#### Particle collisions near a three-dimensional warped AdS black hole

Eur. Phys. J. C
Particle collisions near a three-dimensional warped AdS black hole
Ramón Bécar 2
P. A. González 1
Yerko Vásquez 0
0 Departamento de Física y Astronomía, Facultad de Ciencias, Universidad de La Serena , Avenida Cisternas 1200, La Serena , Chile
1 Facultad de Ingeniería y Ciencias, Universidad Diego Portales , Avenida Ejército Libertador 441, Casilla 298-V, Santiago , Chile
2 Departamento de Ciencias Matemáticas y Físicas, Universidad Católica de Temuco , Montt 56, Casilla 15-D, Temuco , Chile
In this paper we consider the warped AdS3 black hole solution of topologically massive gravity with a negative cosmological constant, and we study the possibility that it acts as a particle accelerator by analyzing the energy in the center of mass (CM) frame of two colliding particles in the vicinity of its horizon, which is known as the Bañnados, Silk and West (BSW) process. Mainly, we show that the critical angular momentum (L c) of the particle decreases when the warping parameter(ν) increases. Also, we show that despite the particle with L c being able to exist for certain values of the conserved energy outside the horizon, it will never reach the event horizon; therefore, the black hole cannot act as a particle accelerator with arbitrarily high CM energy on the event horizon. However, such a particle could also exist inside the outer horizon, with the BSW process being possible on the inner horizon. On the other hand, for the extremal warped AdS3 black hole, the particle with L c and energy E could exist outside the event horizon and, the CM energy blows up on the event horizon if its conserved energy fulfills the condition E 2 > (3ν(2ν+2−3)1l)2 , with the BSW process being possible.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2 Three-dimensional warped AdS black holes . . . . . .
3 Motion of particles in the three-dimensional warped
AdS black hole background . . . . . . . . . . . . . .
4 The CM energy of two colliding particles . . . . . . .
5 Radial motion of the particle with critical angular
momentum . . . . . . . . . . . . . . . . . . . . . . .
6 Final remarks . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
Bañados, Silk and West (BSW) [
1
] demonstrated some years
ago that two particles colliding near the degenerate horizon
of an extreme Kerr black hole could create a large center of
mass (CM) energy if one of the particles has a critical angular
momentum; thus, extreme Kerr black holes can act as
natural particle accelerators. Nowadays, this process is known as
the BSW mechanism, which was found for the first time by
Piran, Shaham and Katz in 1975 [
2–4
]. Then, based on the
infinite acceleration being able to occur not only for extremal
black holes but also for non-extremal ones [
5
], it was shown
that an infinite energy in the CM frame of colliding
particles is a universal property of rotating black holes as long
as the angular momentum of one of the colliding particles
approaches the critical value [
6
]. The requirement that the
black hole be rotating seems to be an essential ingredient
in obtaining ultra-high CM energy; however, it was shown
that the similar effect exists for non-rotating charged black
holes [
7
]. Moreover, the extension of the BSW mechanism to
non-extremal backgrounds shows that particles cannot
collide with arbitrarily high energies at the outer horizon and
that ultra-energetic collisions can only occur near the Cauchy
horizon of a Kerr black hole with any spin parameter [
8
]. The
non-extremal Kerr-de Sitter black holes could also act as
particle accelerators with arbitrarily high CM energy if one of
the colliding particles has the critical angular momentum [
9
].
The BSW mechanism has been studied for different black
hole geometries; for instance, rotating charged cylindrical
black holes were studied in Ref. [
10
] and an extreme rotating
black holes in Horava–Lifshitz gravity in Ref. [
11
]. However,
the fundamental parameter of Horava–Lifshitz gravity avoids
an infinite value of the CM energy being obtained.
Higherdimensional rotating black holes have been studied in Refs.
[
12–14
] and lower-dimensional black holes in Refs. [
15–19
].
The collision of two neutral particles on the vicinity of the
extremal Kerr black hole horizon considered as a complete
vacuum spacetime in its own right was studied in [
20
].
Particle collision in the strong gravitational field of a rotating
black hole in a Randall–Sundrum brane with a
nonvanishing cosmological constant was studied in [
21
]. The
particle collisions near the cosmological horizon of non-extremal
Reissner–Nordstrom de Sitter black holes were studied in
Ref. [
22
], charged dilatonic black holes in Ref. [
23
],
nonrotating and rotating regular black holes in Refs. [
24–27
],
and extremal modified Hayward and Bardeen rotating black
holes in Ref. [28]. On the other hand, charged particles in
general stationary charged black holes was considered in Ref.
[
29
], and on string black holes in Ref. [
30
]. The collisions of
spinning particles on rotating black holes in Refs. [
31, 32
],
and on Schwarzschild black holes was considered in Ref.
[33]; however, the unavoidable appearance of superluminal
motion and the change of trajectories from timelike to
spacelike can be avoided as the energy in the CM frame can grow
unbounded provided that one of the particles is not exactly
critical but slightly deviates from the critical trajectory [
34
].
On the other hand, the formation of black holes through the
BSW mechanism was investigated in [
35
].
