Doctoral thesis

High Energy Particle Collision
and Energy Extraction Processnear a Rotating Black Hole
回転ブラックホール近傍における高エネルギー粒子衝突とエネルギー引き抜き過程

Kota Ogasawara

Department of Physics, Graduate School of Science,
Rikkyo University

Abstract
We consider a collision of two geodesic particles and dust thin shells with ahigh center-of-mass (CM) energy in the Bañados-Teitelboim-Zanelli spacetime.
We show that the CM energy of both the particles and shells can be arbitrarily
large if either of the colliding objects satisfies a certain critical condition.
The rest mass and energy of colliding objects are usually assumed to be
suﬃciently small compared to the mass of the background black hole. However,if the CM energy becomes comparable with the gravitational energy of the blackhole, the self-gravity of colliding objects cannot be neglected. An analyticaltreatment of the self-gravity eﬀects on the particle collision is much complicatedand diﬃcult but it is easily possible in the shell collision.
We compare the dust thin shell collision and the particle collision in order toinvestigate the eﬀects of the self-gravity of colliding objects on the high energy
collision. The self-gravity of the shells aﬀects the position of an event horizon
and it covers the high energy collisional event. Therefore, we conclude that the
self-gravity of colliding objects suppresses its CM energy and that any observer
who stands outside of the event horizon cannot observe the collision with anarbitrary high CM energy. We find a test shell limit, where the CM energy
and the eﬀective potentials for shells in the limit are very similar to the onesof particles. The test shell limit would help us to understand the eﬀect of theself-gravity of the thin shells on the collisions.
We also investigate a particle collision and energy extraction in the Kerr
spacetime. We consider the reaction of particles 1 and 2 into particles 3 and 4
in the ergoregion, where particle 3 escapes to infinity after the collision, while
particle 4 falls into the black hole possibly with negative energy due to theexistence of the ergosphere. If the energy of particle 4 is negative, the energy ofthe escaping particle can be larger than the total energy of the injected particles
because of energy conservation. The energy extraction eﬃciency is defined as
the energy ratio of the escaping particle to the injected particles.
We present an analytic formulation to investigate the energy extraction efficiency and the escape probability. We consider a collision of two particles
which started from infinity and follow geodesics in the equatorial plane. We
focus on a collision with arbitrarily large CM energy, which occurs if eitherof the colliding particles satisfies a certain critical condition. We show that
if this√ particle√is ingoing on the collision, the upper limit of the eﬃciency is(2 + 3)(2 − 2) ≃ 2.186, while if √it is bounced back before the collision, theupper limit of the eﬃciency is (2 + 3)2 ≃ 13.93.
Focusing on more general head-on collisions, where one of the two colliding particles is generated in the ergoregion and initially moves outwardly, weshow that a produced massless particle can escape to infinity with arbitrarily
large energy in the near-horizon limit of the collision point. Furthermore, if we
assume that the emission of the produced massless particles is isotropic in thecenter-of-mass frame but confined to the equatorial plane, the escape probability approaches 5/12 and almost all escaping particles have arbitrarily large
energy at infinity.

2

Contents
1 Introduction

3

2 Kerr black holes as particle accelerators and energy sources
2.1 Geodesic motion in the Kerr spacetime . . . . . . . . . . . . . . .
2.1.1 Kerr metric . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 General geodesic . . . . . . . . . . . . . . . . . . . . . . .
2.2 Kerr black holes as particle acceletators . . . . . . . . . . . . . .
2.2.1 Particle collision in the equatorial plane . . . . . . . . . .
2.2.2 BSW process . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Energy extraction process . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Penrose process . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Collisional and super-Penrose process . . . . . . . . . . .

8
8
8
10
11
11
12
13
14
15

3 High energy particle collision near a BTZ black hole
16
3.1 Particle motion in the BTZ black hole . . . . . . . . . . . . . . . 16
3.1.1 BTZ spacetime . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2 General geodesic of a particle . . . . . . . . . . . . . . . . 17
3.1.3 Motion of a critical particle . . . . . . . . . . . . . . . . . 19
3.1.4 Motion of a subcritical particle . . . . . . . . . . . . . . . 20
3.2 CM energy for particle collisions . . . . . . . . . . . . . . . . . . 21
3.2.1 General colliding particles . . . . . . . . . . . . . . . . . . 21
3.2.2 Critical angular momentum limit . . . . . . . . . . . . . . 22
4 Collision of two thin shells in the BTZ spacetime
24
4.1 Collision of two particles with vanishing angular momenta . . . . 24
4.2 Thin shell and its motion in the BTZ spacetime . . . . . . . . . . 26
4.3 Collision of two dust thin shells . . . . . . . . . . . . . . . . . . . 31
4.3.1 CM energy . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3.2 Observable CM energy . . . . . . . . . . . . . . . . . . . . 35
4.3.3 Test shell limit . . . . . . . . . . . . . . . . . . . . . . . . 35
5 Collisional Penrose process
37
5.1 Particle motion in the Kerr black hole . . . . . . . . . . . . . . . 37
5.1.1 Geodesic in the equatorial plane . . . . . . . . . . . . . . 37

1

5.2

5.3

5.1.2 Radial turning points . . . . . . . . . . . . . . . . . . . .
Particle collision and reaction . . . . . . . . . . . . . . . . . . . .
5.2.1 Particle collision on the horizon . . . . . . . . . . . . . . .
5.2.2 Near horizon and near critical behaviors . . . . . . . . . .
5.2.3 Expansion of the radial momentum . . . . . . . . . . . . .
Energy extraction eﬃciency . . . . . . . . . . . . . . . . . . . . .
5.3.1 First order of the radial momentum conservation . . . . .
5.3.2 Second order of the radial momentum conservation . . . .
5.3.3 Maximum eﬃciency . . . . . . . . . . . . . . . . . . . . .

38
39
40
41
42
43
43
44
45

6 Arbitrarily large energy extraction and escape probability
47
6.1 Particle collision and escape cone . . . . . . . . . . . . . . . . . . 47
6.2 Super-Penrose process and escape probability . . . . . . . . . . . 51
7 Conclusions

55

2

Chapter 1

Introduction
In 2009, Bañados, Silk, and West (BSW) [1] pointed out that the centerof-mass (CM) energy of two colliding particles can be arbitrarily large, if thecollision occurs near the event horizon of an extremal Kerr black hole and eitherof the colliding particles satisfies a certain critical condition. This is now called“BSW process” or “BSW collision”.
In fact, a particle collision with an arbitrarily large CM energy had alreadybeen noticed by Piran, Shaham, and Katz [2] in the study of energy extraction
from a rotating black hole by using a particle collision, which is called “collisional
Penrose process”. See Harada and Kimura [3] and Schnittman [4] for a brief
review and references therein for further details.
A lot of high energy phenomena in the universe, such as ultra-high-energy
cosmic rays and jets from AGN, have been observed. However, there are still unknown physics about these acceleration mechanisms. Terrestrial particle accelerators, such as the Large Hadron Collider, and leading mechanisms of particle
acceleration proposed by astrophysics make use of electromagnetic interaction
for charged particles.
Recently, the first detection of gravitational waves has been reported by the
LIGO Scientific and the Virgo Collaborations [5], the existence of black holes
was ensured in our universe. Since black holes have strong gravity and thisgravitational force works on both charged and neutral particles, black holes canaccelerate not only charged particles but also neutral particles. The phenomena
in a strong gravitational field near black holes such as the BSW process and thecollisional Penrose process would be important in astronomy and astrophysics.
If the BSW process works, the energy of the particle collision can reach Plankscale and it can probe the physics of Plank-scale by observing near the event
horizon black holes. In addition, if the collisional Penrose process works, black
holes can be the origins of ultra-high-energy cosmic rays and energy sources ofjets from AGN, etc.

3

After the rediscovery by Bañados et al. [1], several critical comments on the
BSW process were given in Refs. [6, 7, 8]. We pick up following relevant ones:
Although a spin parameter for astrophysical black holes has an upper bounda ≤ 0.998M , 1 the BSW process requires the extremal Kerr black hole, i.e.,a = M . The conserved energy and angular momentum of either of two colliding
particles must be fine tuned-to the critical condition. It needs arbitrarily longproper time for a particle, which satisfies the critical condition, to reach theevent horizon for a maximally rotating black hole. A self-gravity of colliding
particles and an emission of gravitational waves would significantly aﬀect the
BSW process.
The details of the BSW process have been investigated to answer the criticisms. Harada and Kimura showed that the critical condition is naturally realized in a collision of an innermost stable circular orbit particle in an extremal
Kerr black hole [10] and estimated a bounded CM energy on a particle collision
including the eﬀect of emission of gravitational waves [11]. Patil et al. [12]considered a finite CM energy of a collision of two particles with a finite proper
time. The BSW process has been further investigated in a nonequatorial planeof a Kerr black hole [13], a weak electromagnetic field [14], and the near-horizon
geometry of an extremal Kerr spacetime [15].
The particle collision with a high CM energy occurs not only in a Kerr
black hole spacetime but also in a Kerr naked singularity spacetime [16], a
Kerr-Newmann spacetime [17], a Kerr-(anti-)de Sitter spacetime [18], lowerdimensional spacetimes [19, 20, 21, 22, 23, 24], and higher dimensional spacetimes [25, 26]. As the electromagnetic counterpart of the BSW process, acharged particle collision with a high CM energy has been investigated in an extremal Reissner-Nordström spacetime [27] and in a higher-dimensional ReissnerNordström spacetime or an electrical charged Myers-Perry spacetime [28].
The rest mass and energy of each colliding particle are usually assumed to be
suﬃciently small compared to the mass of the background black hole. However,if the CM energy becomes comparable with the gravitational energy of the blackhole, the self-gravity of colliding particles cannot be neglected. The self-gravity
caused by the high-energy collision will aﬀect the BSW process and the CMenergy will be limited to a finite value. Unfortunately, it is much complicatedand diﬃcult to treat the self-gravity eﬀects on a particle collision.
Kimura et al. [29, 30, 31] considered a collision of two thin shells in a
Reissner-Nordström spacetime instead of a Kerr spacetime for simplicity. They
calculated a collision of two shells including their self-gravity and estimated theupper limit of the CM energy of the shell collision analytically.
Can we treat a shell collision in stationary and axisymmetric spacetime? The
analytical treatment of the shell collision in the Kerr spacetime is very diﬃcult[32, 33] but the diﬃculty of the technical problem depends on the dimension of
1 This is called Thorne’s bound [9]. Thorne’s bound is based on the standard accretion
disk model. The spin parameter is related to the angular momentum with respect to therotational axis as J = M a, where J and M are the angular momenta and mass parameters ofa Kerr spacetime, respectively.

4the spacetime [34, 35, 36]. Mann et al. [34] investigated the collapse of a shellin a 2 + 1 dimensional stationary and axisymmetric spacetime and they showedthat the motion of the shell in 2 + 1 dimension is tractable. Note that again
the BSW collision occurs not only in 4-dimensional spacetimes but also in lowerand higher dimensional spacetimes. Therefore, it would be worth considering acollision of two thin shells in lower and higher dimensional spacetimes.
A black hole solution in 2 + 1 dimension was obtained by Bañados, Teitelboim, and Zanelli (BTZ) [37, 38]. The BTZ black hole is considered as a typical
black hole in 2 + 1 dimension because of the existence of a no-go theorem forasymptotically flat and stationary black holes satisfying the dominant energy
condition in 2 + 1 dimensions in Einstein gravity [39]. One may suspect that thenegative cosmological constant aﬀects the particle collision, and it is very different from the BSW process in the Kerr spacetime. The eﬀect of the negative
cosmological constant on the BSW process will be negligible since the collision
with the high CM energy occurs near an extremal event horizon.
In the BTZ spacetime, particle motions [40], the BSW collision [19, 20, 21,24], gravitational perturbations induced by falling particles [41], and thermodynamics of thin shells [42, 43, 44] have been investigated.
The high CM energy collision of particles can produce high energy and/or
very massive particles. We should keep in mind that an observer distant froma black hole may not see particles with high energy and/or very massive even
if the CM energy is very large because the produced particles must be highly
red-shifted. If such energetic particles can escape to infinity, extra energy gainis necessary. In this context, it is known that the rotating energy of black holes
can be extracted. For this process, the existence of ergoregion, where the energyof the particle E 2 can be negative, plays an important role.
In 1969, Penrose [45] pointed out the energy extraction process by using
particle division, which is called “Penrose process”. Since we are interested in
the energy extraction by using the particle collision, let us consider a collisionof two particles. This process, i.e., the collisional Penrose process [2, 46], is
typically as follows.
We consider the reaction of particles 1 and 2 into particles 3 and 4 in theergoregion, where particle 3 escapes to infinity after the collision, while particle
4 falls into the black hole possibly with negative energy due to the existence ofthe ergosphere. For each particle i (i = 1, 2, 3, 4), we write E as Ei . In thiscase, if the energy of particle 4 E4 is negative, we have
E3 = E1 + E2 − E4 > E1 + E2 ,(1.1)

where we have used the energy conservation equation. The energy extraction

2 If spacetime has the time-transitional Killing vector ξ µ , the energy of particle E is definedas E ≡ −ξ µ pµ , where pµ is the 4-momentum of the particle.

