粒子加速器としての回転 bh high-velocity collision...

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粒子加速器としての回転 BHHigh-velocity collision of particles around a Kerr black hole

Tomohiro Harada

Department of Physics, Rikkyo Univesity

第 5回「BH磁気圏勉強会」＠名古屋大学 28/2-1/3/2012This talk is based on the collaboration with M. Kimura (YITP).

T. Harada (Rikkyo U) Kerr BHs as accelerators 28/2-1/3/2012 1 / 37

Table of contents

.. .1 Introduction

.. .2 BHs as particle accelerators: basics

.. .3 Astrophysical relevance

Dependence on the BH spinsRelevance of the ISCOCollision of two general geodesic particlesRobustness against gravitational radiation reactionMagnetised Schwarzschild BHs

.. .4 Summary


Introduction

Table of contents

.. .1 Introduction




.. .4 Summary


Introduction

Astrophysical black holes (BHs)

BH candidatesX-ray binary: ∼ several-10MGalactic centre: 106 − 109 M

Towards “direct” observationShadows with sub mm - mm VLBISpacetime geometries by gravitational wave observation

Physical processes in the vicinity of the horizon will be important.


Introduction

BHs as particle accelerators

as? in?

BHs as particle accelerators (Banados, Silk & West 2009)BHs may act as particle accelerators, based on classical general relativity.

Cf. BHs in particle accelerators (Giddings & Thomas 2002)BHs may be produced in particle colliders, based on higher-dimensionaltheories.


BHs as particle accelerators: basics

Table of contents

.. .1 Introduction




.. .4 Summary



Rotating BHs

BHs are usually rotating.

Rotating BHs are uniquely described bya Kerr metric, if ....

The Kerr spacetime is parametrized bythe mass M and the spin a.

0 ≤ |a| ≤ M: BH, |a| > M: nakedsingularity

a∗ = a/M: nondimensional Kerrparameter



Set up

Let’s consider particles of mass m0 at restat infinity from a rotating BH of mass M onthe equatorial plane.For such a particle to plunge into thehorizon, the angular momentum L isrestricted to

lL = −2(1+√

1 + a∗) < l < 2(1+√

1 − a∗) = lR,

where l = L/(m0 M).Such two particles can collide near the horizon. We can calculate thecentre-of-mass (CM) energy of them.



Centre-of-mass (CM) energy

Suppose particles 1 and 2 are at the same spacetime point.The sum of the two momenta

patot = pa

1+ pa

2.

Centre-of-mass energy

E2cm = −pa

tot ptota = m21 + m2

2 − 2gab pa1

pb2.

This is coordinate-independent and in principle observable.



Evaluation of the CM energy

a∗ = 0 a∗ = 1BSW (2009)

For a Kerr Black hole with a∗ = 1, Ecm blows up at the horizon for l1 = 2.



Blow-up of the CM energy at the horizon

Let’s take the limit a∗ → 1, then

Ecm

2m0=

√12

(2 − l12 − l2

+2 − l22 − l1

).

(BSW 2009)

If we further fine-tune l1 → lR = 2, Ecm → ∞.

Distinct from the Penrose process. No need for negative energy.

The energy of any ejecta will be strongly redshifted.



Kinematic explanation (Zaslavskii 2011)

Locally NonRotating Frame (LNRF)

ds2 = −e2νdt2 + e2ψ(dφ − ωdt)2 + e2µ1 dr2 + e2µ2 dθ2.

r = const, θ = const, φ = ωt + const, ω = −gφt/gφφ.

ω(t) = −eνdt, ω(r) = eµ1dr, ω(θ) = eµ2dθ, ω(φ) = eψ(dφ − ωdt).

3-velocities: uµ = u(α)ω(α)µ, v(i) ≡ v(i) ≡ u(i)/u(t), where δi jv(i)v( j) ≤ 1.

As a result, the relative velocity becomes the light speed.



Critical views on the BSW process

Criticisms by Berti et al. (2009) and Jacobson & Sotiriou (2010)Needs arbitrarily long proper time.The test particle approximation will break down.No maximally rotating BH in the universe.

OK. Infinite CM energy is unrealistic. But the collision can still besignificantly energetic.

The age of the universe is much longer than the BH dynamical time.The test particle approximation is good if the mass ratio is small.No universal spin bound smaller than Kerr’s.


Astrophysical relevance

Table of contents

.. .1 Introduction




.. .4 Summary


Astrophysical relevance Dependence on the BH spins

Table of contents

.. .1 Introduction




.. .4 Summary



CM energy for the near-horizon collision

Assume m1 = m2 = m0 > 0 for simplicity.