The aim of this work is to consider the special class of
well-known three-dimensional warped anti-de Sitter black
holes solutions [
36–39
] and to study, via the BSW
mechanism, the possibility of obtaining unbounded energy in the
CM frame of two colliding particles and to analyze the effect
of the warped parameter that controls the stretching
deformation on this. It is worth noting that warped AdS3 black
holes can be viewed as discrete quotients of warped AdS3
spacetime just like the BTZ black holes as discrete quotients
of the AdS3. Also, when the warped parameter ν = 1, the
metric reduces to the metric of a BTZ black hole in a
rotating frame. An important feature of these black holes is that
the Killing vector ∂t is spacelike everywhere in spacetime
and consequently its ergoregion extends to infinity. Also, it
is known that two particles in the ergosphere lead to infinity
growth of the energy of the CM frame, provided the angular
momentum of one of the two particles has a large negative
angular momentum and a fixed energy at infinity for the Kerr
black holes [5], which was subsequently proven to be a
universal property of the ergosphere [
40
].
It worth mentioning that the collision of two particles near
the horizon of a BTZ black hole was studied in Refs. [
16–18
].
In Refs. [
16, 17
] the authors found that the particle with the
critical angular momentum could exist inside the outer
horizon of the BTZ black hole regardless of the particle energy
with the BSW process being possible on the inner horizon
for the non-extremal BTZ black hole. Also, the BSW process
could also happen for the extremal BTZ black hole, where the
particle with the critical angular momentum could only exist
on the degenerate horizon. On the other hand, in Ref. [18],
the authors studied the collision of two particles on an event
horizon and outside of the BTZ black hole, and they showed
that although in principle the CM energy of two ingoing
particles can be arbitrarily large on an event horizon, if either
of the two particles has a critical angular momentum and the
other has a non-critical angular momentum, the critical
particles never reach the event horizon. However, the motion of a
particle with a subcritical angular momentum is allowed near
an extremal rotating BTZ black hole and that a CM energy
for a tail-on collision at a point can be arbitrarily large in a
critical angular momentum limit.
On the other hand, the warped AdS3 space has non-AdS
asymptotics and is not a solution of pure three-dimensional
gravity, and it appears as a submanifold, at fixed polar
coordinate, of the near horizon extreme Kerr black hole [
41, 42
].
So, it is interesting to study if the BSW effect, which was
first noted for the extreme Kerr black holes, also occurs in
the warped AdS3 black hole in analogy with the extremal
Kerr geometry. In fact, we will show that the BSW effect is
possible on the outer horizon in the extremal warped AdS3
black hole, and the particle with critical angular
momentum can reach the degenerate horizon when a condition on
its energy is fulfilled, which resembles to what occurs in
the extremal Kerr-AdS black hole; however, in the extremal
Kerr-AdS black hole two conditions must be fulfilled [9];
besides, this effect is also possible on the inner horizon for
the non-extremal warped AdS3 black hole. In addition, we
describe the kinds of orbits in this background for timelike
and null geodesics by analyzing the effective potential. It is
worth mentioning that the study of null geodesics in
backgrounds with and without horizon are very important in
studies about reconstruction of scalar field in the bulk from data
defined on the boundary. It is known that for an
asymptotically AdS spacetime, the reconstruction of a scalar field in
the bulk from data defined on the boundary fails if there
exist null geodesic originating in the bulk that do not reach
the boundary [
43–45
], which is analogous to what occurs
in flat spacetime, where a necessary condition for a
continuous bulk reconstruction of a field from data defined on a
boundary is that every null geodesic originating in the bulk
intersects the boundary [
46
]. We will show that, similar to the
BTZ black hole, not all the outgoing null geodesics outside
the warped AdS3 black hole can reach infinity. On the other
hand, the warped AdS3/CFT2 correspondence was proposed
in [
38
], the asymptotical symmetry group analysis was
studied in [
47–50
], and some studies of quasinormal modes and
real-time correlators were carried out in [
51, 52
].
The manuscript is organized as follows: In Sect. 2 we
give a brief review of the three-dimensional warped AdS
black hole. Then, we study the particle’s motion in the
threedimensional warped black hole background in Sect. 3. In
Sect. 4 we obtain the CM energy of two colliding particles,
and in Sect. 5 we study the radial motion of a particle with
critical angular momentum and we investigate the possibility
that the black hole acts as a particle accelerator. Finally, our
conclusions are in Sect. 6.
2 Three-dimensional warped AdS black holes
The models of gravity in three spacetime dimensions and
their modifications have attracted remarkable interest in the
last decade. One of them is topologically massive gravity
(TMG), which modifies the theory of general relativity (GR)
by adding a Chern–Simons gravitational term [
53–55
] to the
Hilbert-Einstein action. The action is described by
1
I = 16π G
M
3 √
d x
2
−g R + l2
l
+ 96π Gν M
d3x√−g λμν λrσ ∂μ rσν + 23 μστ ντr ,
where τ σ μ = 1/√−g is the Levi–Civita tensor and ν is
a dimensionless coupling constant, which is related to the
graviton mass μ by ν = μl/3. This model, in contrast to
GR in three spacetime dimensions, has a propagating degree
of freedom which corresponds to a massive graviton. Also,
the possibility of constructing a chiral theory of gravity at
a special point in the space of parameters [
56
] are some of
its important characteristics. Accordingly, it was conjectured
that a consistent quantum theory of the so-called chiral
gravity can be defined at μl = 1 or ν = 1/3 [
57
]. However,
for non-chiral values of μl, it has been shown that there are
other two warped AdS3 vacuum solutions for every value
of μl = 3 [
38
]. The warped AdS3 geometry corresponds
to a one parameter-stretched deformation of AdS3. Further
aspects of TMG can be found in [
58–64
] and references
therein.