5

eﬃciency η in the collisional Penrose process is defined asη≡E3energy of the escaping particle=.
total energy of the injected particles
E1 + E2(1.2)

The energy extraction (η > 1) from the black hole is possible provided E4 < 0.
Recently, the interplay between the particle acceleration and energy extraction has been intensively investigated by several authors. Bejger et al. [47]showed that the maximum eﬃciency only amounts to about 1.4 for rear-end collisions if either of the colliding particles satisfies a certain critical condition. This
result was confirmed analytically by Harada et al, [48]. However, Schnittman[49] numerically showed that the maximum eﬃciency
can reach about 14 for√head-on collisions, which is exactly given by (2 + 3)2 [50, 51, 52]. Berti, Brito,
and Cardoso [53] showed that an arbitrarily large eﬃciency is possible by moregeneral head-on collisions, which is called “super-Penrose process”. In the superPenrose process, a radially outward particle must be created near the horizon
by some preceding process. An escape probability of the produced particle has
been investigated in an extremal Kerr black hole spacetime [54] and in a Kerr
naked singularity spacetime [55].
This thesis is organized as follows:
Chapter 2
We briefly review the general geodesic motion, BSW process, and Penrose process in the Kerr black hole spacetime.
Chapter 3
We consider a particle collision in the BTZ black hole spacetime with the negative cosmological constant and angular momentum motivated by next investigations. This is based on
N. Tsukamoto, K. Ogasawara, and Y. Gong, “Particle collision with an arbitrarily high center-of-mass energy near a Bañados-Teitelboim-Zanelli black hole,”
Phys. Rev. D 96, 024042 (2017) [24].
Chapter 4
We consider a collision of two dust thin shells in the BTZ spacetime and compare
the shell collision and the particle collision in order to investigate the eﬀects ofthe self-gravity of colliding objects on the high CM energy collision. This isbased onK. Ogasawara and N. Tsukamoto, “Eﬀect of the self-gravity of shells on a highenergy collision in a rotating Bañados-Teitelboim-Zanelli spacetime,”
Phys. Rev. D 99, 024016 (2019) [56].

6

Chapter 5
We present an analytic formulation to investigate the upper limits of the energyof the escaping particle and of the energy extraction eﬃciency in the collisional
Penrose process. This is based on
K. Ogasawara, T. Harada, and U. Miyamoto, “High eﬃciency of collisional
Penrose process requires heavy particle production,” Phys. Rev. D 93, 044054(2016) [51] andT. Harada, K. Ogasawara, and U. Miyamoto, “Consistent analytic approach to
the eﬃciency of collisional Penrose process,” Phys. Rev. D 94, 024038 (2016)[52].
Chapter 6
We also present an analytic formulation to investigate an arbitrarily large energy
extraction and escape probability in the super-Penrose process. This is basedon
K. Ogasawara, T. Harada, U. Miyamoto, and T. Igata, “Escape probability ofthe super-Penrose process,” Phys. Rev. D 95, 124019 (2017) [54].
Chapter 7
We show the conclusions of this thesis.
In this thesis, we use the units in which the speed of light is unity. We set
Newton’s constant in four dimensions unity except for Chaps. 3 and 4. We set
8G = 1 in Chaps. 3 and 4 as in Sec. III in Ref. [34], where G is Newton’sconstant in three dimensions.

7

Chapter 2

Kerr black holes as particle
accelerators and energy
sourcesIn this Chapter, we investigate the general geodesic motion of the particle
in the Kerr black hole and make a brief review of the BSW process [1] and thePenrose process [45].

2.1

Geodesic motion in the Kerr spacetime

2.1.1

Kerr metric

The line element ds2 = gµν dxµ dxν in the Kerr spacetime [57] in the BoyerLindquist coordinates is given by()
2M r
4M ar sin2 θρ2 2
ds2 = − 1 − 2
dt2 −
dtdφ
+
dr
ρ
ρ2
∆
(
)
2M a2 r sin2 θ+ ρ2 dθ2 + r2 + a2 +sin2 θdφ2 ,
(2.1)
ρ2
where
ρ2 ≡ ρ2 (r, θ) = r2 + a2 cos2 θ,(2.2)∆ ≡ ∆(r) = r − 2M r + a .

(2.3)

2

2

The spacetime is parametrized by two parameters, namely mass M and spina. The spin parameter is related to the angular momentum J with respect tothe rotational axis of the black hole as a = J/M . We note that the metric,8

given by Eq. (2.1), reduces to the Schwarzschild black hole spacetime if thespin parameter vanishes, i.e., a = 0.
The Kerr spacetime is stationary and axisymmetric with corresponding two
Killing vectors ξ µ and ψ µ , where ξ µ ∂µ = ∂t and ψ µ ∂µ = ∂φ . We assume a ≥ 0
without loss of generality because taking φ → −φ eﬀectively changes the signof a. When a ≤ M is satisfied the spacetime has an event horizon at√r = r+ ≡ M + M 2 − a2 ,(2.4)
where ∆(r) vanishes at r = r+ , and it is known as an extremal black hole for
a = M . For a > M , it has a naked singularity at r = 0 and ∆(r) = 0 has noreal roots.
Although the time translation Killing vector ξ µ is timelike at infinity, it neednot be timelike everywhere outside of the event horizon. The norm of ξ µ is given
by
(
)
2M r
ξ µ ξµ = gtt = − 1 − 2,(2.5)
r + a2 cos2 θ
so ξ µ is nontimelike whenr2 − 2M r + a2 cos2 θ ≤ 0,is satisfied. The boundary of this region, i.e., the hypersurface√r = rE ≡ M + M 2 − a2 cos2 θ,

(2.6)

(2.7)

is the ergosphere. The ergosphere intersects the event horizon at θ = 0 and π.
The region, where ξ µ becomes spacelike is called ergoregion. See Fig. 2.1 forthe schematic picture.
We find that the vectorχ µ ≡ ξ µ + ΩH ψ µ ,(2.8)

is null on the event horizon, where
ΩH ≡

dφ
a
a
√
(
),
= 2
=
2
dt r=r+r+ + a2M M + M 2 − a2

(2.9)

is the angular velocity of the event horizon. Because it is a linear combinationof two Killing vectors ξ µ and ψ µ , χµ is also a Killing vector and a null generatorof the event horizon. We note a diﬀerence between a static (Schwarzschild) andstationary (Kerr) black hole. For a static black hole, ξ µ becomes null on theevent horizon. For a stationary black hole, ξ µ becomes null on the ergosphereand χµ becomes null on the event horizon.

9Figure 2.1: The schematic picture showing a side view of the Kerr black hole.
The radial
position of the event√ horizon and ergosphere are given by r+ ≡√M + M 2 − a2 and rE ≡ M + M 2 − a2 cos2 θ, respectively. The ergosphereintersects the event horizon at θ = 0 and π. The region r+ < r < rE , where theKilling vector ξ µ becomes spacelike, is called the ergoregion.

2.1.2

General geodesic

The conserved energy E and angular momentum L of a particle with the 4momentum pµ are defined as
E ≡ −gµν ξ µ pν ,(2.10)L ≡ gµν ψ µ pν ,(2.11)

andrespectively. These two conserved quantities are constant along a geodesic forthe particle. Furthermore, there exists an additional constant Q, which is known
as a Carter constant. Because the suﬃcient number of constants of motion, theequations of motion of this system are completely integrable.
By using the Hamilton-Jacobi method [58], the components of the 4-momentum10are given by[]1(r2 + a2 )P2,
p = 2 −a(aE sin θ − L) +ρ∆σ√pr = 2 R,ρθ√σpθ = 2 Θ,
ρ
[ (
)
]
1
L
aP
φ
p = 2 − aE −+,ρ∆sin2 θ
t(2.12)(2.13)(2.14)(2.15)

where σ ≡ sgn(pr ), σ θ ≡ sgn(pθ ),P ≡ (r2 + a2 )E − aL,
[
]
L2
2
2
2
2
Θ ≡ Q − cos θ a (m − E ) +,sin2 θ
[]R ≡ P 2 − ∆ m2 r2 + (L − aE)2 + Q ,(2.16)(2.17)(2.18)and m denotes a rest mass of the particle.

2.2

Kerr black holes as particle acceletators

If two particles 1 and 2 are at the same spacetime point, the energy observedby an observer whose 4-velocity is parallel to the sum of 4-momenta pµ1 and pµ2of two particles 1 and 2, respectively, at that point is the center-of-mass (CM)energy. The CM energy is defined as2≡ −gµν (pµ1 + pµ2 ) (pν1 + pν2 ) ,Ecm(2.19)

where gµν is the spacetime metric. We concentrate on a collision of two particles
that come from infinity.

2.2.1

Particle collision in the equatorial plane

We consider the situation where two particles move inward on the equatorial
plane and collide rear-end near the event horizon. In this case, the CM energy
for a near-horizon collision is given by [10, 11]2
lim Ecm= m21 + m22 +

r→r+

+

2
m21 r++ (L1 − aE1 )2 E2 − ΩH L1
2
r+
E1 − ΩH L2

2
m22 r++ (L2 − aE2 )2 E1 − ΩH L2
2(L1 − aE1 )(L2 − aE2 )
−
,
2
2
r+
E2 − ΩH L1
r+(2.20)

where Ei , Li , and mi are E, L, and m for particle i (i = 1, 2), respectively.
Therefore, the CM energy is divergent if either of particles 1 and 2 satisfies
11

a condition Ei − ΩH Li = 0. We call the condition Ei − ΩH Li = 0 a criticalcondition, and particles which satisfy the critical condition are called criticalparticles. When both the particles satisfy the critical condition, the CM energy
in the near horizon becomes finite.
For such a high energy collision to occur, the critical particle must reach thenear horizon from infinity. We concentrate on a geodesic motion of a massive
particle in the equatorial plane. From Eq. (2.13), it reduces to a simple onedimensional potential problem as
1 r 2(p ) + V (r) = 0,2(2.21)

where()
L2 − a2 E 2 − m2
M m2
M (L − aE)2E 2 − m2
V (r) = −
+
,
−
−
2
3
r
2r
r
2

(2.22)

is the eﬀective potential for the particle.
For simplicity, we consider a particle which is initially at rest at infinity, i.e.,
E = m. In this case, a particle which satisfies a condition
2r2 − M l2 r + 2M 2 (l − a∗ )2 > 0,(2.23)

can approach the event horizon from infinity, where l ≡ L/(M m) and a∗ ≡ a/M .
Solving this condition for l, we obtain
(
)
(
)
√
√
−2 1 + 1 + a∗ ≡ lL < l < lR ≡ 2 1 + 1 − a∗ .
(2.24)
We introduce a nondimensional critical angular momentum lc ≡ E/(ΩH M m)as the critical value of l. The explicit form of lc with E = m is given by)(√
2 1 + 1 − a∗.
(2.25)lc =a∗
We note that lR ≤ lc and equality holes only for a∗ = 1. This implies thatthe critical particle can approach the event horizon for a∗ = 1 but cannot fora∗ < 1.

2.2.2

BSW process

We consider a particle motion with the 4-velocity uµ and a rest mass m in theequatorial plane θ = π/2 of the Kerr black hole. The conserved specific energy
and angular momentum of the particle are given bye ≡ −gµν ξ µ uν and ℓ ≡ gµν ψ µ uν ,12

(2.26)

respectively. Using these constants of motion and the normalization conditionof the 4-velocity, gµν uµ uν = −1, the components of the 4-velocity are given by[]
1 (r2 + a2 )T
ut = 2− a(ae − ℓ) ,(2.27)
r∆1√
ur = ± 2 T 2 − ∆ [r2 + (ℓ − ae)2 ],(2.28)
r[]
1 aT
uφ = 2− (ae − ℓ) ,(2.29)
r∆
where T ≡ e(r2 + a2 ) − ℓa.
Let us consider a particle collision of two massive particles, named particles
1 and 2, with equal mass m. For both particle i (i = 1, 2), we write uµ , e, and ℓas uµi , ei , and ℓi , respectively. We assume that each particle is at rest at infinity,i.e., ei = 1. In this case, the CM energy is given by
[
2m2
2
=
Ecm
2a2 (r + M ) − 2M a(ℓ1 + ℓ2 ) − ℓ2 ℓ1 (r − 2M ) + 2(r − M )r2r∆]√√− 2M (a − ℓ1 )2 − ℓ21 r + 2M r2 2M (a − ℓ2 )2 − ℓ22 r + 2M r2 .
(2.30)In the previous subsection, we have seen that the CM energy diverge if eitherof particles 1 and 2 satisfies the critical condition. However, the critical particle
can approach the event horizon only when a∗ = 1, i.e., the extremal Kerr blackhole spacetime. In the extremal case, the CM energy at the horizon r+ becomes
simpler as√√ℓ2 − 2Mℓ1 − 2M
lim Ecm = 2m
(2.31)+,
r→r+ℓ1 − 2Mℓ2 − 2M
and the critical condition is expressed as ℓi − 2M = 0.
Here we showed that the CM energy can be arbitrarily large, even if the massand energy of two colliding particles are finite and they are at rest at infinity.
This process is so-called the “BSW process.”

2.3

Energy extraction process

Since the Killing vector ξ µ becomes spacelike inside the ergoregion, the conserved energy E ≡ −ξ µ pµ can be either positive or negative. We note thatparticles with negative energy can exist only inside of the ergoregion and E < 0means the energy that would be observed at infinity if the particle could be
carried there. This existence of the negative energy particle can use the energy
extraction from the Kerr black hole.

