 1.0 B=0.0 B=0.1 0.6 0.6 Ω 0.8 Ω 0.8 0.4 0.4 0.2 0.2 0.0 0.0 0 2 4 6 8 10 0 2 4 r 6 8 10 r 1.0 1.0 B=0.3 B=1 0.6 0.6 Ω 0.8 Ω 0.8 0.4 0.4 0.2 0.2 0.0 0.0 0 2 4 6 8 10 0 r 2 4 6 8 10 r Figure 11: The shaded gray (light gray) regions represent (r, ω) positions of the stable (unstable) circular orbits of charged particle of the minus family having L = L− in the field of KS naked singularities and black holes in axisymmetric, asymptotically magnetic field. The figures are given for the same values of the parameter B as in preceding figure. 24 8 Collisions of charged particles at equatorial circular orbits We consider here only two simple cases of the collisional processes of charged test particles that could well represent the role of the magnetic field added to the gravitational field of the KS naked singularities, as has been demonstrated in our preceding paper [97]. 8.1 Collisions of charged particles at circular orbits with infalling neutral particles The four-velocity of charged test particles moving along circular orbits has the following form −1/2 uα , 0, 0, γv/r), (95) c = (γf where v is the velocity of the charged particle at the circular orbit at radius r and γ is the relativistic Lorentz factor, related to the static observers. Using the conditions uα uα = −1 and dφ/dλ = vγ/r, one can easily find that γ 2 = 1 + β 2 r2 , where β= v=p βr 1 + β 2 r2 , Lc − B, r2 (96) (97) where the Lc is the specific angular momentum of the charged particle at the circular orbit. For the neutral particle incoming from infinity with specific energy En and specific angular momentum Ln , the 4-velocity is given by the relation s 2 L L E n , En 2 − f 1 + n , 0, n (98) uα n = f r2 r2 Considering particles of identical rest mass m1 = m2 = m, and the neutral particle with En = 1, we find the centre-of-mass energy to be given by 2 ECM 2m2 β = 1 − gαβ uα c un p = 1 − Ln β + 1 + β 2 r2 f −1/2 . (99) The results obtained for some characteristic values of the parameters B and ω are illustrated in Figure 12 demonstrating a critical role of the magnetic field enabling a significant enhancement of the acceleration of particles in the combined gravitational and electromagnetic fields of the KS naked singularities immersed in an uniform magnetic field. 25 B = 1, Ln = 0 3.5 B = 1, Ln = 2 7 6 3.0 2 Ecm 2m2 2 Ecm 2m2 5 2.5 2.0 4 3 2 1.5 1 0 1.0 2 3 4 5 6 2 3 4 rM 5 6 5 6 5 6 rM B = 10, Ln = 0 B = 10, Ln = 2 60 20 50 40 2 Ecm 2m2 2 Ecm 2m2 15 10 30 20 5 10 0 0 2 3 4 5 6 2 3 4 rM rM B = 100, Ln = 0 200 B = 100, Ln = 2 600 500 150 2 Ecm 2m2 2 Ecm 2m2 400 100 300 200 50 100 0 0 2 3 4 5 6 rM 2 3 4 rM 2 Figure 12: The radial dependence of the CM energy ECM /2m2 of the collisions of charged particle on circular orbits of the plus family, and neutral particles coming from infinity is given for typical values of the parameter B and the specific angular momentum of the neutral particle. The different lines correspond to the different values of ω: ω = 0.15 (solid line), 0.2 (dashed line), 0.25 (dotted line), 0.31 (dot-dashed line). 8.2 Head-on collisions of oppositely charged particles at circular orbits As a second example we discuss head-on collisions of two oppositely charged particles having the same mass m and following the same circular orbit in opposite direction, i.e., having the same energy but opposite angular momentum. After the collision, the resulting 4-momentum of the system takes the form P α = 2mγf −1/2 ξtα . 26 (100) Ω=0.3 Ω=0.2 50 0.0 0.1 0.5 1.0 100 50 2 ECM 2m2 20 2 ECM 2m2 200 0.0 0.1 0.5 1.0 10 5 20 10 5 2 0 1 2 3 4 5 2 0 6 1 2 rM Ω=0.4 10 3 4 0.0 10 6 0.0 0.1 0.1 0.5 0.5 1.0 104 2 ECM 2m2 2 ECM 2m2 6 Ω=0.45 6 100 0 5 rM 1.0 104 100 1 2 3 4 5 6 0 1 rM 2 3 4 5 6 rM Figure 13: Radial profiles of the CM energy of the head-on collisions of two oppositely charged particles following the circular orbits of the plus family with angular momentum L = L+ . The profiles are constructed for the characteristic values of the parameter ω = 0.2, 0.3, 0.4, 0.45, the dependence on the magnetic field is represented for the magnetic parameter B = 0.0, 0.1, 0.3 and 1.0. The 4-velocities of the colliding particles are given by −1/2 uα , 0, 0, −γv/r), 1 = (γf (101) −1/2 uα , 0, 0, γv/r), 2 = (γf (102) which implies that the squared CM energy is given by 2 ECM /2m2 = 2γ 2 , (103) and the CM energy then takes a simple intuitively expected form β ECM = 1 − gαβ uα 1 u2 = 2mγ. (104) The results of calculations of the centre-of-mass energy are given for characteristic values of the the magnetic field intensity parameter B and the Hořava parameter ω in Figure 13 for colliding particles of the plus family, while Figure 14 illustrates results of the collisions of the particles of the minus family. We can see clearly that the role of the magnetic field can be very strong and increases with increasing magnetic parameter B. Note that there is a qualitative difference in the behaviour of the radial profiles of the CM energy due to the magnitude of the ω parameter that is related to the existence of the photon circular geodesics. In the KS naked singularity spacetimes √ with the