The metric describing the spacelike stretched black holes
[
36–39
] in ADM form is given by
ds2 = −N 2(r )dt 2 +
2 R2(r ) dφ + Nφ (r )dt 2
4dr 2
+ 4R2(r )N 2(r ) ,
where the metric functions are
r
R2(r ) = 4
3(ν2 − 1)r + (ν2 + 3)(r+ + r−)
N 2(r ) =
Nφ (r ) =
− 4ν r+r−(ν2 + 3) ,
2(ν2 + 3)(r − r+)(r − r−) ,
4R2
2νr −
r+r−(ν2 + 3)
2 R2
with ν2 ≥ 1 being the parameter that controls the
stretching deformation, and r+ and r− being the outer and inner
horizon, respectively. For ν = 1 the metric reduces to the
metric of BTZ black holes in a rotating frame. An
important feature of these black holes is that the Killing vector ∂t
is spacelike everywhere in the spacetime and consequently
its ergoregion extends to infinity; therefore, observers cannot
follow the orbits of ∂t in the exterior region. Also, the energy
of a particle can have negative energy in the exterior region.
It is worth noting that the warped AdS3 space also arises in
other contexts, for instance see [
41,65–74
].
3 Motion of particles in the three-dimensional warped
AdS black hole background
The equations of the geodesics can be derived from the
Lagrangian of a test particle, which is given by [
75
]
(1)
1
L = 2
d x μ d x ν
gμν dτ dτ
So, for the three-dimensional warped AdS black hole (2), the
Lagrangian reads
2L =
−N 2(r ) +
2 R2(r )Nφ2(r ) t˙2 + 2 2 R2(r )Nφ (r )t˙φ˙
4
2
+ 4R2(r )N 2(r ) r˙ +
2 R2(r )φ˙ 2,
where q˙ = dq/dτ , and τ is an affine parameter along
the geodesic that we choose as the proper time. Since the
Lagrangian (7) is independent of the cyclic coordinates (t, φ),
their conjugate momenta ( t , φ ) are conserved. The
equations of motion are obtained from ˙ q − ∂∂Lq = 0, where
q = ∂L/∂q˙ are the conjugate momenta to the coordinate
q, and are given by
t =
−N 2(r ) +
2 R2(r )Nφ2(r ) t˙
+ 2 R2(r )Nφ (r )φ˙ ≡ −E ,
4
r ,
r = 4R2(r )N 2(r ) ˙
φ =
2 R2(r )Nφ (r )t˙ +
2 R2(r )φ˙ ≡ L ,
where E and L are dimensionless integration constants
associated with each of them. The Hamiltonian is given by
H = t t˙ + φ φ˙ + r r˙ − L,
4
2H = − E t˙ + L φ˙ + 4R2(r )N 2(r ) ˙ ≡ −m2.
r 2
By normalization, we shall consider that m2 = 1 for massive
particles (m2 = 0 for photons). We solve the above
equation for r˙2 in order to obtain the radial equation, t˙, and φ˙ ,
which allows us to characterize the possible movements of
the test particles without an explicit solution of the equations
of motion, which yields
a =
b =
c =
−3E 2(ν2 − 1) + m2 2(ν2 + 3)
4
,
,
−4L2 + (ν2 + 3)r+r−m2 2 + 4E L r+r−(ν2 + 3)
4
be used in the next section to obtain the CM energy of two
colliding particles falling freely from rest with the same rest
mass m0 in the warped AdS3 black hole background. We will
assume t˙ > 0 for all r > r+ so that the motion is forward in
time outside the horizon. So, the following condition must
be fulfilled
E + L Nφ > 0, for all r > r+.
Now, in order to see if a particle can reach the event
horizon, we will analyze the effective potential of the
threedimensional warped AdS black hole by using the equation
of motion of the particle in the radial direction given by
r˙2 + V = 0,
where V is the effective potential of the particle in the radial
direction. Hence by comparing Eq. (15) with Eq. (17), we
obtain:
4
V (r ) = − 4 R2(r )
− N 2(r ) m2
E + L Nφ (r ) 2
L2
+ 2 R2(r )
.
The motion of the particle is allowed in regions where V (r ) ≤
0, and it is prohibited in regions where V (r ) > 0. It is clear
that the particle can exist on the event horizon r = r+ because
N 2(r+) = 0, and then the effective potential is negative. On
the other hand, when r → ∞ it is easy to show that the
effective potential is given by
V (r → ∞) ≈
((ν2 + 3)m2 2 − 3E 2(ν2 − 1))r 2
4
.