Ecm is bounded except in the near-horizon limit.In the near-horizon limit r → rH, we obtain

Ecm

2m0=

√√√√1 +

4M2m20[(E1 − ΩH L1) − (E2 − ΩH L2)]2 + (E1L2 − E2L1)2

16M2m20(E1 − ΩH L1)(E2 − ΩH L2)

.

If we fine-tune L1 → L1c ≡ E1/ΩH, Ecm → ∞. We call particles withL = Lc ≡ E/ΩH critical particles.

The BSW process: E1 = E2 = m0 and ΩH = 1/2M.



Maximum CM energy for large spins

Assume particles are at rest at infinity orE1 = E2 = m0.

Let’s fine tune l1 = lR and l2 = lL, then

Ecm

2m0≈ 1√

4 − 2√

2

√2 − l2

4√

1 − a2∗

for a∗ ≈ 1.

(Jacobson & Sotiriou 2010, Harada & Kimura 2011)

If we further take the limit a∗ → 1, Ecm → ∞.



Critical particles and maximal rotation

The effective potentials for critical particles

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

1 10 100

Effec

tive

pote

nti

al

r

a∗ = 1a∗ = 0.99a∗ = 0.9

For a∗ < 1, lc > lR (lc = Lc/(m0 M)): critical particles are bounceddue to the barrier.For a∗ = 1, lc = lR: critical particles can reach the horizon.


Astrophysical relevance Relevance of the ISCO

Table of contents

.. .1 Introduction




.. .4 Summary



Astrophysical significance of the ISCO

The inner edge of the standard accretion disk is given by the ISCO.

A compact object around a supermassive BH inspirals adiabaticallyand begins to plunge into the horizon at the ISCO.

The counterpart of an ISCO for general geodesic particles is called alast stable orbit (LSO).



Natural fine-tuning for the ISCO particle

As a∗ → 1, we find

rISCO → rH,

EISCO → m0/√

3,

LISCO → 2m0 M/√

3,ΩH → 1/2M.

Thus,LISCO → EISCO/ΩH.

Therefore, the ISCO particle asymptotically satisfies the criticalcondition in the maximal rotation limit.



Near-horizon collision of an ISCO particle

Substitute E1 and L1 for the ISCO particle into the formula.

For a∗ ≈ 1,

Ecm

2m0≈ 1

21/231/4

√2e2 − l2

4√

1 − a2∗

, where e2 =E2

m0and l2 =

L2

m0 M.



On-ISCO collision of an ISCO particle

Since rISCO → rH as a∗ → 1, we don’t need to take the near-horizonlimit. Let’s consider the collision on the ISCO.

Near-horizon collision On-ISCO collisionThe different diverging behaviour for a∗ → 1.

Ecm

2m0≈ 1

21/631/4

√2e2 − l2

6√

1 − a2∗

for a∗ ≈ 1.



CM energy of an ISCO particle for general BH spins

1

2

5

10

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Ecm/(

2m0)

a∗

(a)(b)(c)

1

2

5

10

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Ecm/(

2m0)

a∗

(a)(b)(c)

Near-horizon collision On-ISCO collision

For a∗ = 0.998 (Thorne’s bound), γ = Ecm/(2m0) ' 3.86 − 6.95 for thenear-horizon collision and 2.43 − 4.11 for the on-ISCO collision.


Astrophysical relevance Collision of two general geodesic particles

Table of contents

.. .1 Introduction




.. .4 Summary



Inclined motion

Important not only because realistic in astrophysics but also becausehelps us to get a physical insight.

The conserved quantities are m, E, L and the Carter constant Q.



CM energy of colliding general geodesic particles

The CM energy for the general collision can be calculated. It isbounded except near the horizon.In the near-horizon limit,

E2cm = m2

1 + m22 +

1r2

H+ a2 cos2 θ

[(m2

1r2H + K1)

E2 − ΩH L2

E1 − ΩH L1

+(m22r2

H + K2)E1 − ΩH L1

E2 − ΩH L2

−2(L1 − a sin2 θE1)(L2 − a sin2 θE2)

sin2 θ− 2σ1θ

√Θ1(θ)σ2θ

√Θ2(θ)

,where Ki ≡ Qi + (Li − aEi)2 and

Θi(θ) = Qi − cos2 θ

a2(m2i − E2

i ) +L2

i

sin2 θ

.This includes the equatorial result.If we fine-tune L1 → L1c ≡ E1/ΩH, Ecm → ∞.T. Harada (Rikkyo U) Kerr BHs as accelerators 28/2-1/3/2012 27 / 37


High-velocity collision belt

E2 < m2/3: prohibitedE2 = m2/3: the prograde ISCO on the equatorial plane

E2 = m2: between latitudes ±acos√

2/3 ' ±35.26

E2 → ∞: between latitudes ±acos(√

3 − 1) ' ±42.94

The high-velocity collision belts lie between ±acos√

2/3 ' ±35.26

for bound particles and ±acos(√

3 − 1) ' ±42.94 with no restriction.