13

Figure 2.2: The schematic picture of the Penrose process for extracting energy
from the Kerr black hole.

2.3.1

Penrose process

In 1969, Penrose [45] pointed out the energy extraction process by using particle
division. Let us consider the following situation. A particle with E0 > 0 comesfrom infinity and it decays into two particles inside the ergoregion. One of theparticles falls into a black hole with negative energy E4 < 0 while the otherparticle escapes to infinity with E3 > 0. See Fig. 2.2. From the conservation ofenergy E0 = E3 + E4 , we obtain
E3 = E0 − E4 > E0 .

(2.32)

Because the energy of the escaping particle E3 is larger than the energy of theinitial particle E0 , the black hole loses own energy, i.e., the energy of the blackhole is extracted. This process is so-called “Penrose process.”
The Penrose process has an upper limit of the extraction energy because it
makes spin-down of the Kerr black hole. As a result, when the spin parametera becomes zero, the ergoregion is no longer present, and no further the energy
extraction occurs. To see this upper limit of the extraction energy in detail, we
use the fact that the Killing vector χµ , defined by Eq. (2.8), is timelike justoutside the event horizon. Hence for any particle, which includes all negative
energy particles, the combination −χµ pµ must be positive−χµ pµ = − (ξ µ + ΩH ψ µ ) pµ = E − ΩH L > 0.

(2.33)

Thus, we obtain
L<

E
.
ΩH

14

(2.34)

This implies that L must be negative if E < 0. Since the negative energy particle
carries negative angular momentum to the black hole, the black hole will loseangular momentum during the Penrose process. The angular momentum of theblack hole will go to zero, i.e., a → 0, the ergoregion will disappear, and finally,
the Penrose process will stop [59, 60].
The quantitative energy ranges of the decaying particle in the Penrose process is shown in Refs. [61, 62]. We consider a case where the particle with initialenergy E and mass m decays into a particle with energy E ′ and mass m′ on theevent horizon at the equatorial plane of the extremal Kerr black hole. In thiscase, we have
√
√
E
E2
E′
E
E2
γ − γv
+
1
≤
≤
γ
+
γv
+ 1,
(2.35)
mm2
m′mm2
where γ and v are the Lorentz factor and velocity of the decaying particle in
the rest frame of the original particle. When the original particle is at rest at
infinity (E/m = 1) and decays
into two particles with m′ , the lower energy limit√
can be negative if v > 1/ 2. This is equivalent to two decaying particles eachwith the rest mass of only 35% of the original particle. If the original particle
decay into two massless particles, we obtain
√
1+ 2
′
E ≤
E ≃ 1.207E.
(2.36)
2This is the upper limit of the energy extraction in the Penrose process.

2.3.2

Collisional and super-Penrose process

A bit modified process of the Penrose process called the collisional Penrose
process [2, 46], in which two particles collide in the ergoregion instead of a single
particle decay. The maximum energy extraction eﬃciency√ can reach about 14
for head-on collisions [49], which is exactly given by (2 + 3)2 [50, 51, 52]. Moreeﬃcient energy extraction is possible by more general head-on collisions, whichis called the super-Penrose process [53]. In this thesis, we distinguish between
the two processes according to the origin of the colliding particles described
below:
Collisional Penrose process: Two colliding particles come from infinity
and collide in the ergoregion. We consider both a rear-end collision and a headon collision. This process is discussed in Chap. 5.
Super-Penrose process: One of the two collision particles is generated
in the ergoregion and initially moves outwardly. The trajectory of this particle
cannot be realized with particles which came from infinity. In the super-Penrose
process, a radially outward particle must be created near the horizon by somepreceding process but we will not discuss the details. This process is discussed
in Chap. 6.

15

Chapter 3

High energy particle
collision near a BTZ blackhole
We consider a particle collision with an arbitrarily large CM energy near arotating BTZ black hole motivated by further investigations for the eﬀect of theself-gravity on a high energy collision, see Chap. 4. We obtain the CM energyof two general colliding geodesic particles in the BTZ black hole spacetime. We
show that the CM energy of two ingoing particles can be arbitrarily large on theevent horizon if either of the two particles has a critical angular momentum andthe other has a noncritical angular momentum. We also show that a motionof a particle with a subcritical angular momentum is allowed near the extremal
BTZ black hole and that the CM energy for a rear-end collision at a point canbe arbitrarily large in a critical angular momentum limit.
One may suspect that the negative cosmological constant aﬀects the particle collision, and it is diﬀerent from the BSW process. However, the eﬀect of
the negative cosmological constant on the collision will be negligible since thecollision with a high CM energy occurs near an extremal event horizon.
In this chapter, we use the units in which the speed of light and 8G are unity
as in Sec III in Ref. [34], where G is Newton’s constant in three dimensions.

3.1

Particle motion in the BTZ black hole

3.1.1

BTZ spacetime

The line element in the BTZ spacetime [37, 38] is given by
ds2 = −f (r)dt2 +dr2
2+ r2 [dφ − Ω(r)dt] ,
f (r)
16

(3.1)

where
r2
J2
+ 2,
2
ℓ
4r
J
gtφ
= 2,
Ω(r) ≡ −
gφφ
2r
√
1
ℓ≡
.
−Λ
f (r) ≡ −M +

(3.2)
(3.3)
(3.4)

Here M , J, and Ω(r) are the mass, angular momentum, and angular velocityof the spacetime, respectively, and ℓ 1 is the scale of a curvature related to thenegative cosmological constant Λ < 0. Without loss of generality, we assume
that the angular momentum J is nonnegative. In this thesis, we concentrate on
the case where M is positive.
If J ≤ ℓM is satisfied, the BTZ spacetime is a black hole spacetime and f (r)vanishes at
v (
)
√
u
uM
J2
t
1± 1− 2 2 ,(3.5)
r = r± ≡ ℓ2ℓ M
where r+ and r− correspond to outer and inner horizons, respectively. The BTZ
black hole with J = ℓM is known as an extremal black hole and the outer andinner horizons of the extremal BTZ black hole coincide,√M
r+ = r− = ℓ
.
(3.6)
2
If J > ℓM is satisfied, the BTZ spacetime is a naked singularity spacetime andf (r) = 0 has no real roots.
Using the horizon radius r± , f (r), Ω(r), M , and J are expressed as2
2))(r2 − r−(r2 − r+r+ r−
, Ω(r) =
,
2
2
ℓ r
ℓr2
r2 + r2
2r+ r−
M = + 2 − , and J =,ℓℓf (r) =

(3.7)

respectively.

3.1.2

General geodesic of a particle

The BTZ spacetime is stationary and axisymmetric with corresponding two
Killing vectors ξ µ and ψ µ , where ξ µ ∂µ = ∂t and ψ µ ∂µ = ∂φ . The conserved
energy E and angular momentum L of a particle with 3-momentum pµ are
defined as
E ≡ −gµν ξ µ pν ,(3.8)

1 In this chapter, ℓ denote the scale of a curvature rather than a specific angular momentum.

17

and
L ≡ gµν ψ µ pν ,(3.9)

respectively. These two conserved quantities are constant along a geodesic forthe particle.
From the normalization condition of the 3-momentum pµ pµ = −m2 and Eqs.
(3.8) and (3.9), the components of the 3-momentum are given by
S(r)
,
f (r)
√
pr = σ R(r),
pt =

pφ =

(3.10)(3.11)Ω(r)S(r)
L
+ 2,
f (r)
r

(3.12)

where σ ≡ sgn(pr ),S(r) ≡ E − Ω(r)L,
(
)
L2
2
2
R(r) ≡ S (r) − m + 2 f (r),r

(3.13)(3.14)and m denotes a rest mass of the particle.
We assume the forward-in-time condition pt ≥ 0 for the particle motion.
This condition implies that the coordinate t increases along the particle motion.
We call a condition S(r+ ) = 0 a critical condition. The critical condition isrewritten as
E − ΩH L = 0,

(3.15)

whereΩH ≡ Ω(r+ ) =

J
r−
=
,
2
2r+
ℓr+

(3.16)

is the angular velocity of the horizon. We define a critical angular momentum
Lc as
Lc ≡

ℓr+
E.
r−

(3.17)

We call a particle a critical particle if it has the critical angular momentum, forwhich satisfies Eq. (3.15). Accordingly, we call a particle with L < Lc (L > Lc )a subcritical (supercritical) particle.
The particle motion is restricted to a region where R(r) is nonnegative. The
particle can exist at the event horizon because of R(r+ ) = S 2 (r+ ) ≥ 0. If
particle has a mass, it cannot exist at infinity because
lim R(r) = lim −

r→∞

r→∞

18

m2 r 2< 0.
ℓ2(3.18)

For a massless particle, R(r) is expressed as
[
]
L 2
L2
L
2
2
R(r) = E − 2 + 2 −2Er+ r− + (r+ + r− )ℓℓrℓ

(3.19)

If −Eℓ ≤ L ≤ Eℓ is satisfied, R(r) is nonnegative in the region r+ ≤ r. On theother hand, if L < −Eℓ or Eℓ < L is satisfied, R(r) is nonnegative in the regionr+ ≤ r ≤ r0 , where√ℓS 2 (r+ )r0 ≡ r+ 1 − 2 2> r+ .
(3.20)
E ℓ − L2

3.1.3

Motion of a critical particle

We investigate the motion of a particle with the critical angular momentum
Lc . Substituting Eq. (3.17) into Eqs. (3.10), (3.11), and (3.12), we obtain thecomponents of the 3-momentum of the critical particle asEℓ2
2 ,r 2 − r−√pr = σ R(r),Eℓr+pφ = 22 )r ,(r − r−
−
pt =

where
R(r) = Rc (r) ≡ −2 [ 2 22
2 ]r2 − r+E (r+ − r−
) m2 (r2 − r−
)
+
.
2
2
2
r
r−
ℓ

(3.21)(3.22)(3.23)(3.24)

The above expression shows Rc (r+ ) = 0 and Rc (r) < 0 in the region r+ < r.
Therefore, the critical particle cannot exist outside of the event horizon r+ < r.
The derivative of Rc (r) with respect r is given by
[
]
2
2
2
4
−E 2 ℓ2 (r+− r−) + r−
m2
2m2 r 2r+′
Rc (r) = − 2 +,(3.25)
2 r3ℓℓ2 r−
where a prime denotes a derivative with respect to r and it becomesRc′ (r+ ) = −

2
2
2
m2 )
2(r+
− r−
)(E 2 ℓ2 + r−
≤ 0,
2
2
ℓ r+ r−(3.26)

on the event horizon.
In the extremal case, i.e., r+ = r− , we obtain2 2
m2 (r2 − r+
)
,
2
2
ℓ r
2
2
2m2 (r2 + r+)(r2 − r+)Rc′ (r) = −
.
2
3
ℓ r

Rc (r) = −Therefore, we obtain Rc (r+ ) = Rc′ (r+ ) = 0 on the event horizon.
19(3.27)(3.28)

3.1.4

Motion of a subcritical particle

We consider the motion of a particle with the subcritical angular momentum
L = Lc −ℓr+ℓr+ (E − δ)
δ=
,
r−
r−

where δ is a positive constant. From Eq. (3.14), R(r) is expressed as
[ (
) ( 2
) ]
2
2
2
r2 δ 2E r2 − r+
− r − r+− r−
δR(r) = Rc (r) + +
.
2 r2
r−

(3.29)(3.30)The particle with the subcritical angular momentum can exist on the event
horizon and near the black hole sinceR(r+ ) = δ 2 > 0.
The derivative of R(r) with respect r is given by
) ]
( 2
[
2
2
2
δ
+ r−
− r+
δ 2Er+
2r+′′
R (r) = Rc (r) +
,
2
3
r− r
and it becomes2δ
R (r+ ) = Rc′ (r+ ) +

[

′

) ]
( 2
2
2
δ
+ r−
− r+
2Er+
2
r+ r−

(3.31)(3.32)(3.33)

on the event horizon.
From Eqs. (3.26) and (3.33), Rc′ (r) is a quadratic equation with respect to δ.
After some straightforward calculation, we obtain the following results. WhenE < Em , Rc′ (r) is negative, where√4 − r4
m r+
−
Em ≡
.
(3.34)ℓr−
When E ≥ Em , Rc′ (r) is nonnegative if and only if δL ≤ δ ≤ δR is satisfied.
Here δL and δR are defined as√()
2
2 − r 4 − r 4 m2
Eℓr+
− r− E 2 ℓ2 r−
+
−
( 2
)
δL ≡
,
(3.35)
2
ℓ r+
+ r−
and
δR ≡

2
Eℓr+
+ r−

√
(
)
2 − r 4 − r 4 m2
E 2 ℓ2 r−
+
−
( 2
)
,
2
ℓ r+
+ r−

(3.36)

respectively. We note that an inequality 0 ≤ δL ≤ δR ≤ E is satisfied bydefinitions.
20

If E ≥ Em and δL ≤ δ ≤ δR are satisfied, R′ (r) = 0 has only two real solutions. Either of the two solutions is positive and the other solution is negative.
The positive solution is given by(r = rm ≡ℓ2 R′ (r+ )1+
2r+ m2) 41

r+ ≥ r+ .

(3.37)As r increases from r+ to infinity, R(r) begins with R(r+ ) = δ 2 , monotonically increases to a local maximum R(rm ) > δ 2 at r = rm , and monotonicallydecreases to negative infinity.
In the extremal BTZ black hole case, i.e., r+ = r− , we obtain
Em = 0, δL = 0, and δR = E,

(3.38)

rm → r+ , R(r+ ) → 0, and R′ (r+ ) → 0,

(3.39)

furthermore

in the critical angular momentum limit δ → 0.