(19)
t˙ =
φ˙ = −
r 2
˙ =
,
,
where the above equations represent all nonzero 3-velocity
components u = (t˙, r˙, φ˙ ) for the geodesic motion that will
This expression shows that the existence of massive particles
at infinity depends on the warped parameter ν and on its
energy E . Therefore, massive particles can exist at infinity
when the following condition is fulfilled, with ν = 1:
E 2 > (ν2 + 3) 2
.
3(ν2 − 1)
Note that the effective potential (18) is a parabola, and it
can be written asV (r ) = ar 2 + br + c, where
(20)
(21)
(22)
(23)
In the caseν = 1 (BTZ black hole in a rotating frame), a
simplifies to a = 4m2/ 2; thus, for m = 0, the effective
potential is a straight line. In this case there are trapped null
geodesics and the condition for the existence of these null
geodesics outside the event horizon that cannot reach spatial
infinity can be obtained by studying the effective potential,
which yields
E (√r r + − −
+ − − r+) < L < E √r r
E (r+ + r−) .
2
(24)
Otherwise, the outgoing null geodesics can reach infinity (see
Fig. 1), which could have holographic implications in the
studies of bulk reconstruction, see for instance [
43–46
].
However, for ν = 1 and massless particles (m = 0), a reduces
to a = −3E 2(ν2 − 1)/ 4 < 0. So, the effective potential
is a convex parabola, and thus there are massless particles
that can reach infinity (see Fig. 1). However, there are also,
trapped null geodesics, in fact, by studying the behavior of
the potential V (r ) with ν > 1, yields the condition for the
existence of null geodesics outside the event horizon that
cannot reach spatial infinite, the condition can be written as
which reduces to (24) when ν = 1. Additionally, in contrast
to the BTZ black hole, there are unstable circular orbits, this
occurs when the effective potential has a unique real root r0
bigger than r+. A unique real root exists when the following
condition is satisfied
0.5
behavior of the effective potential between r = 0 and r = 3, while the
right figure shows the behavior of the effective potential between r = 0
and r = 100
On the other hand, for massive particles (m = 1),
the effective potential is a concave parabola (a > 0)
for ν = 1 (see Fig. 2). So, massive particles cannot
reach infinity. However, for ν = 1 and m = 1, a =
−3E 2(ν2 − 1) + 2(ν2 + 3) / 4, and when E 2 < (3ν(2ν+2 −3)1)2
the parabola is concave (a > 0). So, the massive particles
cannot reach infinity in this case, whereas for E 2 > (ν2+3) 2 ,
3(ν2−1)
the parabola is convex (a < 0) and there are massive
particles that can reach spatial infinity, which is different to the
BTZ black hole (see Fig. 2).
4 The CM energy of two colliding particles
In this section we calculate the CM energy of two colliding
particles in the warped AdS3 black hole. To achieve this, we
must derive the 2+1 dimensional 3-velocity components to
obtain the CM energy of the colliding particles. We consider
that the particles have the same rest mass m0, energies E1 and
E2 and angular momentum L 1 and L 2, respectively. From
the relation EC M = √2m0 1 − gμν u1μuν2, where u1 and u2
1
2
3
with i = 1, 2. Also, when the particles arrive to the horizon
r = r+, N 2(r+) → 0, H1 → K12 and H2 →
CM energy (27) at the horizon yields:
K22, the
E 2
1
2mCM2 (r → r+) = ( R2)2 N 2
0
K1 K2 −
K 2
1
K 2
2
. (29)
Note that K1 K2 < 0 the E C2 M on the horizon will be a
negative infinity therefore the energy center of mass will be
imaginary and then it is not a physical solution. In fact, we can
set K1 < 0 and K2 > 0 without loss of generality; however,
K1 < 0 outside the event horizon is in contradiction with
condition (16). However, when K1 K2 ≥ 0, the numerator
of this expression will be zero and the value of EC M will
be undetermined. Now, in order to find the limiting value
of the CM energy at the horizon we use the L’Hopital rule,
obtaining
Note that the numerator of the above expression is finite at
the horizon and when Ki (r+) = 0 the CM energy of two
colliding particles on the horizon could be arbitrarily high,
EC M Ki =0 → ∞. So, from Ki (r+) = 0 we obtain that the
critical angular momentum is given by:
On the other hand, when K1(r+) and K2(r+) are both zero,
then EC M is finite at the horizon. In this case H1(r+) =
H2(r+) = 0 and
Therefore, in order to obtain an infinite CM energy only one
of the colliding particles should have the critical angular
momentum, making the BSW process possible. In Fig. 3,
we show the behavior of EC M versus L1 with the other
parameters fixed. We observe that there is a critical value
of angular momentum for the particle 1 at which the CM
energy blows up. Note that in order to get positive EC M the
asymptotic value of critical angular momentum has to be
reached from the right. Additionally, in Fig. 4, we have
plotted Lc in terms of warped parameters ν for different values of
energy E . Notice that when the energy of particle 1 and the
warped parameter increases, the critical angular momentum
Lc decreases. By a similar analysis, it is possible to evaluate
the EC M on the inner horizon, finding that this is also infinity
as long as one of the two particles has the following critical
angular momentum:
(30)
,
(31)
(32)
ECM2 r
In this section, we will study the radial motion of the particle
with critical angular momentum and energy E . As we have
mentioned, the particle reaches the event horizon of the black
holes if the square of the radial component of the 3-velocity
r˙2 in Eq. (15) is positive or V is negative in the neighborhood
of the black hole horizon. We will denote the explicit form of
r˙2 with critical angular momentum as Rc(r ), which is given
by
E 1
Rc =
(r − r+) −m2l2(r − r−)(3 + ν2) + E 2(3r (−1 + ν2) + r−(3 + ν2) − 4r+ν2)
l4
,
Lci =
i = 1, 2.