Hořava parameter satisfying the condition 1/2 > ω > ωph = 2/(3 3) = 0.3849, two photon circular orbits can exist, the outer one being unstable relative to the 27 Ω=0.2 Ω=0.3 200 200 100 20 10 5 2 0 0.0 0.1 0.5 1.0 50 2 ECM 2m2 2 ECM 2m2 100 0.0 0.1 0.5 1.0 50 20 10 5 2 4 6 8 2 0 10 2 4 rM Ω=0.4 10 200 100 100 0.0 0.1 0.5 1.0 20 0.0 0.1 0.5 1.0 50 2 ECM 2m2 50 2 ECM 2m2 8 Ω=0.45 200 10 5 2 0 6 rM 20 10 5 2 4 6 8 2 0 10 rM 2 4 6 8 10 rM Figure 14: Radial profiles of the CM energy of the head-on collisions of two oppositely charged particles following the circular orbits of the minus family with angular momentum L = L− . The profiles are constructed for the characteristic values of the parameter ω = 0.2, 0.3, 0.4, 0.45, the dependence on the magnetic field is represented for the magnetic parameter B = 0.0, 0.1, 0.3 and 1.0. 28 radial perturbations, while the inner one is stable relative to perturbations. In such spacetimes, unstable circular orbits with diverging energy exist. In vicinity of the outer photon circular orbits, the charged particle orbits have to be unstable, but near the inner photon circular orbits stable particle orbits with extremely large energy can exist. Collisions of oppositely charged particles following the orbits in vicinity of the photon circular orbits leads to diverging CM energy as demonstrated in Figures 13 and 14. Our results indicate again a significant role of the magnetic field in the enhancement of the acceleration of the colliding particles. For collisions of test particles on the plus family orbits the CM energy increases with increasing magnetic parameter, mapping the radial profile of the collisions of uncharged particles. Moreover, collisions occur also on circular orbits of charged particles located under the photon circular orbit. The CM energy diverges for orbits approaching the photon circular orbits. On the other hand, for collisions of charged particles on the minus family orbits the behavior of the radial profile of the CM energy is more complex and generally it is not following the radial profiles of the CM energy of the uncharged particles, if the magnetic parameter B is large enough. A detailed study of this phenomenon is behind the scope of the present paper and is planned for a future work. 9 Conclusions We have demonstrated that at the KS naked singularity spacetimes, at least three kinds of collisions of particles falling freely from infinity could lead to ultrahigh-energy observed in the CM system. Two types of the collisions can occur quite naturally in the field of near-extreme KS naked singularities at the specific radius r = M , in close analogy to the collisions in the Kerr naked singularity spacetimes [86]. There are only small quantitative differences between the KS and Kerr cases for the purely radial collisions. At this point the notion of the naked singularity is decisive qualitatively. Moreover, we can see that the local efficiency of the collisions of the particles in purely radial direction is identical to the efficiency obtained in the case of purely axial motion of the particles if they collide at the turning point of their radial motion. However, the efficiency of the ultra-high-energy processes, occurring at the specific radius r = M , and related to the distant observers, differ substantially in the KS and Kerr naked singularity spacetimes. In the near-extreme KS case, the efficiency is strongly restricted for both the escaping and the frequency shift, and is similar to the black hole case. In fact, the gravitational redshift cancels the energy excess obtained due to the collisional process and only the covariant energy of the colliding particles (rest energy for particles freely falling from rest at infinity) is relevant for energy observed at infinity. Such a system works as a standard accelerator of particles. On the other hand, in the near-extreme Kerr case, both these phenomena can be enhanced relatively very strongly, if the collisions occurs near the equatorial plane of the Kerr geometry where the rotational effects of the Kerr background corresponding to efficient Penrose processes are strongest [86]. We can conclude that for the high efficiency of the ultra-high-energy processes relative to distant observers, both the non-existence of the horizon, and the strong rotational effects are necessary, if we do not consider relevance of electromagnetic phenomena. 29 We have to stress that significant magnification of the efficiency of the ultrahigh energy collisions is possible due to additional electromagnetic phenomena influencing collisions of charged particles, as demonstrated here for simple situations related to the circular motion of charged particles orbiting near the KS naked singularities immersed in an asymptotically uniform magnetic field. Of course, this phenomenon occurs also in the collisions near black holes [39, 1, 2, 97]. Acknowledgments Z.S. and J.S. acknowledge the Albert Einstein Centre for gravitation and astrophysics supported by the Czech Science Foundation No. 14-37086G. 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