0

5

10

15

20

25

30

35

40

45

0 0.5 1 1.5 2 2.5 3 3.5 4

Lat

itude

(deg

ree)

Specific energy


Astrophysical relevance Robustness against gravitational radiation reaction

Table of contents

.. .1 Introduction




.. .4 Summary



Gravitational radiation reaction

We consider extreme-mass ratio inspirals (EMRIs) in the equatorialplane. We adopt the basic assumption:

−(

dEdt

)= EGW =

325η2Ω10/3E,

where η = µ/M 1 is the mass ratio, the right-hand side isestimated through circular orbits, Ω is the nondimensional angularvelocity and E is the correction factor.T. Harada (Rikkyo U) Kerr BHs as accelerators 28/2-1/3/2012 30 / 37


Method

Due to gravitational radiation reaction, the particle no longer movesalong a geodesic of the Kerr metric.

The transition from adiabatic inspiral to plunge was formulated by Ori& Thorne (2000) and refined by Kesden (2011).

Large spin limit: For δ = 1 − a∗ → 0, gravitational radiation power isvanishing. In fact, the numerical results by the GREMLIN code can befit by

E = Aδm,

where A ' 1.80 and m ' 0.317 (Hughes, Kesden 2011).

The Ori-Thorne-Kesden formalism can be justified ifε ' η2/5δ2m/5−1/3 . 1.



Maybe bounded but significantly high

The radiative effects are subdominant for ε . 1.The near-horizon collision (assuming m = 1/3)

Ecm ' 2.6 × 1030GeV( µ2

100GeV

)1/2 (M

10M

)1/2

ε5/4

' 4.6 × 1058erg(µ2

M

)1/2 (M

108 M

)1/2

ε5/4,

The on-ISCO collision (assuming m = 1/3)

Ecm ' 1.3 × 1021GeV( µ1

100GeV

)1/6 ( µ2

100GeV

)1/2 (M

10M

)1/3

ε5/6

' 2.3 × 1057erg(µ1

M

)1/6 (µ2

M

)1/2 (M

108 M

)1/3

ε5/6.

Both are bounded for ε . 1, but still high enough of physical interest.


Astrophysical relevance Magnetised Schwarzschild BHs

Table of contents

.. .1 Introduction




.. .4 Summary



Effect of an astrophysical magnetic field

Magnetic fields around astrophysical black holesB ∼ 108 Gauss for stellar mass BHsB ∼ 104 Gauss for supermassive BHs

Very weak effect on the geometry

B Bmax =c4

G3/2 M∼ 1019

(M

M

)−1

Gauss

Very strong effect on the orbits of charged particles

b =qBGM

mc4∼ 1011

(B

108Gauss

) (M

10M

)



Magnetised Schwarzschild BHs as particle accelerators

Let’s consider a uniform magnetic field around a Schwarzschild BH.

As b → ∞, rISCO → rH but EISCO < ∞ for a charged particle.

Let’s consider the collision of such a particle of mass m0 with aneutral particle of mass m0 which is initially at rest at infinifty. The CMenergy becomes (Frolov 2011)

Ecm

m∼ 1.74b1/4.

This factor is ∼ 103 for stellar mass BHs and ∼ 104 for supermassiveBHs.

Want to know more? → Igata-san’s talk


Summary

Table of contents

.. .1 Introduction




.. .4 Summary


Summary

Summary

An arbitrarily high CM energy for two colliding geodesic particles nearthe horizon in the maximal rotation limit

The natural fine-tuning of the angular momentum realised by aparticle orbiting the ISCO

The high-velocity collision belt at the latitude between ±43 near thehorizon in the maximal rotation limit of the BH.

Gravitational radiation reaction will not so significantly constrain theCM energy at least within the validity of the transition formalism.