Ei r−
3r+ + ν2r+ + 4ν2r− − 4ν r−r+(3 + ν2)
2 −2νr− +
r−r+(3 + ν2)
,
(33)
and it vanishes on the event horizon. Also, for some values
of the parameters, Rc can be positive, which implies that
particles with critical angular momentum can exist outside
the event horizon; as we shall see, however they cannot reach
the event horizon unless r+ = r−. Particles with critical
angular momentum can reach the event horizon if
Fig. 5 The behavior of Rc(r ) as a function of r with r− = √1, r+ = 2,
ν = 1 (Lc = −0.586), ν = 1.5 (Lc = −1.380), ν = 3 (Lc =
−1.732), ν = 2.5 (Lc = −2.894), E = 1 and l = 1
d Rc
dr r=r+
> 0,
which yields
(35)
Rc
40
20
0
20
40
2
4
6
8
r
d Rc
dr r=r+ = −
(E 2 + l2m2)(r+ − r−)(ν2 + 3)
l4
< 0. (36)
Therefore, if particles with critical angular momentum exist
outside the black hole such particles cannot reach the event
horizon. In Fig. 5 we plot the behavior of Rc as a function of
r for the three-dimensional warped black hole. Notice that if
dd2rR2c = −2 m2l2(3+ν2l4−3E2(ν2−1)) > 0, Rc(r ) has a zero also
at r0 = (ν2+3)(ν22r+−+3)(32r−−3+E(r2−(ν−2−4r1+))ν2)E2 which is greater than
r+, and particles with critical angular momentum can exist
outside the event horizon at r > r0 > r+, see Fig. 5.
On the other hand, we notice that the particle with the
critical angular momentum can exist inside the event horizon r+.
This can be shown by replacing the critical angular
momentum Eq. (33) in the square of the radial component of the
3-velocity Eq. (15), obtaining the analog of Eq. (34), whose
derivative evaluated on the internal horizon r− is positive
ddRrc |r=r− > 0; therefore, the particle with critical angular
momentum can reach the inner horizon and the center of
mass energy can be arbitrarily high, with the BSW process
being possible on the inner horizon.
On the other hand, in the extremal case r+ = r− we obtain
Rc = −
(r − r+)2 m2l2(3 + ν2) − 3E 2(ν2 − 1)
l4
,
(37)
= −
2(r − r+) m2l2(3 + ν2) − 3E 2(ν2 − 1)
l4
. (38)
and
d Rc
dr
Fig. 6 The behavior of Rc(r ) as a function of r for the extremal warped
AdS3 black hole for different values of the warped parameter ν =
1, ν = 1.5 (Lc = −0.709), ν = √3 (Lc = −1.015), ν = 2.5 (Lc =
−1.959), E = 1 and l = 1
Then, clearly Eqs. (37) and (38) are zero on the event horizon,
then it is necessary to calculate dd2rR2c |r=r+ :
d2 Rc
dr 2 r=r+ =
−2 m2l2(3 + ν2 − 3E 2(ν2 − 1))
l4
.
(39)
If dd2rR2c
r=r+
tum will reach the degenerate horizon. Our case it is fulfilled
if:
> 0, the particle with critical angular
momen
,
which is the same expression given in Eq. (20). In Fig. 6 we
plot the behavior of the Rc as a function of r for the
threedimensional extremal warped AdS black hole. Notice that
the particle with critical angular momentum can reach the
degenerate horizon if the condition (40) is satisfied, and thus
the BSW process is possible. This result is similar to that
found in [
9
] for Kerr-AdS black holes. Notice that for ν = 1
the particle with critical angular momentum cannot exist
outside the event horizon; however, it can exist on the
degenerate horizon, that corresponds to the behavior observed for
the extremal BTZ black hole [
17
].
6 Final remarks
In this paper we considered two colliding particles in the
vicinity of the horizon of a three-dimensional warped AdS
black hole, and we analyzed the energy in the CM frame of
the colliding particles in order to investigate the possibility
that the three-dimensional warped AdS black hole can act as
a particle accelerator. We found for the warped AdS3 black
hole that despite the particle with critical angular momentum
being able to exist for certain values of the conserved energy
outside the black hole, it will never reach the event horizon;
1
1.5
3
(40)
therefore, the black hole cannot act as a particle accelerator
with arbitrarily high CM energy on the event horizon.