3.2

CM energy for particle collisions

We consider a collision of two particles, named particles 1 and 2, in the BTZ
spacetime. For both particles i (i=1,2), we write pµ , m, E, L, and σ as pµi ,mi , Ei , Li , and σi , respectively. From Eq. (2.19), the CM energy of the twoparticles is given by√
S1 (r)S2 (r) − σ1 σ2 R1 (r)R2 (r) 2L1 L2
2
2
2
Ecm (r) = m1 + m2 + 2−,(3.40)f (r)r2
where Si (r) and Ri (r) are defined as
(
)
r+ r−
L2i
2
2
Si (r) ≡ Ei −Li and Ri (r) ≡ Si (r) − mi + 2 f (r),ℓr2
r(3.41)

respectively.

3.2.1

General colliding particles

We consider a rear-end collision of two ingoing particles, i.e., we choose σ1 =σ2 = −1. Since both the numerator and denominator of the third term in Eq.
(3.40) vanish at the horizon, we use l’Hopital’s rule to estimate it and we obtain
√
(
(
)
)
S1 (r)S2 (r) − R1 (r)R2 (r)S2 (r+ ) L21
S1 (r+ ) L22
2
2
=
lim 2
2 + m1 + S (r )2 + m2 .
r→r+
f (r)S1 (r+ ) r+r+
2 +(3.42)

21

Therefore, the CM energy in the near horizon r → r+ is obtained as()()
S1 (r+ ) L22
2L1 L2S2 (r+ ) L21
2
2
2
+
m
+
+
m
lim Ecm(r) = m21 + m22 +
.
1
2 −
2
2
r→r+
S1 (r+ ) r+
S2 (r+ ) r+
r2
(3.43)
This show that the CM energy of the rear-end collision can be arbitrarily large
if and only if either of the colliding particles satisfies the critical condition.
When both the particles satisfy the critical condition, the CM energy in thenear horizon r → r+ becomes finite as
(
)
(
)
E2
E1
2
lim Ecm(r) = 1 +m21 + 1 +m22 .
(3.44)
r→r+
E1
E2

3.2.2

Critical angular momentum limit

Next, we consider a rear-end collision of two ingoing particles in the extremal
BTZ black hole. We assume that particle 1 has a subcritical angular momentum
L1 = ℓ(E1 − δ) ≤ Lc1 ,(3.45)

where δ satisfies 0 = δL1 ≤ δ ≤ δR1 = E1 , and particle 2 has a subcriticalangular momentum L2 < Lc2 . We consider a particle collision at r = rm where()1ℓ2 R1′ (r+ ) 4
rm = 1 +r+ ≥ r+ .
2r+ m21(3.46)

Here, δL1 , δR1 , and R1′ (r) are δL , δR , and R′ (r) for particle 1, respectively, and
Lci is the the critical angular momentum for particle i.
From Eq. (3.40), the CM energy of the rear-end collision at r = rm in thecritical angular momentum limit δ → 0 is given by2(rm ) = m21 + m22 + 2Γ −lim Ecm

δ→0

2E1 L2 ℓ,
2r+(3.47)

where Γ is defined as√
S1 (rm )S2 (rm ) − R1 (rm )R2 (rm )Γ ≡ lim.
δ→0f (rm )

(3.48)

Both the numerator and denominator of Γ vanish in the critical angular momentum limit. Using l’Hopital’s rule with respect to δ, Γ is expressed as)(
S2 (rm ) ˙Ṙ1 (rm ),Γ = limS1 (rm ) − √(3.49)
δ→0 f˙(rm )2 R1 (rm )where a dot denotes a derivative with respect to δ. Since both the numerator anddenominator of the second term in the above equation vanish, we use l’Hopital’s
22rule again to estimate it and we obtain, after a straightforward calculation,√√Ṙ1 (rm )R̈1 (rm )
lim √
= lim
= lim S˙1 (rm ),(3.50)
δ→0 2 R1 (rm )
δ→0
δ→0
2
where
lim S˙1 (rm ) = 1 +δ→0

E12 ℓ22 m2 .
r+2(3.51)

Thus, Γ is obtained as
(
)
√
2
m41
S2 (rm ) S˙1 (rm ) − S˙1 (rm ) r+
Γ = limδ→0

E12 ℓ2 δ→ ∞.

(3.52)

Therefore, the CM energy Ecm (rm ) of the rear-end collision at the point r = rm
diverges in the critical angular momentum limit δ → 0.

23

Chapter 4

Collision of two thin shells
in the BTZ spacetime
We consider a collision of two dust thin shells with a high CM energy including their self-gravity in a rotating BTZ spacetime. The shells divide the BTZ
spacetime into three domains and the domains are matched by the DarmoisIsrael junction conditions. We treat only the collision of two shells which corotate with a background BTZ spacetime because of the junction conditions. The
counterpart of the corotating shell collision is a collision of two particles withvanishing angular momenta. We compare the dust thin shell collision and theparticle collision in order to investigate the eﬀects of the self-gravity of colliding
objects on the high CM energy collision. We show that the self-gravity of theshells aﬀects the position of an event horizon and it covers the high-energy collisional event. Therefore, we conclude that the self-gravity of colliding objects
suppresses its CM energy and that any observer who stands outside of the event
horizon cannot observe the collision with an arbitrary high CM energy.
In this chapter, rH denote the horizon radius instead of r+ and we use theunits in which the speed of light and 8G are unity as in Sec III in Ref. [34],where G is Newton’s constant in three dimensions.

4.1

Collision of two particles with vanishing angular momenta

First, we consider a collision of two particles with vanishing angular momenta
as the counterpart of the shell collision.
The angular velocity of a particle ω(r) is defined as
ω(r) ≡

pφ
dφ
= t.
dt
p

24

(4.1)

When a particle has a zero conserved angular momentum L = 0, from Eqs.
(3.10), (3.12), and (4.1), the angular velocity of the particle coincides with theangular velocity of the spacetime, i.e., ω(r) = Ω(r). This means that the particle
with L = 0 corotates with the background spacetime.
From Eq. (3.11), the energy equation which describes the radial motion ofof the particle is given by
(

dr
dτ

)2
+ V (r) = 0,

(4.2)

where V (r) is the eﬀective potential for the radial motion of the particle and τ
is its proper time. Here we have used relations pµ = muµ and uµ = dxµ /dτ ,where m and uµ are the rest mass and 3-velocity, respectively. The particle
motion is restricted to the region where V (r) ≤ 0. Using a dimensionless radial
coordinate x ≡ r/ℓ, the eﬀective potential with the specific energy e ≡ E/m
and the position of the event horizon are expressed as
V (x) = −e2 + f (x) = x2 − M − e2 +

j2
,
4x2

(4.3)

and
√
xH ≡

H

r
=
ℓ

√

M 2 − j2
,
2

(4.4)

j2
J
, j≡ .
4x2
ℓ

(4.5)

M+

respectively, wheref (x) = x2 − M +As x increases from 0 to infinity, V (x) and f (x) begin√ with infinity, monotonically decrease to a local minimum at x = xm ≡ j/2, and monotonicallyincrease to infinity. The positive solutions of V (x) = 0 are given by√√
M + e2 ± (M + e2 )2 − j 2
±
x=x ≡
.
(4.6)
2
Therefore, a motion of the particle is restricted to the region x− ≤ x ≤ x+ . We
notice that a relationx− ≤ xm ≤ xH ≤ x+ ,(4.7)

is satisfied and we obtain xm = xH (xH = x+ ) in the extremal (critical) caseand x− = xm = xH = x+ in the extremal and critical case.
We consider a collision of two particles, named particle 1 and 2, with vanishing conserved angular momenta L1 = L2 = 0. For both particle i (i = 1, 2),25

we write e, L, m, σ, V (r), and x± as ei , Li , mi , σi , Vi (r), and x±
i , respectively.
The CM energy Ecm (x) of the particles is given by√e1 e2 − σ1 σ2 V1 (x)V2 (x)
2
(4.8)
Ecm
(x) = m21 + m22 + 2m1 m2.
f (x)
We consider a rear-end collision, i.e., σ1 = σ2 = −1, and assume that a collisionoccur in a region xH ≤ x ≤ x+
i where is seen by an observer who stands at theoutside of the horizon xH .
In this case, as x increases from xH to x+
i , the CM energy begins with
√
(
)
e2
e1
H
2
2
Ecm (x ) = m1 + m2 + m1 m2
+
,
(4.9)
e1
e2
and monotonically increases to√Ecm (x+i )=m21 + m22 + 2m1 m2

e1 e2
.
e2i

(4.10)

Here we have used l’Hopital’s rule to estimate Ecm (xH ). We get Ecm (x) =m1 + m2 if the both particles have a same specific energy e1 = e2 .
From Eq. (3.15), the critical condition for particle i (i = 1, 2) with Li = 0
is given by
Ei = 0, i.e., ei = 0.

(4.11)We are interested in the collision of the inner particle 1 with a critical limit e1 →0 and the outer particle 2 which is not critical in the extremal BTZ spacetime
since the collision will correspond with the BSW collision in the extremal Kerr
spacetime [1] and in the extremal Reissner-Nordström spacetime [27]. In thiscase, the collisional point must be x → xH + 0 because of inequality (4.7) andHx±
1 → x ± 0 and the CM energy Ecm diverges there.
′′′
We note that a critical particle has V1 (xH ) = V1 (xH ) = 0 and V1 (xH ) > 0,
where a prime denotes a derivative with respect to x, in the extremal BTZ′′′
spacetime while one has V1 (xH ) = V1 (xH ) = 0 and V1 (xH ) < 0 in an extremal
Kerr black hole spacetime and an extremal Reissner-Nordström spacetime. The′′
positive sign of V1 (xH ) is caused by the negative cosmological constant and it
will not aﬀect on the BSW collision strongly since the collision happens on thehorizon.

4.2

Thin shell and its motion in the BTZ spacetime

In this section, as a preparation to study a shell collision, we review the DarmoisIsrael junction conditions [60, 63, 64] and a motion of a dust thin shell in the
BTZ spacetime [34].
26

Figure 4.1: The schematic picture of the BTZ spacetime divided into two domains D1 and D2 by the hypersurface Σ. Both domains DA (A = 1, 2) are the
BTZ spacetime with the same scale of the curvature ℓ, but diﬀerent masses MA
and angular momenta JA . A thin shell on the hypersurface Σ has a proper mass
µ and specific energy E ≡ (M2 − M1 )/µ.

We consider a two-dimensional hypersurface Σ which divides the BTZ spacetime into an interior domain D1 and an exterior domain D2 . For simplicity, we
assume that both domains DA (A = 1, 2) are the BTZ spacetime with the same
scale of the curvature ℓ, but diﬀerent masses MA and angular momenta JA , wereMA and JA are M and J for DA , respectively. See Fig. 4.1 for the schematic
picture of the BTZ spacetime divided into two domains D1 and D2 by the hypersurface Σ. Hereinafter, we use xµ denoting coordinates in every domain andy i denoting same coordinates in both the side of Σ for simplicity.
The projection operator from the ambient spacetime to Σ is defined aseµAi ≡

∂xµA
.
∂y i

(4.12)The induced metric on Σ and the extrinsic curvature are defined as
A µ ν
hij ≡ gµνeAi eAj ,(4.13)AKij≡ eµAi eνAj ∇Aµ nν ,(4.14)

and27

respectively, where ∇Aµ is the covariant derivative within DA and nµ is the unit
normal vector, which is directed from D1 to D2 . The first and second junction
conditions are given by[hij ] = 0 and [Kij ] = 0,

(4.15)

respectively. Here, the bracket is defined as[Ψ] ≡ Ψ(D2 )|Σ − Ψ(D1 )|Σ ,(4.16)

where Ψ is any quantity defined on the both sides of Σ.
We assume that the thin shell is at r = R(t) since we are interested in a
motion of the shell on Σ. It is useful to convert to a frame that is comoving withthe shell, i.e., a corotating frame, by making the following change of azimuth
coordinatesJAdt.
2R2 (t)(4.17)

[
(
) ]2
dr2
JA
1
1
+ r2 dϕ +−dt ,fA (r)
2 R2 (t) r2(4.18)

dφ → dϕ +The metric in DA then becomesAdxµ dxν
ds2A = gµν= −fA (r)dt2 +wherefA (r) ≡ −MA +

2
r2
JA
+
.
ℓ2
4r2

(4.19)The trajectory of the shell is parametrized by the proper time τ ast = T (τ ) and r = R(T (τ )) = R(τ ).

(4.20)

From the first junction condition [hij ] = 0, the induced metric hij on Σ is givenbyds2Σ = hij dxi dxj = −dτ 2 + R2 (τ )dϕ2
[
]
Ṙ2
2
= −fA (R)Ṫ +dτ 2 + R2 (τ )dϕ2 ,fA (R)
where a dot denotes a derivative with respect to τ . This implies√fA (R)Ṫ = Ṙ2 + fA (R) ≡ βA .

