Download Ultra-high-energy collisions of particles in the field of near

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
rM
35
30
25
Ecm
20
15
10
5
0
1
2
3
rM
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
rM
5
6
5
6
5
6
rM
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
rM
rM
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
rM
2
3
4
rM
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
rM
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
rM
1.0
104
100
1
2
3
4
5
6
0
1
rM
2
3
4
5
6
rM
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
rM
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
rM
20
10
5
2
4
6
8
2
0
10
rM
2
4
6
8
10
rM
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. The authors
further acknowledge the project Supporting Integration with the International
Theoretical and Observational Research Network in Relativistic Astrophysics
of Compact Objects, reg. no. CZ.1.07/2.3.00/20.0071, supported by Operational Programme Education for Competitiveness funded by Structural Funds
of the European Union and state budget of the Czech Republic.
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