However, the particle with the critical angular momentum could
also exist inside the outer horizon of the warped AdS3 black
hole, with the BSW process being possible on the inner
horizon. Also, we showed that the critical angular momentum
decreases when the parameter that controls the stretching
deformation increases. At ν = 1, the non-extremal warped
AdS3 black hole solution is given by the BTZ black hole
in the rotating frame, and we showed that there are trapped
null geodesics, when the condition E (√r+r− − r+) < L <
E √r+r− − E(r+2+r−) , is satisfied. That corresponds to the
behavior observed for the BTZ black hole [
76
]. Otherwise,
the outgoing null geodesics can reach infinity. However, for
ν = 1 there are massless particles that can reach
infinity, as well as, there are trapped null geodesics when the
condition 21 E
3E 2(r+ − r−)2(ν2 − 1) + 2E − (r+ + r−)ν
+ r+r−(ν2 + 3) is satisfied. Also, we found that in
contrast to the BTZ black hole, there are unstable circular orbits.
On the other hand, massive particles cannot reach infinity for
ν = 1. However, for ν = 1, when E 2 < (3ν(2ν+2 −3)1)2 the massive
particles cannot reach infinity. Otherwise, there are massive
particles that can reach spatial infinity, which is different to
the BTZ black hole.
On the other hand, for the extremal warped AdS3 black
holes, we found that the particle with critical angular
momentum could exist outside the event horizon and reach a high CM
energy on the event horizon as long as its conserved energy
fulfills the condition E 2 > (ν2+3)l2 , with the BSW process
3(ν2−1)
being possible. At ν = 1 the particle with critical angular
momentum cannot exist outside the event horizon; however
it can exist on the degenerate horizon, that corresponds to
the behavior for the extremal BTZ black hole. It would be
interesting to explore the possible holographic implications
of the results found here.
Acknowledgements We would like to thank the anonymous referees
for valuable comments which help us to improve the quality of our
paper. This work was partially funded by the Comisión Nacional de
Ciencias y Tecnología through FONDECYT Grant 11140674 (PAG)
and by the Dirección de Investigación y Desarrollo de la Universidad
de La Serena (Y.V.). P. A. G. acknowledges the hospitality of the
Universidad de La Serena, National Technical University of Athens and
Pontificia Universidad Católica de Valparaíso, where part of this work
was undertaken.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
1. M. Bañados , J. Silk , S.M. West , Phys. Rev. Lett . 103 , 111102 ( 2009 ). arXiv: 0909 .0169 [hep-ph]
2. T. Piran , J. Shaham , J. Katz , Astrophys. J. 196 , L107 ( 1975 )
3. T. Piran , J. Shaham , Phys. Rev. D 16 , 1615 ( 1977 ). https://doi.org/ 10.1103/PhysRevD.16.1615
4. T. Piran , J. Shaham , Astrophys. J. 214 , 268 ( 1977 )
5. A.A. Grib , Y.V. Pavlov , Int. J. Mod. Phys. D 20 , 675 ( 2011 ). arXiv: 1008 .3657 [gr-qc]
6. O.B. Zaslavskii , Phys. Rev. D 82 , 083004 ( 2010 ). arXiv: 1007 .3678 [gr-qc]
7. O.B. Zaslavskii , JETP Lett. 92 , 571 ( 2010 ) [Pisma Zh . Eksp. Teor. Fiz . 92 , 635 ( 2010 ) ] . arXiv: 1007 .4598 [gr-qc]
8. S. Gao , C. Zhong , Phys. Rev. D 84 , 044006 ( 2011 ). arXiv: 1106 .2852 [gr-qc]
9. Y. Li , J. Yang , Y.L. Li , S.W. Wei , Y.X. Liu , Class. Quantum Gravity 28 , 225006 ( 2011 ). arXiv: 1012 .0748 [hep-th]
10. J.L. Said , K.Z. Adami , Phys. Rev. D 83 , 104047 ( 2011 ). arXiv: 1105 .2658 [gr-qc]