(4.21)(4.22)The induced basis vectors eµi and the unit normal vector nµ to Σ are givenby()
eµτ = Ṫ , Ṙ, 0 , eµϕ = (0, 0, 1),28(4.23)

and()
nµ = −Ṙ, Ṫ , 0 ,(4.24)

respectively. Substituting the above expressions into Eq. (4.14), we obtain theA
components of the extrinsic curvature and its trace K A ≡ hij Kij
asKτAτ = −

β˙A
JA
A
, Kϕϕ
= RβA , KτAϕ =,2R
Ṙ

(4.25)

βA
β˙A
+
,
R
Ṙ

(4.26)

andKA =respectively. We note that the second junction condition [Kij ] = 0 is violated
unless we consider a trivial case. Therefore we introduce a thin shell on Σ
following equations
πSij = − ([Kij ] − hij [K]) ,(4.27)

where Sij is the surface stress-energy tensor of the thin shell. We assume thatthe thin shell is a dust thin shell, i.e., the surface stress-energy tensor Sij isgiven by
Sij = ρui uj ,(4.28)

where ρ and ui = (−1, 0) are the surface energy density and the 2-velocity ofthe shell, respectively.
By using Eqs. (4.25), (4.26), and (4.28), we write down Eq. (4.27) andobtain[β] + πρR = 0,(4.29)[β̇] = 0,[J] = 0.

(4.30)(4.31)

From Eqs. (4.29) and (4.30), we obtaind(πρR) = 0.
dτ(4.32)

Therefore, we can define a constant µ as
µ ≡ 2πρR.

(4.33)

This constant µ is interpreted as the proper mass of the shell and it is conserved
along a trajectory. We define the specific energy E of the shell as
E≡

M2 − M1
[M ]
=
.
µ
µ
29

(4.34)

We assume that the proper mass µ and specific energy E are positive. Thisimplies that the masses satisfy the relation M1 < M2 . From Eq. (4.31), thedeviation of the angular momentum must be zero, i.e.,
J ≡ J1 = J2 .

(4.35)

Employing expression Eq. (4.22), we obtain an equation for the radial motion of the shell as()2dR+ V (R) = 0,(4.36)dτ
where V (R) is the eﬀective potential of the shell motion. When we define thedimensionless parameter x ≡ R/ℓ, the eﬀective potential is expressed as( 2)j2
M2 + M1µ2
2
2
V (x) = −Z + f (x) = x −++ E + 2,(4.37)
16
2
4x
where
Z≡

µ
j2
− E, f (x) ≡ x2 − M2 + 2 .
4
4x(4.38)As x increases from 0 to infinity, V (x) and f (x) begin√ with infinity, monotonically decrease to a local minimum at x = xm ≡ j/2, and monotonicallyincreases to infinity. The positive solutions of V (x) = 0 are given by√√
B ± B2 − j2±x=x ≡(4.39),
2
where
B ≡ Z 2 + M2 =µ2
M2 + M1++ E 2.
162(4.40)

Therefore, the shell motion is restricted to the region x− ≤ x ≤ x+ .
Here we compare the eﬀective potential of a thin shell and particle since thecounterpart of the corotating shell motion is a motion of a particle with vanishingangular momentum. From Eqs. (4.3) and (4.37), the eﬀective potential of theparticle VP (x) and shell VS (x) are given by
VP (x) = x2 − M − e2 +and(VS (x) = x2 −

j2
,
4x2

µ2
M2 + M1
+
+ E2
16
2

(4.41)

)
+

j2
.
4x2

(4.42)

By comparing the above expressions, we can see that the diﬀerences between
VS (x) and VR (x) are regarded as the self-gravity interaction terms.
30

If we take limits µ → 0 and M1 → M ≡ M2 with E = (const) ̸= 0, VS (x)becomeVS (x) → x2 − M − E 2 +

j2
.
4x2

(4.43)

Since we can see that the above expression has almost the same as the eﬀective
potential of the particle, we may call these limits a test shell limit.

4.3

Collision of two dust thin shells

Here we investigate a collision of two dust thin shells in the BTZ spacetime.
We assume that shell 1 and shell 2 are on an inner hypersurface Σ1 and anouter hypersurface Σ2 , respectively. These two hypersurfaces divide the BTZ
spacetime into an interior domain D1 , a middle domain D2 , and an exterior
domain D3 . We assume that all domains DA (A = 1, 2, 3) are the BTZ spacetime
with same ℓ, but diﬀerent masses MA , where MA are masses in DA . From thejunction condition Eq. (4.31), all the domains have the same j ≡ J/ℓ, see Fig.
4.2. As in the Sec. 4.2, we assume M1 < M2 < M3 below. We consider fivecases according to the value of j as shown in Table 4.1: case I (II) for j < M1
(j = M1 ), case III for j = M2 , and case IV (V) for j = M3 (j > M3 ).
case

D1

D2

D3

j ≡ J/ℓX ≡ f (xm )

I

S

S

S

j < M1

X<0

II

E

S

S

j = M1

X<0

III

O

E

S

j = M2

X=0

IV

O

O

E

j = M3

X>0

V

O

O

O

j > M3

X>0

Table 4.1: Five cases of the spacetime according to the value of j ≡ J/ℓ. Symbols
S, E, and O denote the sub-extremal, extremal, and over-spinning spacetime,
respectively. X is defined as the minimum of f2 (x).
When j ≤ MA is satisfied, an observer in DA seems to have the event horizon
at x = xH
A , where√√
MA + MA2 − j 2H
xA =.
(4.44)
2
We note that a position of xH
A is not always in DA . It depends on where shells
are located.

31

Figure 4.2: The schematic picture of the BTZ spacetime divided into threedomains D1 , D2 , and D3 by the two hypersurfaces Σ1 and Σ2 . All domains
DA (A = 1, 2, 3) are the BTZ spacetime with diﬀerent masses MA but the sameangular momentum J. A thin shell a (a = 1, 2) on the hypersurfaces Σa has aproper mass µa and specific energy Ea ≡ (Ma+1 − Ma )/µa .

The eﬀective potential of shell a (a = 1, 2) is given by
Va (x) = −Za2 + f (x),(4.45)

whereµ2µ1− E1 , Z2 ≡+ E2 ,
42
j2
f (x) ≡ f2 (x) = x2 − M2 + 2 .
4xZ1 ≡(4.46)(4.47)

Here µa and Ea are the proper mass and specific energy for shell a, respectively.
As x increases from 0 to infinity, Va (x) begins with infinity, monotonicallydecreases to a local minimum
Va (xm ) = −Za2 + X,

(4.48)

X ≡ f (xm ) = j − M2 ,(4.49)

where√at x = xm ≡ j/2, and monotonically increase to infinity. We note that xm
for shell 1 and 2 are the same.

32

A motion of shell a is restricted to the region where the eﬀective potential
Va (x) is nonpositive. In order for such the region to exist, the condition
X ≤ Za2 ,(4.50)

should be satisfied. If the above condition is satisfied, Va (x) = 0 has two positive
solutions x = x±
a , where√√
Ba ± Ba2 − j 2±x = xa ≡,(4.51)
2andBa ≡ Za2 + M2 .

(4.52)+
Therefore, the motion of shell a is restricted to the region x−
a ≤ x ≤ xa .
In Sec. 4.1, we have seen that the critical particle is important in order torealize a particle collision with high CM energy. In this sense, we consider acritical condition for a shell. We can easily show that′ H
V1 (xH2 ) = V1 (x2 ) = 0,(4.53)

is satisfied if and only if Z1 = 0 and j = M2 are satisfied. Therefore, we call acondition
Z1 = 0,(4.54)a critical condition for shell 1.

4.3.1

CM energy

The CM energy Ecm (x) of two shells is given by [29]2(x) ≡ −gµν (µ1 U1µ + µ2 U2µ ) (µ1 U1ν + µ2 U2ν )Ecm()
Ṙ1 Ṙ2
2
2= µ1 + µ2 + 2µ1 µ2 Ṫ1 Ṫ2 f (x) −,f (x)(4.55)

where gµν is the metric in D2 ,)() ( |Z |√a
Uaµ = Ṫa , Ṙa , 0 =, σa −Va (x), 0 ,f (x)(4.56)

is the 3-velocity of the shell a, and σa ≡ sgn(Ṙa ) = ±1.
We concentrate on a rear-end collision of two dust thin shells. In this case,we should choose σ1 = σ2 = −1 and the CM energy is expressed as√|Z1 |Z2 − V1 (x)V2 (x)2
2
2,(4.57)Ecm (x) = µ1 + µ2 + 2µ1 µ2
f (x)
33

We note that if |Z1 | = Z2 is satisfied, the CM energy becomes the total mass ofthe two shells and it is constant with respect to x, i.e., Ecm (x) = µ1 + µ2 .
In order for shell 2 to catch up with shell 1, Ṙ22 − Ṙ21 ≥ 0, i.e., V1 (x) ≥ V2 (x),should be satisfied at a collision point. From the above condition, we obtain+
Z12 ≤ Z22 and x+
1 ≤ x2 .

(4.58)

From Eq. (4.50), in order for the region where the eﬀective potential V1 (x) is
nonpositive to exist, the condition
X ≤ Z12 ,(4.59)

should be satisfied. We note that since X is nonpositive in the cases I, II, andIII, Eq. (4.59) is satisfied trivially. On the other hand, since X is positive in thecases IV and V, Eq. (4.59) gives the minimum of Z12 .
Cases I, II, and III+
In these cases, we consider the region xH
2 ≤ x ≤ x1 as in the case of the particle+
Hcollision. As x increases from x2 to x1 , the CM energy begins with
√
(
)
Z2
|Z1 |
H
Ecm (x2 ) = µ21 + µ22 + µ1 µ2(4.60)
+
,
|Z1 |
Z2
max
and monotonically increases to Ecm (x+1 ) = Ecm , where√Z2
max
Ecm ≡ µ21 + µ22 + 2µ1 µ2.
|Z1 |(4.61)We have used the l’Hopital’s rule to estimate Ecm (xH
2 ). In a critical limit++
Z1 → 0, both Ecm (xH2 ) and Ecm (x1 ) become arbitrarily large and x1 coincides
Hwith x2 . This behavior of the CM energy and results are similar to those ofparticle collision in the critical limit ei → 0.
The significant diﬀerence between the shell collision and the particle collision
is that the event horizon xH
3 exists outside the collision point because of the selfgravity of shell 2. Thus, an arbitrarily large CM energy cannot be seen by anobserver who stands outside the event horizon xH
3 .
Cases IV and V
In these cases, the event horizon xH
2 does not exist. Thus, we consider the+−+region x−
1 ≤ x ≤ x1 . As x increases from x1 to x1 , the CM energy begins with−max
Ecm (x1 ) = Ecm , monotonically decreases to√√|Z1 |Z2 − (Z12 − X)(Z22 − X)
(4.62)Ecm (xm ) = µ21 + µ22 + 2µ1 µ2
,
X
34

max
and monotonically increases to Ecm (x+1 ) = Ecm .
From Eqs. (4.59),√the critical shell with Z1 = 0 is forbidden, since X ispositive. When Z1 → X, i.e., V1 (xm ) = 0, shell 1 can be only at x = xm = x±
1.
Then, the CM energy there is given by√
2µ1 µ2 Z2Ecm (x = xm = x±,µ21 + µ22 + √(4.63)
1)=X

and it is finite.

4.3.2

Observable CM energy

Here, we assume that the D3 is the BTZ black hole spacetime, i.e., except theclass V, and also we assume that an observer stands outside the event horizon
xH
3 . We are interested in an observable collision, i.e., a collision occurs in the+Hregion xH
3 ≤ x. A condition x3 ≤ x1 must be satisfied for the existence of shell+
1. From Eqs. (4.44) and (4.51), the condition xH
3 ≤ x1 is expressed asµ2 E2 ≤ Z12 .

(4.64)

From Eqs. (4.61) and (4.64), the finite upper bound of the observable CMenergy is given by√µ1 µ2 (µ2 + 4E2 )+√Ecm (x = xH.
µ21 + µ22 +(4.65)
3 = x1 ) =2 µ2 E2
This implies that the self-gravity caused by the collision of two shells suppresses
the CM energy.
For an equal mass µ ≡ µ1 = µ2 case, the upper bound becomesµ3/4 ( √√ )Ecm = √ 1/4 2 E2 + µ ,2E2(4.66)and it becomesEcm ≃√ 1/4 3/4
2E2 µ ,

(4.67)

for a small mass case, where µ ≪ E2 .

4.3.3

Test shell limit

We consider test shell limit for shells 1 and 2µ1 → 0 and M1 → M− ≡ M2 with E1 = (const) ̸= 0,(4.68)µ2 → 0 and M2 → M+ ≡ M3 with E2 = (const) ̸= 0,(4.69)

and35

respectively.
Here we show that there are two eﬀects of the self-gravity of thin shells.
First, the mass of inner shell aﬀects its critical condition:
Z1 =µ1− E1 = 0.
4(4.70)In the test shell limit for shell 1, the critical condition becomes E1 → 0 and it
corresponds to the critical condition for particles Eq. (4.11). Second, the radialH
position of the event horizon behaves as xH
a → xa+1 in the test shell limit forshell a.
We can also see the correspondence to the CM energy of particles and shells.
From Eq. (4.61), the CM energy of two thin shells with the equal mass µ atx = x+
1 in the test shell limit for shells 1 and 2 is given by
2
Ecm
(x+
Z2
E2
1)
=1+
→1+ .
2µ2
|Z1 |
E1

(4.71)

From Eq. (4.10), the CM energy of two particles with the equal mass m at
x = x+
1 is given by
2
Ecm
(x+
e2
1)
=1+ .
2m2
e1

(4.72)

Furthermore, from Eq. (4.67), the CM energy behaves as
√ 1/4
Ecm
≃ 2E2 µ−1/4 ≫ 1,µ

(4.73)

for a small mass case, where µ ≪ E2 . Therefore, when we estimate the observable CM energy, we cannot ignore the self-gravity caused by the colliding
objects.
These imply that the self-gravity of the shell influences the critical condition,
the radial position of the event horizon, and the CM energy all of them. The
test shell limit would help us to understand the eﬀect of the self-gravity of thethin shells on the collisions.
For technical reasons, we discussed the shell collision in the BTZ spacetime
instead of the particle collision in the Kerr spacetime. However, the BSW
like collision occur not only in 4-dimensional spacetimes but also in the BTZ
spacetime and the eﬀect of the negative cosmological constant will be negligible
since the collision with a high CM energy occurs near an extremal event horizon,
we have shown in Chap. 3.