Magnetic fields can be very important for particle accelerations.References

Banados, Silk and West, PRL 103 (2009) 111102T. Harada and M. Kimura, PRD 83 (2011) 024002T. Harada and M. Kimura, PRD 83 (2011) 084041T. Harada and M. Kimura, PRD 84 (2011) 124032V. P. Frolov, PRD 85 (2012) 024020


Dependence on the BH spins (details)

Table of contents

.. .5 Dependence on the BH spins (details)

.. .6 Relevance of the ISCO (details)

.. .7 ISCO (details)

.. .8 Collision of two general geodesic particles (details)

.. .9 Robustness against gravitational radiation reaction (details)

...10 Summary (details)



Kerr BH

Kerr metric

ds2 = −(1 − 2Mr

ρ2

)dt2 − 4Mar sin2 θ

ρ2dφdt +

ρ2

∆dr2 + ρ2dθ2

+

(r2 + a2 +

2Mra2 sin2 θ

ρ2

)sin2 θdφ2,

where ρ2 = r2 + a2 cos2 θ and ∆ = r2 − 2Mr + a2.

We assume 0 ≤ a2 ≤ M2. ∆ vanishes at r = r± = M ±√

M2 − a2.The horizon radius is rH = r+.Two commuting Killing vectors: ξa = (∂/∂t)a and ψa = (∂/∂φ)a.Horizon null generator: χa = ξa + ΩHψ

a

ΩH =a

r2H+ a2

=a

2M(M +√

M2 − a2).



Geodesic motion on the equatorial plane (1)

Conserved quantities for the equatorial motion:

m2 = −pa pa, E = −pt = −ξa pa, L = pφ = ψa pa

The geodesic equation reduces to the first-order form

t =1∆

[(r2 + a2 +

2Ma2

r

)e − 2Ma

rL],

φ =1∆

[(1 − 2M

r

)L +

2Mar

e],

12

r2 + V(r) = 0,

where the dot is the derivative w.r.t. the affine parameter.



Geodesic motion on the equatorial plane (2)

The effective potential

V(r) = −m2 Mr+

L2 − a2(E2 − m2)

2r2−

M(L − aE)2

r3− E2 − m2

2.

‘Forward in time’ condition: to have t > 0 near the horizon

L ≤ Lc ≡ E/ΩH.


Relevance of the ISCO (details)

Table of contents









Innermost Stable Circular Orbit (ISCO)

No stable circular orbit near the horizon.

A particle inside the ISCO plunges into the horizon.



Significance of the high-velocity collision

γ = Ecm/(2m0) ∼ 4 − 7 may not be so high as a particle accelerator,but ...The proton collision with ∼ 10 GeV occurs near the inner edge of theaccretion disk, which might be observable.The high-velocity collision of compact objects can occur around asupermassive BH. This is currently under investigation by numericalrelativity.

Sperhake et al. 2008 Sperhake et al. 2009



相対論的なコンパクト天体衝突

数値相対論によると、BH正面衝突で γが大きい極限で Ecmの約 14%、回って合体する場合は 25-35 %程度重力波として放出される。ただし γ ' 3まで (Shibata et al. 2008, Sperhake et al. 2008, 2009)。超大質量 BH周りのコンパクト天体は試験粒子近似できるから、

EGW ∼ 1055erg(ε/0.2)(γ/5)(m0/10M), fGW ∼ 1kHz(m0/10M)−1

ただし BHに吸い込まれる部分がある。強い赤方偏移も受ける。T. Harada (Rikkyo U) Kerr BHs as accelerators 28/2-1/3/2012 45 / 37

ISCO (details)

Table of contents








ISCO (details)

ISCO particle

The circular orbit on the equatorial plane in the Kerr metric is given byVeff(r) = V′

eff(r) = 0. The condition implies (s = ±1)

e =r1/2(r − 2M) + saM1/2

r3/4(r3/2 − 3Mr1/2 + s2aM1/2)1/2,

L = sM1/2(r2 + a2 − s2M1/2ar1/2)

r3/4(r3/2 − 3Mr1/2 + s2aM1/2)1/2.

The ISCO is determined by the condition de/dr = dL/dr = 0.rISCO

M= 3 + Z2 − s[(3 − Z1)(3 + Z1 + 2Z2)]1/2,

Z1 = 1 + (1 − a2∗)

1/3[(1 − a∗)1/3 + (1 + a∗)1/3],Z2 = (3a2

∗ + Z21)1/2.

rISCO/rH decreases from 3 to 1 as a∗ is increased from 0 to 1.e → 1/

√3, l → 2/

√3 and hence l → lH as a∗ → 1


ISCO (details)

ISCO particle

The circular orbit is given by V(r) = V′(r) = 0. Then, we findE = E(r) and L = L(r) as a function of r.The ISCO is determined by dE(r)/dr = dL(r)/dr = 0. The radius ofthe prograde ISCO is given by (Bardeen, Press & Teukolsky 1972)

rISCO

M= 3 + Z2 − [(3 − Z1)(3 + Z1 + 2Z2)]1/2,

Z1 = 1 + (1 − a2∗)

1/3[(1 − a∗)1/3 + (1 + a∗)1/3],Z2 = (3a2

∗ + Z21)1/2.