11. A. Abdujabbarov , B. Ahmedov , B. Ahmedov , Phys. Rev. D 84 , 044044 ( 2011 ). arXiv:1107.5389 [astro-ph.SR]
12. A. Abdujabbarov , N. Dadhich , B. Ahmedov , H. Eshkuvatov, Phys. Rev. D 88 , 084036 ( 2013 ). arXiv: 1310 .4494 [gr-qc]
13. U. Debnath, arXiv: 1508 .02385 [gr-qc]
14. O.B. Zaslavskii , Int. J. Mod. Phys. D 26 ( 10 ), 1750108 ( 2017 ). arXiv: 1602 .08779 [gr-qc]
15. J. Sadeghi , B. Pourhassan , H. Farahani , Commun. Theor. Phys . 62 ( 3 ), 358 ( 2014 ). arXiv: 1310 .7142 [hep-th]
16. K. Lake, Phys. Rev. Lett . 104 , 211102 ( 2010 ) [ Erratum: Phys. Rev. Lett . 104 , 259903 ( 2010 ) ] . https://doi.org/10.1103/PhysRevLett. 104.259903, https://doi.org/10.1103/PhysRevLett.104.211102, arXiv: 1001 .5463 [gr-qc]
17. J. Yang , Y.L. Li , Y. Li , S.W. Wei , Y.X. Liu , Adv. High Energy Phys . 2014 , 204016 ( 2014 ). arXiv: 1202 .4159 [hep-th]
18. N. Tsukamoto , K. Ogasawara , Y. Gong , Phys. Rev. D 96 ( 2 ), 024042 ( 2017 ). arXiv: 1705 .10477 [gr-qc]
19. S. Fernando, Mod. Phys. Lett. A 32 , 1750074 ( 2017 ). arXiv: 1703 .00373 [gr-qc]
20. A. Galajinsky, Phys. Rev. D 88 , 027505 ( 2013 ). https://doi.org/10. 1103/PhysRevD.88.027505, arXiv: 1301 .1159 [gr-qc]
21. S.R. Shaymatov , B.J. Ahmedov , A.A. Abdujabbarov , Phys. Rev. D 88 ( 2 ), 024016 ( 2013 ). https://doi.org/10.1103/PhysRevD.88. 024016
22. C. Zhong , S. Gao , JETP Lett. 94 , 589 ( 2011 ). arXiv: 1109 .0772 [hep-th]
23. P. Pradhan, Astropart. Phys. 62 , 217 ( 2015 ). arXiv: 1407 .0877 [grqc]
24. P. Pradhan, arXiv: 1402 .2748 [gr-qc]
25. S.G. Ghosh, P. Sheoran , M. Amir , Phys. Rev. D 90 ( 10 ), 103006 ( 2014 ). arXiv: 1410 .5588 [gr-qc]
26. S.G. Ghosh , M. Amir , Eur. Phys. J. C 75 ( 11 ), 553 ( 2015 ). arXiv: 1506 .04382 [gr-qc]
27. M. Amir , S.G. Ghosh , JHEP 1507 , 015 ( 2015 ). arXiv: 1503 .08553 [gr-qc]
28. B. Pourhassan , U. Debnath, arXiv: 1506 .03443 [gr-qc]
29. Y. Zhu , S. Fengwu , Y. Xiao Liu , Y. Jiang , Phys. Rev. D 84 , 043006 ( 2011 )
30. S. Fernando, Gen. Relativ. Gravity 46 , 1634 ( 2014 ). arXiv: 1311 .1455 [gr-qc]
31. M. Guo , S. Gao , Phys. Rev. D 93 ( 8 ), 084025 ( 2016 ). arXiv: 1602 .08679 [gr-qc]
32. Y.P. Zhang , B.M. Gu , S.W. Wei , J. Yang , Y.X. Liu , Phys. Rev. D 94 ( 12 ), 124017 ( 2016 ). arXiv: 1608 .08705 [gr-qc]
33. C. Armaza , M. Bañados , B. Koch , Class. Quantum Gravity 33 ( 10 ), 105014 ( 2016 ). arXiv: 1510 .01223 [gr-qc]
34. O.B. Zaslavskii , EPL 114 ( 3 ), 30003 ( 2016 ). arXiv: 1603 .09353 [grqc]
35. F. Atamurotov , B. Ahmedov , S. Shaymatov , Astrophys. Space Sci . 347 , 277 ( 2013 )
36. K.A. Moussa , G. Clement, C. Leygnac , Class. Quantum Gravity 20 , L277 ( 2003 ). https://doi.org/10.1088/ 0264 -9381/20/24/L01, arXiv:gr-qc/ 0303042
37. A. Bouchareb , G. Clement, Class. Quantum Gravity 24 , 5581 ( 2007 ). https://doi.org/10.1088/ 0264 -9381/24/22/018
38. D. Anninos , W. Li , M. Padi , W. Song , A. Strominger , JHEP 0903 , 130 ( 2009 ). arXiv: 0807 .3040 [hep-th]
39. K.A. Moussa , G. Clement, H. Guennoune , C. Leygnac , Phys. Rev. D 78 , 064065 ( 2008 ). https://doi.org/10.1103/PhysRevD.78. 064065, arXiv: 0807 .4241 [gr-qc]
40. O.B. Zaslavsky , Mod. Phys. Lett. A 28 , 1350037 ( 2013 ). arXiv: 1301 .4699 [gr-qc]
41. I. Bengtsson, P. Sandin , Class. Quantum Gravity 23 , 971 ( 2006 ). arXiv:gr-qc/0509076
42. J.M. Bardeen , G.T. Horowitz, Phys. Rev. D 60 , 104030 ( 1999 )
43. R. Bousso , B. Freivogel , S. Leichenauer , V. Rosenhaus , C. Zukowski , Phys. Rev. D 88 , 064057 ( 2013 ). https://doi.org/10. 1103/PhysRevD.88.064057, arXiv: 1209 .4641 [hep-th]