36

Chapter 5

Collisional Penrose process
We present an analytic formulation to investigate the energy extraction eﬃciency in the collisional Penrose process. We focus on a collision with arbitrarily
large CM energy, which occurs if either of the colliding particles satisfies a certain critical condition. We show that √if this particle
is ingoing on the collision,√
the upper limit of the eﬃciency is (2+ 3)(2− 2) ≃ 2.186, while if√it is bounced
back before the collision, the upper limit of the eﬃciency is (2 + 3)2 ≃ 13.93.
For simplicity, we assume that the motion and collision of particles are confinedin the equatorial plane of the extremal Kerr black hole, where θ = π/2 anda = M.

5.1

Particle motion in the Kerr black hole

5.1.1

Geodesic in the equatorial planeFirst of all, we consider a particle motion in the equatorial plane of the extremal
Kerr black hole. It reduces to a simple one-dimensional potential problem Eq.
(2.21) and the radial momentum is expressed as√
pr = σ −2V (r),(5.1)
where V (r) is the eﬀective potential of the particle motion and σ ≡ sgn(pr ) =±1. The explicit form of V (r) is given by()
L2 − M 2 E 2 − m2
M m2
M (L − M E)2E 2 − m2
V (r) = −+−−, (5.2)r2r2
r32
where E, L, and m are the energy, angular momentum, and rest mass of theparticle, respectively. The particle motion is restricted to a region where V (r)is nonpositive.

37

The forward-in-time condition pt ≥ 0 near the horizon r → r+ + 0 reducestoE − ΩH L ≥ 0,

(5.3)

where r+ = M and ΩH = 1/(2M ) are the horizon radius and the angularvelocity of the horizon, respectively. We define a critical angular momentum Lc
as
E
Lc ≡
.
(5.4)
ΩH
We call a particle a critical particle if it has the critical angular momentum.
Accordingly, we call a particle with L < Lc (L > Lc ) a subcritical (supercritical)
particle.
The forward-in-time condition for general position r+ ≤ r is expressed as[]
1 ( r )3r+
+ 2 E ≥ l,
(5.5)
2
M
M
where l ≡ L/M and the critical angular momentum lc ≡ Lc /M becomeslc = 2E.

5.1.2

(5.6)

Radial turning points

Next, we are concerned with a particle that comes from or escapes to infinity,
which requires the eﬀective potential is nonpositive for large r. This requires
E ≥ m.
massless particles
For a massless particle (m = 0), solving V (r) = 0 for the impact parameterb ≡ L/E, we obtain b = b± (r), where()
4M 2
b+ (r) ≡ r + M, b− (r) ≡ − r + M +(5.7).
r − 2M
The numerical plot of b = b± (r) is given in Fig. 5.1 (a). This implies that a
particle with b = b± (r) has a turning point at r.
As r increases from M to infinity, b+ (r) begins with 2M and monotonicallyincreases to infinity. As r increases from M to 2M , b− (r) begins with 2M andmonotonically increases to infinity. As r increases from 2M to infinity, b− (r)begins with negative infinity, monotonically increases to a local maximum −7M
at r = 4M and monotonically decreases to negative infinity.
Therefore, for 2M < b ≤ b+ (r∗ ), the particle can escape to infinity irrespective of the sign of the initial velocity, which is shown by the yellow region in
Fig. 5.1 (a), where we denote the radial position of the collision as r∗ . On theother hand, for M < r∗ < 4M and −7M < b ≤ 2M , the particle can escapeto the infinity only if it moves initially outwardly, which is shown by the blue
region in Fig. 5.1 (a).
38

Figure 5.1: The radial turning points are plotted for a massless particle (a)with b = b± (r) and for a massive particle (b) with l = l± (r, E, m), where we set
E = m = 1. The negative energy particles are confined to the gray regions.

massive particles
For a massive particle (m > 0), solving V (r) = 0 for l = L/M , we obtainl = l± (r, E, m), where√−2M 2 E ± r(r − M ) E 2 − m2 + 2M m2 /r
l± (r, E, m) ≡.
(5.8)
M (r − 2M )The numerical plot of l = l± (r) is given in Fig. 5.1 (b). This implies that a
particle with E, m, and l = l± (r, E, m) has a turning point at r.
As r increases from M to infinity, l+ (r, E, m) begins with 2E and monotonically increases toinfinity. As r increases from M to 2M , l− (r, E, m) begins
with 2E and monotonically increases to infinity. As r increases from 2M toinfinity, l− (r, E, m) begins with negative infinity, monotonically increases to a
local maximum lm (< 0) at r = rm and monotonically decreases to negative
infinity.
Therefore, for 2E < l ≤ l+ (r∗ , E, m), the particle can escape to infinity
irrespective of the sign of the initial velocity, which is shown by the yellow
region in Fig. 5.1 (b). On the other hand, for M < r∗ < rm and lm < l ≤ 2E,
the particle can escape to infinity only if it moves initially outwardly, which isshown by the blue region in Fig. 5.1 (b).

5.2

Particle collision and reaction

Let us consider the reaction of two colliding particles, named particles 1 and
2, to two produced particles, named particles 3 and 4. For each particle i(i = 1, 2, 3, 4), we write pµ , E, l, m and σ as pµi , Ei , li , mi , and σi , respectively.
We assume that particle 1 and 2 come from infinity, particle 3 escape to infinity,and particle 4 is a negative energy particle, i.e., E4 < 0.
39

The local conservation of the 4-momenta can be written as
pµ1 + pµ2 = pµ3 + pµ4 .

(5.9)

The t- and φ-components of Eq. (5.9) represent the conservations of energy andangular momentum(5.10)

E1 + E2 = E3 + E4 , and l1 + l2 = l3 + l4 ,respectively. The r-component represents the conservation of the radial momentum
σ1 |pr1 | + σ2 |pr2 | = σ3 |pr3 | + σ4 |pr4 |.

(5.11)

We note that there are four parameters (E, l, m, σ) for each particle. We
have three conservation equations and so 4 × 4 − 3 = 13 degrees of freedom areremaining. If we specify the parameters of the incident particles, we can fix(E1 , l1 , m1 , σ1 ) and (E2 , l2 , m2 , σ2 ). Thus, 13 − 8 = 5 degrees of freedom areremaining. Moreover, we can fix (m3 , σ3 ) and (m4 , σ4 ) for a particle reaction weknow. Thus, only one degree of freedom is remaining. We express this degree of
freedom as parameter δ and use it for an escape particle, where δ parameterizes
the ratio of E and l, i.e.,l = E(2 + δ).

(5.12)

Clearly, we cannot control δ, which corresponds to the direction of the initial
velocities of the product particles.

5.2.1

Particle collision on the horizon

From Eqs. (5.1) and (5.2) , we obtain the r-component of 4-momentum on theevent horizon as|pri | = 2Ei − li ,(5.13)

where we have used the forward-in-time condition to open the square root.
To investigate the collisional Penrose process with an arbitrarily large CM
energy, we assume particle 1 to be critical, i.e., 2E1 − l1 = 0, and particle 2 tobe subcritical, i.e., 2E2 − l2 > 0. We also assume that particle 2 moves inwardly,
i.e., σ2 = −1, because a subcritical particle does not have a turning point near
the event horizon 1 , see Fig. 5.1.
Then, Eq. (5.11) is expressed as σ (2E − l )(for σ3 = σ4 )32
2,−(2E2 − l2 ) =(5.14)
 σ [2(2E − l ) − (2E − l )] (for σ = −σ )
3

3

3

2

2

3

4

and several situations are possible depending on the values of σ3 and σ4 .
1 We consider an outgoing subcritical particle in Chap. 6.

40

• If σ3 = σ4 = 1, we obtain 2E2 − l2 = 0, which contradicts our assumption.
• If σ3 = σ4 = −1, we obtain a trivial equation, which implies that particle 3
can be either critical or subcritical. Nevertheless, we treat only the criticalcase because a subcritical ingoing particle cannot escape to infinity.
• If σ3 = −σ4 = 1, we obtain 2E3 − l3 = 0, i.e., particle 3 is critical.
• If σ3 = −σ4 = −1, we obtain 2E2 − l2 = 2E3 − l3 , which implies thatparticle 3 is subcritical. It is not interesting since a subcritical ingoingparticle cannot escape to infinity again.
From the above considerations, the following assumptions are interesting forenergy extraction.
• Particle 1 is critical 2E1 − l1 = 0, come from infinity, and both σ1 = ±1.
• Particle 2 is subcritical 2E2 − l2 > 0, come from infinity, and moves
inwardly σ2 = −1.
• Particle 3 is critical on the horizon, escape to infinity, and both σ3 = ±1.
• Particle 4 is a negative energy particle, falls into the black hole, and moves
inwardly σ4 = −1.

5.2.2

Near horizon and near critical behaviors

We assume that the particle collision occurs near the event horizon. We introduce the radial position of near-horizon collision r∗ asr∗ =M, 0 < ϵ ≪ 1.
1−ϵ(5.15)

We take the energy and angular momentum of the escaping particle as functions of those of the incident particles, the collision point r∗ , and δ, i.e.,
E3 = E3 (E1 , l1 , m1 , σ1 ; E2 , l2 , m2 , σ2 ; m3 , σ3 ; m4 , σ4 ; δ; r∗ ) ,l3 = l3 (E1 , l1 , m1 , σ1 ; E2 , l2 , m2 , σ2 ; m3 , σ3 ; m4 , σ4 ; δ; r∗ ) .

(5.16)(5.17)

Since E3 and l3 are function of r∗ , we can assume that E3 and l3 are expandable
in terms of ϵ as
E3 = E3(0) + E3(1) ϵ + E3(2) ϵ2 + O(ϵ3 ),23l3 = l3(0) + l3(1) ϵ + l3(2) ϵ + O(ϵ ),(5.18)(5.19)We expand Ei and li in terms of ϵ for the product particles i = 3, 4, but notfor the incident particles i = 1, 2. This looks asymmetric but is suitable for thepresent physical setting. Equivalently, instead of l3 , it is more convenient toexpand δ as
δ = δ(1) ϵ + δ(2) ϵ2 + O(ϵ3 ).
41(5.20)

Therefore, we regard particle 3 as near-critical in the near-horizon collision andthe near-critical angular momentum is given byl3 = E3 (2 + δ)()
= 2E3(0) + E3(0) δ(1) + 2E3(1) ϵ()+ E3(0) δ(2) + E3(1) δ(1) + 2E3(2) ϵ2 + O(ϵ3 ).

(5.21)

From Eqs. (5.8), (5.15), and (5.18), we obtain()√
2
l+ (r∗ , E3 , m3 ) = 2E3(0) + 2E3(0) + 2E3(1) − E3(0)− m23 ϵ + O(ϵ2 ). (5.22)
If particle 3 moves initially inwardly with E3 ≥ m3 and to escape to infinity, it
must be bounced back by the eﬀective potential, i.e., 2E3 < l3 ≤ l+ (r∗ , E3 , m3 )must be satisfied. This implies√2
2E3(0) − E3(0)− m23.
(5.23)
0 < δ(1) ≤ δ(1),max ≡E3(0)If particle 3 moves initially outwardly with E3 ≥ m3 , it always can escape toinfinity.
From Eqs. (5.5), (5.15), and (5.18), the forward-in-time condition for particle
3 in the near-horizon implies()(5.24)
l3 < 2E3(0) + 2 E3(0) + E3(1) ϵ + O(ϵ2 ).
Therefore, when l3 ≤ l+ (r∗ , E3 , m3 ), the forward-in-time condition is always
satisfied.

5.2.3

Expansion of the radial momentum

We consider the series expansion of the radial momentum in powers of ϵ foreach particle. Since we have assumed particle 1 to be critical, particle 2 to be
subcritical, and particle 3 to be near-critical, the radial momentum for each42

particles are expanded as|pr1 | =√3E12 − m21 ϵ − √E12
3E12 − m21

ϵ2 + O(ϵ3 ),(5.25)(3E2 − l2 )(E2 − l2 ) − m22 2|pr2 | = (2E2 − l2 ) − 2(E2 − l2 )ϵ +ϵ + O(ϵ3 ), (5.26)
2(2E2 − l2 )
√
(
)(
)
2
|pr3 | = E3(0)3 − δ(1) 1 − δ(1) − m23 ϵ
[
(
)(
)]
(
)(
)
2
E3(0)
−1 + 2 − δ(1) 2δ(1) − δ(2) + E3(0) E3(1) 3 − δ(1) 1 − δ(1) 2
√
+
ϵ
(
)(
)
2
3 − δ(1) 1 − δ(1) − m23
E3(0)+ O(ϵ3 ),(5.27)[()]|pr4 | = (2E2 − l2 ) + 2E1 − 2(E2 − l2 ) − E3(0) 2 − δ(1) ϵ
[
(
)
(
)
2E2 − l2+
− E3(0) 2δ(1) − δ(2) − E3(1) 2 − δ(1)2](
)2
E1 + E2 − E3(0) + m24 2−
ϵ + O(ϵ3 ).
2(2E2 − l2 )

5.3

(5.28)

Energy extraction eﬃciency

Let us estimate the maximum value of the energy-extraction eﬃciency by using
the expanded radial momentum and its conservation. Since we have assumed
that particle 2 and 4 move inwardly, i.e., σ2 = σ4 = −1, the radial momentum
conservation is expressed asσ1 |pr1 | − |pr2 | = σ3 |pr3 | − |pr4 |,(5.29)

where σ1 = −1 and σ1 = 1 correspond to the BSW process [1] and theSchnittman process [49], respectively. Since we have expanded E3 in termsof ϵ, η is expressed asη (σ1 )(σ3 ) =E3(0),E1 + E2(5.30)

where superscripts denote the signs of σ1 and σ3 . For example, η (+)(−) denotethe eﬃciency when choose σ1 = 1 and σ3 = −1.

5.3.1

First order of the radial momentum conservation

Since the terms of O(1) in Eq. (5.29) is already considered, we proceed to theterms of O(ϵ) in the same equation. From Eqs. (5.25)–(5.28), we obtain
√
(
)
(
)(
)
2
A − E3(0) 2 − δ(1) = σ3 E3(0)(5.31)
3 − δ(1) 1 − δ(1) − m23 ,43where A ≡ 2E1 + σ1
find√
3E12 − m21 > 0. Squaring the both sides of Eq. (5.31), we
2 − δ(1) =2
A2 + E3(0)+ m23

2AE3(0).

(5.32)

Substituting the above expression into Eq. (5.31), we obtain
A−

2
E3(0)
+ m23

A

= 2σ3

√
(
)(
)
2
E3(0)
3 − δ(1) 1 − δ(1) − m23 .

(5.33)√

This implies E3(0) ≤ λ0 ≡ A2 − m23 (E ≥ λ0 ) for σ3 = 1 (σ3 = −1).
If we choose σ3 = −1, δ(1) ≥ 0 must be satisfied for particle 3 to escape toinfinity. Supposing δ(1) ≥ 0 in Eq. (5.32), we have2
E3(0)− 4AE3(0) + A2 + m23 ≤ 0.

Thus, we findλ− ≤ E3(0) ≤ λ+ , λ± ≡ 2A ±√3A2 − m23 ,where E3(0) = λ+ is realized only for δ(1) = 0.
Therefore, the maximum eﬃciency is given by√()2√
2E1 + σ1 3E12 − m21 − m23
η (σ1 )(+) =,E1 + E2(5.34)(5.35)(5.36)for σ3 = 1 while
) √ (
)2
(
√
√
2 2E1 + σ1 3E12 − m21 + 3 2E1 + σ1 3E12 − m21 − m23
η (σ1 )(−) =,E1 + E2(5.37)for σ3 = −1. We note that λ+ is always greater than λ0 , i.e., the upper limit ofη for σ3 = −1 is always greater than that for σ3 = 1.

5.3.2

Second order of the radial momentum conservation

The terms of O(ϵ2 ) in Eq. (5.29) yield
E2(3E2 − l2 )(E2 − l2 ) − m22
σ1 √ 2 1+2
2(2E2 − l2 )3E1 − m1
[
(
)(
)]
(
)(
)
2
E3(0) −1 + 2 − δ(1) 2δ(1) − δ(2) + E3(0) E3(1) 3 − δ(1) 1 − δ(1)
√
= σ3
(
)(
)
2
E3(0)
3 − δ(1) 1 − δ(1) − m23
(
)2
(
)
(
)
E1 + E2 − E3(0) + m24
2E2 − l2
− E3(0) 2δ(1) − δ(2) − E3(1) 2 − δ(1) −.
+2
2(2E2 − l2 )(5.38)
44

From the above equation, if we fix E3(0) and δ(1) , we find the relation between
E3(1) and δ(2) . Since both E3(1) and δ(2) appear only linearly, we can always
solve the above equation for E3(1) in terms of δ(2) . Thus, we do not obtain any
additional condition to the lower-order terms.

5.3.3

Maximum eﬃciency

The BSW process: σ1 = −1
From Eqs. (5.36) and (5.37), the eﬃciency for σ3 = 1 and σ3 = −1 are given by√()2√
2E1 − 3E12 − m21 − m23
η (−)(+) =,(5.39)
E1 + E2and)√2 2E1 − 3E12 − m21 +(η (−)(−) =√ (3 2E1 −

)2√
3E12 − m21 − m23, (5.40)

E1 + E2

respectively. Here we still take mi as fixed parameters but E1 and E2 as free
ones in the ranges E1 ≥ m1 and E2 ≥ m2 , respectively. The maximum eﬃciency
is attained for E1 = m1 and E2 = m2 . In this case, we obtain√(√ )2
2 − 2 m21 − m23(−)(+)
η=(5.41),
m1 + m2
which is less than unity, and
√ (
√ )
√ )2
2 2 − 2 m1 + 3 2 − 2 m21 − m23(

η (−)(−) =m1 + m2.

(5.42)

From the above expressions, we consider
several cases as following. For per√7−4 2(−)(−)
fectly elastic collision, η=< 1. For pair annihilation, η (−)(−) =2√√(2− 2)(2+ 3)≃ 1.093, which agrees very well with the numerical result in Ref.
2√√[47]. For inverse Compton with m2 = m3 = 0, η (−)(−) = (2 − 2)(2 + 3) ≃2.186.
The Schnittman process: σ1 = 1
From Eqs. (5.36) and (5.37), the eﬃciency for σ3 = 1 and σ3 = −1 are given by√()2√
2E1 + 3E12 − m21 − m23
η (+)(+) =(5.43),
E1 + E2

45

and
)
√
2 2E1 + 3E12 − m21 +(η(+)(−)=√ (3 2E1 +

E1 + E2√

3E12 − m21

)2

− m23
, (5.44)

respectively. For the Schnittman process, the situation is very diﬀerent andmuch simpler. The maximum eﬃciency can be attained for E1 ≫ max(m1 , m3 )and E1 ≫ E2 . In this case, we find√(+)(+)= 2 + 3 ≃ 3.732,
ηmax(5.45))2(√(+)(−)
ηmax= 2 + 3 ≃ 13.93.
(5.46)The latter is a universal maximum eﬃciency irrespective of the details of theparticle reaction or the masses of the particles.

46

Chapter 6

Arbitrarily large energy
extraction and escapeprobability
We present an analytic formulation to investigate the energy extraction eﬃciency and the escape probability in the super-Penrose process. We consider thesituation where two massive particles, named particles 1 and 2, collide head-on
in the equatorial plane of the extremal Kerr black hole and produce two massless particles, named particles 3 and 4. Focusing on a typical case, where bothof the colliding particles have zero angular momenta, we show that a masslessparticle produced in such a collision can escape to infinity with arbitrarily large
energy in the near-horizon limit of the collision point.
Furthermore, if we assume that the emission of the produced massless particles is isotropic in the center-of-mass frame but confined to the equatorial plane,
the escape probability of the produced massless particle approaches 5/12, andalmost all escaping massless particles have arbitrarily large energy at infinity
and an impact parameter approaching 2M , where M is the mass of the black
hole.

6.1

Particle collision and escape cone

From Eq. (2.1), the metric in the equatorial plane of the extremal Kerr blackhole is given by
(
)
(
)2
2M
4M 2
r
2
2
ds = − 1 −dt −dtdφ +
dr2 + A(r)dφ2 ,
(6.1)
r
r
r−M
where A(r) ≡ r2 + M 2 + 2M 3 /r. The radius of the horizon and ergoregion
are given by r = r+ ≡ M and r = rE ≡ 2M . From Eq. (2.12)–(2.15), the47components of the 4-momentum pµ are given by
[
]
2M 2
1
t
A(r)E −L ,p =(r − M )2r√
pr = σ −2V (r),θp = 0,pφ =1(r − M )2

[

(
) ]
2M
2M
E+ 1−
L ,
r
r

(6.2)
(6.3)
(6.4)

2

(6.5)

where E, L, m, and V (r) are the energy, angular momentum, rest mass, andeﬀective potential of the particle, respectively, and σ ≡ sgn(pr ) = ±1. Here theexplicit form of V (r) is given by()
L2 − M 2 E 2 − m2
M m2
M (L − M E)2E 2 − m2
V (r) = −+−−. (6.6)
2
3
r
2r
r
2
A locally nonrotating frame (LNRF) is a tetrad basis associated with observers who have zero angular momentum. The transformation that relates thecomponents of an arbitrarily vector v µ to the components in the LNRF v (α) is
given byµv (α) = e(α)
µ v ,

(6.7)

(α)

where eµ is the tetrad basis of the LNRF. According to Refs. [55, 61], the(α)
components of eµ are given by
(
)
r
−
M
√
e(t)
, 0, 0, 0 ,
(6.8)
µ =
A(r)
(
)
r
=
0,
e(r)
(6.9)
,
0,
0
,
µ
r−M
e(θ)
µ = (0, 0, r, 0) ,
(
)
√
2M 2
(φ)
eµ = − √, 0, 0, A(r) .
r A(r)(6.10)(6.11)(α) (β)These satisfy gµν = η(α)(β) eµ eν , where η(α)(β) = diag(−1, 1, 1, 1). From Eqs.
(6.2)–(6.11), the components of the 4-momentum in the LNRF are given by)(√r −2V (r)L
A(r)E − 2M 2 L/r
(µ)
√
, σ
, 0, √.
p =(6.12)
r−M(r − M ) A(r)A(r)
We consider the situation where two massive particles, named particles 1
and 2, collide head-on in the equatorial plane of the extremal Kerr black holeand produce two massless particles, named particles 3 and 4. For each particle(µ)
i (i = 1, 2, 3, 4), we write p(µ) , E, L, m, V (r), and σ as pi , Ei , Li , mi , Vi (r),48

and σi , respectively. The components of the total 4-momentum of two colliding
particles in the LNRF are obtained as
(µ)

(µ)

p1 + p2= (T, R, 0, Φ),(6.13)

where
A(r)(E1 + E2 ) − 2M 2 (L1 + L2 )/r√,(r − M ) A(r)
)
√
r ( √
(r)
(r)
R ≡ p1 + p2 =σ1 −2V1 (r) + σ2 −2V2 (r) ,r−M
L1 + L2(φ)(φ)Φ ≡ p1 + p2 = √
.
A(r)
(t)

(t)

T ≡ p1 + p2 =(6.14)(6.15)(6.16)

We note that T is always positive because of the forward-in-time condition.
Without loss of generality, we assume that particle 1 moves radially outward,
while particle 2 moves radially inward. For simplicity, we assume that particles
1 and 2 have the same mass and zero angular momenta and are marginallybound
σ1 = −σ2 = 1,
L1 = L2 = 0,

(6.17)(6.18)

E1 = m1 = E2 = m2 ≡ m.

(6.19)(I)(I)

In this case, the spatial components of the total 4-momentum p1 + p2 (I =r, θ, φ) vanish. Thus, the LNRF coincides with the center-of-mass frame (CMF)(t)(t)
and p1 + p2 yields the CM energy. When we denote the radial position of thecollision as r∗ , where M < r∗ ≤ 2M , i.e., we assume that the collision occurs in
the ergoregion, the CM energy is expressed as√) 2m√A(r )
)(
(
∗
(ν)
(µ)
(ν)
(µ)
Ecm ≡ −η(µ)(ν) p1 + p2=
p1 + p2.
(6.20)r∗ − M
The local conservation of the 4-momenta can be written as
(µ)

(µ)

p1 + p2(µ)(µ)= p3 + p4= Ecm (1, 0, 0, 0).

(6.21)

From the above equations, the 4-momenta of the two massless particles 3 and4 moving on the equatorial plane can be written as
Ecm(1, cos α, 0, sin α) ,
2
Ecm
(µ)
p4 =
(1, − cos α, 0, − sin α) .
2
(µ)

p3

=

(6.22)(6.23)(r)

This implies that the spatial velocity of particle 3 makes an angle α with eµ ,49

where −π ≤ α < π. From the components of the 4-momentum, we obtain(φ)

p3b(r∗ − M )(6.24))2 ()2 = A(r∗ ) − 2M 2 b/r∗ ,
(r)
(φ)
p3
+ p3
√
√
(r)
σ3 r∗ A(r∗ )p3b2 − M 2
2M (b − M )2
cos α = √(
+
,
2
)2 (
)2 = A(r∗ ) − 2M 2 b/r∗ 1 −
r∗
r∗3
(r)
(φ)
p3
+ p3
sin α = √((6.25)
where we have defined b ≡ L3 /E3 .
Now we consider the escape of particle 3 from near the horizon to infinity.
We define an escape cone S, which is the range of angles in the CMF along which
particle 3 must be emitted to escape to infinity. The condition for particle 3
to escape to infinity is determined by investigating a radial turning point, i.e.,V3 (r) = 0. Solving V3 (r) = 0 for b, we obtain b = b± (r), where()
4M 2
b+ (r) ≡ r + M, b− (r) ≡ − r + M +.
(6.26)
r − 2M
This mean that particle 3 with impact parameter b = b± (r) has the turning
point at r, see Sec 5.1.2 for more detail. The numerical plot of b = b± (r) is
given in Fig. 6.1. For 2M < b ≤ b+ (r∗ ), particle 3 can escape to infinity
irrespective of the sign of σ3 . On the other hand, for −7M < b ≤ 2M , particle
3 can escape to infinity only if σ3 = 1 is chosen.
We define critical angles as the boundaries of the impact parameters toescape to infinity:αI ≡ α(σ3 = −1, r = r∗ , b = 2M ),αII ≡ α(σ3 = −1, r = r∗ , b = b+ (r∗ )),(6.27)(6.28)

αIII ≡ α(σ3 = 1, r = r∗ , b = b+ (r∗ )),αIV ≡ α(σ3 = 1, r = r∗ , b = −7M ).