44. S. Leichenauer , V. Rosenhaus , Phys. Rev. D 88 ( 2 ), 026003 ( 2013 ). https://doi.org/10.1103/PhysRevD.88.026003, arXiv: 1304 .6821 [hep-th]
45. Maria Ioanna Stylianidi Christodoulou , A study In AdS/CFT correspondence: local bulk field reconstruction , Master Thesis . Institute of Physics, University of Amsterdam ( 2013 )
46. C. Bardos , G. Lebeau, J. Rauch, SIAM J. Control Optim . 30 ( 5 ), 1024 - 1065 ( 1992 )
47. G. Compere, S. Detournay , Class. Quantum Gravity 26 , 012001 ( 2009 ) [Erratum: Class. Quantum Gravity 26 , 139801 ( 2009 ) ] . https://doi.org/10.1088/ 0264 -9381/26/1/012001, https://doi.org/ 10.1088/ 0264 -9381/26/13/139801, arXiv: 0808 . 1911 [hep-th]
48. G. Compere, S. Detournay, JHEP 0908 , 092 ( 2009 ). https://doi. org/10.1088/ 1126 - 6708 / 2009 /08/092. arXiv: 0906 .1243 [hep-th]
49. M. Blagojevic , B. Cvetkovic , JHEP 0909 , 006 ( 2009 ). https://doi. org/10.1088/ 1126 - 6708 / 2009 /09/006, arXiv: 0907 .0950 [gr-qc]
50. B. Chen , J.J. Zhang , J.D. Zhang , D.L. Zhong , JHEP 1304 , 055 ( 2013 ). https://doi.org/10.1007/JHEP04( 2013 ) 055 , arXiv: 1302 .6643 [hep-th]
51. B. Chen , Z.B. Xu , JHEP 0911 , 091 ( 2009 ). https://doi.org/10.1088/ 1126 - 6708 / 2009 /11/091, arXiv: 0908 .0057 [hep-th]
52. B. Chen , B. Ning , Z.B. Xu , JHEP 1002 , 031 ( 2010 ). https://doi. org/10.1007/JHEP02( 2010 ) 031 , arXiv: 0911 .0167 [hep-th]
53. S. Deser , R. Jackiw , S. Templeton, Ann. Phys. 140 , 372 ( 1982 ) [Erratum-ibid . 185 ( 1988 ) 406]
54. S. Deser , R. Jackiw , S. Templeton, Ann. Phys. 185 , 406 ( 1988 )
55. S. Deser , R. Jackiw , S. Templeton, Ann. Phys. 281 , 409 ( 2000 )
56. S. Deser , R. Jackiw , S. Templeton, Phys. Rev. Lett . 48 , 975 ( 1982 )
57. W. Li , W. Song , A. Strominger , JHEP 0804 , 082 ( 2008 ). arXiv: 0801 .4566 [hep-th]
58. A. Garbarz , G. Giribet, Y. Vasquez , Phys. Rev. D 79 , 044036 ( 2009 )
59. M. Nakasone , I. Oda , Prog. Theor. Phys . 121 , 1389 ( 2009 )
60. E.A. Bergshoeff , O. Hohm , P.K. Townsend , Phys. Rev. D 79 , 124042 ( 2009 )
61. I. Oda, JHEP 0905 , 064 ( 2009 )
62. N. Ohta , Class. Quantum Gravity 29 , 015002 ( 2012 )
63. K. Muneyuki , N. Ohta , Phys. Rev. D 85 , 101501 ( 2012 )
64. Y. Vasquez, JHEP 1108 , 089 ( 2011 )
65. M. Grses , Class. Quantum Gravity 11 ( 10 ), 2585 ( 1994 )
66. M. Rooman , P. Spindel , Class. Quantum Gravity 15 , 3241 ( 1998 ). arXiv:gr-qc/9804027
67. M.J. Duff , H. Lu , C.N. Pope , Nucl. Phys. B 544 , 145 ( 1999 ). arXiv:hep-th/9807173
68. D. Israel , C. Kounnas , M.P. Petropoulos , JHEP 0310 , 028 ( 2003 ). arXiv:hep-th/0306053
69. T. Andrade, M. Banados , R. Benguria , A. Gomberoff , Phys. Rev. Lett . 95 , 021102 ( 2005 ). arXiv:hep-th/0503095
70. M. Banados , G. Barnich, G. Compere , A. Gomberoff , Phys. Rev. D 73 , 044006 ( 2006 ). arXiv:hep-th/0512105
71. D.T. Son, Phys. Rev. D 78 , 046003 ( 2008 ). arXiv: 0804 .3972 [hepth]
72. K. Balasubramanian , J. McGreevy , Phys. Rev. Lett . 101 , 061601 ( 2008 ). arXiv: 0804 .4053 [hep-th]
73. G. Giribet, Y. Vasquez , Phys. Rev. D 93 ( 2 ), 024001 ( 2016 ). arXiv: 1511 .04013 [hep-th]
74. G. Clement, Class. Quantum Gravity 26 , 105015 ( 2009 ). https:// doi.org/10.1088/ 0264 -9381/26/10/105015, arXiv: 0902 .4634 [hep-th]
75. S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1983 )
76. N. Cruz , C. Martinez , L. Pena, Class. Quantum Gravity 11 , 2731 ( 1994 ). https://doi.org/10.1088/ 0264 -9381/11/11/014, arXiv:gr-qc/ 9401025