(6.29)(6.30)

From Eqs. (6.24)–(6.30), we findsin αI > 0, cos αI < 0,(6.31)

sin αII = sin αIII = 1,
sin αIV < 0, cos αIV > 0.

(6.32)(6.33)

αIV < αIII = αII < αI .

(6.34)

Therefore, we can findThis means that the escape cone S is given byα ∈ (αIV , αI ).
50(6.35)

Figure 6.1: The condition for particle 3 to escape to infinity is determined byinvestigating a radial turning point, i.e., V3 = 0. Solving V3 = 0 for the impact
parameter b, we obtain b = b± (r). The radial turning points for particle 3 withb = b± (r) are plotted by the black solid lines. The range of b where particle
3 can escape from r = r∗ to infinity depends on whether it moves initiallyradially inward or outward. If particle 3 moves initially inward, the maximum
and minimum values of b with which particle 3 can escape to infinity are givenby the blue dashed lines labeled αII and αI . The values of the blue dashed lines
labeled αII and αI are given by b+ (r∗ )/M and 2, respectively. If particle 3 moves
initially outward, the maximum and minimum values of b with which particle 3
can escape to infinity are given by the red dotted lines labeled αIII and αIV . The
values of the red dotted lines labeled αIII and αIV are given by b+ (r∗ )/M and−7, respectively. Please notice that the blue dashed line labeled αII and the red
dotted line labeled αIII take the same value, i.e., αII = αIII .

The escape probability P is given by the ratio of the angle of the escape cone to
2π if we assume that particle 3 is emitted isotropically in the CMF and confinedto the equatorial plane. It is given by
P ≡

6.2

S
αI − αIV=.
2π
2π(6.36)

Super-Penrose process and escape probability

The components of the time-transitional Killing vector ξ µ in the Boyer-Lindquist
coordinates are given by ξ µ = (1, 0, 0, 0). Using Eq. (6.7), its components in51

the CMF are given by
(
ξ

(µ)

=

r−M
2M 2 /r√, 0, 0 − √
A(r)
A(r)

)
.

From Eqs. (6.20), (6.22), (6.24), (6.25), and (6.37), E3 is given by)(2M 2(µ) (ν)sin α .
E3 ≡ −η(µ)(ν) ξ p3 = m 1 +r∗ (r∗ − M )Therefore, the energy extraction eﬃciency is given by
)
(
E3
1
2M 2
η≡
=
1+
sin α .
E1 + E22r∗ (r∗ − M )(6.37)(6.38)(6.39)

Because the Killing vector ξ µ becomes spacelike inside the ergoregion, the conserved energy E4 can be negative. Thus, if α satisfies
sin α >r∗ (r∗ − M ),2M 2(6.40)

the energy extraction eﬃciency can be larger than unity, i.e., the collisional
Penrose process occur.
We assume that the collision point is near the horizon and its radial positionis given by
r∗ =

M
,
1−ϵ

0 < ϵ ≪ 1.

(6.41)The explicit form of the critical angles is given by
5π+ O(ϵ),6π
αII = αIII = ,27αIV = − ϵ + O(ϵ2 ).
18
αI =(6.42)(6.43)(6.44)

See Fig. 6.2 for the schematic diagram of the situation.
From Eqs. (6.36), (6.42), and (6.44), the escape probability is obtained as
P →

5
,
12

(6.45)

in the horizon limit ϵ → 0. From Eqs. (6.39) and (6.41), if sin α = O(1), η isgiven by
η=

sin α
E3
=
+ O(1),2m
ϵ52(6.46)

Figure 6.2: The schematic diagram of the critical angles and the escape cone forparticle 3, i.e., the range of angles in the CMF along which particle 3 must be
emitted to escape from r = r∗ to infinity. The colored arrows denote the spatial
velocities of particle 3 with the critical angles. The colored sectors denote theescape cone for particle 3. The escape probability is given by the ratio of theangle of the escape cone to 2π. If particle 3 moves initially inward, the maximum
and minimum values of α with which particle 3 can escape to infinity are given
by αI and αII . Particle 3 emitted radially inward in the range α ∈ (αII , αI )bounces back at the turning point and can escape to infinity. If particle 3
moves initially outward, the maximum and minimum values of α with which
particle 3 can escape to infinity are given by αIII and αIV . Particle 3 emitted
radially outward in the range α ∈ (αIV , αIII ] can directly escape to infinity. Since
αII = αIII , we represent the spatial velocities of particle 3 with αII and αIII as
the one arrow colored half blue and half red.

and this diverges to infinity in the limit ϵ → 0. Only if sin α = O(ϵ), η turns
out to be finite. if particle 3 is emitted along the angle αIV , η becomesη=E3
1= + O(ϵ).
2m
9(6.47)

This means that the proportion of the particles with finite energy is minuscule.
In other words, almost all the particles which escape to infinity have arbitrarily
large energy.
From Eqs. (6.24) and (6.41), the relation among the emission angle α, impact
parameter b, and ϵ is given by
2M − b =b
ϵ + O(ϵ2 ),2 sin α(6.48)

if sin α = O(1). Therefore, almost all the particles which escape to infinity havean impact parameter b = 2M + O(ϵ).
53Here, we focus on a special set of parameter values of the colliding particles
where particles 1 and 2 have the same mass and zero angular momenta and are
marginally bound. However, we would like to stress that it is not a singularexample but a typical one which provides a picture that we believe is representative of more general collisions in the super-Penrose process. Even if we do
not assume the above assumptions, the escape probability takes a value of theorder of unity and almost all the escaping particles have arbitrarily large energy
except for the sets of fine-tuned parameter values.

54Chapter 7

Conclusions
In this thesis, we have studied a collision of two geodesic particles and dust
thin shells in the Bañados-Teitelboim-Zanelli (BTZ) spacetime and an energy
extraction by using a particle collision in the Kerr black hole spacetime.
In Chap. 3, we have investigated the collision of two particles in the BTZ
black hole spacetime with the negative cosmological constant and angular momentum. We have obtained a general formula for the CM energy of two geodesic
particles in the BTZ black hole spacetime. We have shown that the CM energyof two ingoing particles in the near horizon limit can be arbitrarily large if eitherof the two particles has a critical angular momentum (3.17) and the other has
a noncritical angular momentum. We have shown that the motion of a particle
with a subcritical angular momentum is allowed near an extremal BTZ blackhole and the CM energy for a rear-end collision at a point near the extremal
BTZ black hole can be arbitrarily large in the critical angular momentum limit.
Since colliding particles near horizon have large energy, the self-gravity ofthe particles aﬀects on the collision. The self-gravity caused by the high energy
collision will aﬀect the large CM energy and it will be limited to a finite value.
This consideration is shown in Chap. 4 by using a collision of two shells.
In Chap. 4, we have investigated the rear-end collision of two dust thinshells in the rotating BTZ spacetime to investigate the eﬀects of the self-gravity
of colliding objects on the high energy collision. The shells divide the BTZ
spacetime into three domains and the domains are matched by Darmois-Israel’s
method. From the junction condition, all the domains must have the sameangular momenta J and this implies that the shells and domains corotate.
We have revealed that there are two eﬀects of the self-gravity of thin shells.
First, we have shown that the mass of inner shell aﬀects its critical conditionH(4.54). Second, the position of the event horizon changed from xH
1 to x3 becauseof the masses of two shells.
We have considered the shell collision in five cases according to the value of55

J. The cases I (J < ℓM1 ), II (J = ℓM1 ), and III (J = ℓM2 ) would be especiallyinteresting cases because of following reasons. The case I is a usual astrophysicalsituation as a black hole sub-extremely rotates and two objects collide near its
event horizon. In case II, the black hole extremely rotates initially and thecollision of two falling shells corresponds with the BSW collision of two particles
with an arbitrary high CM energy in the extremal black hole spacetime. In caseIII, inner shell 1 can be satisfied the critical condition that the eﬀective potential′ H
H
for shell 1 becomes V1 (xH
2 ) = V1 (x2 ) = 0 on the extremal event horizon x = x2
as with the BSW process in extremal black hole spacetimes [1, 3]. In cases I-III,
the CM energy of the shells can be arbitrarily large if inner shell 1 satisfies thecritical condition and if outer shell 2 does not. However, an observer outside
the event horizon xH
3 cannot see the products of the collision with the arbitrarylarge CM energy because it occurs inside the event horizon xH
3 .
An observer at x ≥ xH
maysee
products
after
thecollision
if a shell colli3
sion occurs in a region xH
≤
x.
We
have
obtainedthefiniteupperbound of3
the CM energy of the collision in the region xH≤x(4.65)–(4.67).
We have3
concluded that the self-gravity of colliding objects suppresses its CM energy andthe observer in the region xH
3 ≤ x can only see the suppressed collision.
We have also found a test shell limit. We have shown that the CM energy
and the eﬀective potentials for shells in the test shell limit are very similar tothe ones of particles. The test shell limit would help us to understand the eﬀectof the self-gravity of the thin shells on the collisions.
In Chap. 5, we have investigated the particle collision and the energy extraction eﬃciency, where a critical particle (particle 1) and subcritical particle(particle 2) collide near the event horizon of the extremal Kerr black hole andthen two particles are produced, one of which escapes to infinity (particle 3)
and another falls into the black hole (particle 4). We have proposed an analytic approach to the collisional Penrose process with the expansion in powersof small parameter ϵ, which is parameterized the near-horizon collision (5.15).
We have considered the series expansion of the radial momentum in powersof ϵ for each particle. By using the expanded radial momentum and its conservation, we have estimated the energy extraction eﬃciency. We have found thatthe upper limits of the collisional Penrose process restricted√ in the√ equatorial
plane of √the extremal Kerr black hole are given by (2 + 3)(2 − 2) ≃ 2.186
and (2 + 3)2 ≃ 13.93 for the BSW and Schnittman collision, respectively. The
former is realized for inverse Compton scattering, while the latter can be universally attained for the various reaction of particles.
Chap. 6, we have investigated the analytic formulation of the energy extraction eﬃciency and the escape probability in the super-Penrose process. We
have considered the situation where two massive particles, named particles 1
and 2, collide head-on in the equatorial plane of the extremal Kerr black holeand produce two massless particles, named particles 3 and 4.
We have focused a special set of parameter values of the colliding particles,
where particles 1 and 2 have the same mass and zero angular momenta and are56

marginally bound. In this case, the spatial components of the total 4-momentumof the colliding particles vanish in the LNRF, and hence the LNRF coincides
with the CMF. We computed the escape cone, i.e., the range of angles in the
CMF along which particles must be emitted to escape to infinity, and computed
the escape probability. We showed that the escape probability approaches 5/12
and almost all the escaping particles have arbitrarily large energy with an impact
parameter b → 2M in the near-horizon limit of the collision point. We would
suggest that very high-energy particles with impact parameter b ≃ 2M areproduced by the super-Penrose process and can be in principle observed fromrapidly rotating black holes in astrophysics.
In this thesis, we have compared the shell collision and the particle collisionin the BTZ spacetime in order to investigate the eﬀects of the self-gravity ofcolliding objects on the high CM energy collision. Since we have shown the
self-gravity of colliding objects suppresses its CM energy, next we need to verify
the eﬀect on the energy extraction eﬃciency. The energy extraction by using
the collision of two spherical charged shells in the Reissner-Nordström blackhole spacetime is founded by Nakao et al. [65]. Thus, we need to investigate
the energy extraction by using the shell collision in the rotating black holespacetime.

57

Acknowledgements
I would like to express the deepest appreciation to professor Tomohiro Harada
for insightful suggestions, comments, discussions, and collaborations. I havelearned many things from him as a researcher. I am grateful to my collaborators U. Miyamoto, T. Igata, N. Tsukamoto, N. Tanahashi, and Y. Gongfor valuable suggestions, comments and discussions. I am also grateful to theexaminers T. Kobayashi and K. Ieki for useful suggestions and comments. I
would like to thank all the colleagues in Department of Physics in Rikkyo University, especially K. Inagawa, T. Kokubu, S. Yokoyama, T. Hiramatsu, and T.
Hirayama. Many thanks to my friends, especially K. Yamada, R. Kase, and
Y. Tanaka, for conversations, discussions, and drinking. I sincerely thank myfamily, Miki, Yuta, Zenya, and Kiyoko. K.O. was supported by JSPS.

58

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