The energy and angular momentum are given by E = E(rISCO) andL = L(rISCO), respectively.


Collision of two general geodesic particles (details)

Table of contents









General geodesic particle, massive or massless

The conserved quantities are m, E, L and the Carter constant Q.

The tragectories in r and θ are determined by (Carter 1968)

ρ2 t = −a(aE sin2 θ − L) +(r2 + a2)P(r)∆(r)

,

ρ2φ = −(aE − L

sin2 θ

)+

aP(r)∆(r)

,

ρ2 r = σr√

R(r), ρ2θ = σθ√Θ(θ), (σr = ±1, σθ = ±1),

R(r) = P2(r) − ∆(r)[m2r2 + (L − aE)2 + Q],P(r) = (r2 + a2)E − aL,

Θ(θ) = Q − cos2 θ

[a2(m2 − E2) +

L2

sin2 θ

].

‘Forward-in-time’ condition: L ≤ Lc ≡ E/ΩH.



CM energy of two colliding general geodesic particles

The CM energy of two colliding geodesic particles:

E2cm = m2

1 + m22 +

2ρ2

P1 P2 − σ1r√

R1σ2r√

R2

∆

−(L1 − a sin2 θE1)(L2 − a sin2 θE2)

sin2 θ− σ1θ

√Θ1σ2θ

√Θ2

,where and hereafter Ei, Li, Qi, Pi = Pi(r), Ri = Ri(r) and Θi = Θi(θ)are E, L, Q, P = P(r), R = R(r) and Θ = Θ(θ) for particle i,respectively.

If mi, Ei, Li, Qi are bounded, Ecm is also bounded except near thehorizon where ∆ = 0.



Classification of critical particles

Putting V(r) = −R(r)

2r4, we find

12

r2 +r4

(r2 + a2 cos2 θ)2V(r) = 0.

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 10 100

Veff(r

)=−

R(r

)/(2

r4)

r

III

IIIIV

Class R(r) at r = rH BH spin ScenarioI R = R′ = 0, R′′ > 0 a∗ = 1 Direct collisionII R = R′ = R′′ = 0 a∗ = 1 LSO (ISCO) collisionIII R = R′ = 0, R′′ < 0 a∗ = 1 Multiple scatteringIV R = 0, R′ < 0 0 < a∗ < 1 Multiple scattering



Condition for the occurrence of critical particles

We only adopt the direct collision and the LSO (ISCO) collision hereso that we can assume R = R′ = 0 and R′′ ≥ 0 at r = rH.

Then, the BH must be maximally rotating and

R′′(rH) = 2[(3E2 − m2)M2 − Q] ≥ 0

as well as Θ ≥ 0. This implies

cos2 θ

[M2(m2 − E2) +

4M2E2

sin2 θ

]≤ Q ≤ (3E2 − m2)M2.

For Q to exist, we find E2 ≥ m2/3 and

sin θ ≥

√√−(4E2 − m2) +

√12E4 − 4E2m2 + m4

m2 − E2.



Direct collision from infinity

R for the critical particle for the extremal Kerr BH is given by

R = (r − M)2[(E2 − m2)r2 + 2ME2r − Q].

Then, for the marginally bound (E2 = m2) and unbound (E2 > m2)case, we can easily find

R′′(rH) ≥ 0 ⇐⇒ [R(r) > 0 for rH < ∀r < ∞]

Therefore, the result still applies even if we consider direct collisionfrom infinity.



Multiple-scattering scenario

Grib & Pavlov (2010) proposed a possibility of an arbitrarily high CMenergy even for a nonmaximally rotating BH.

A particle with L = Lc − δ (δ > 0) cannot approach the horizon fromwell outside due to the potential barrier.However, a particle might be put near the horizon ‘initially’ throughmultiple scattering with other particles beforehand.


Robustness against gravitational radiation reaction (details)

Table of contents









Gravitational radiation reaction

We consider extreme-mass ratio inspirals (EMRIs) in the equatorialplane. We adopt the basic assumption:

−(

dEdt

)= EGW =

325η2Ω10/3E,

where η = µ/M 1 is the mass ratio, Ω is the nondimensionalangular velocity and E is the correction factor.The particle no longer moves along a geodesic of the Kerr metric.T. Harada (Rikkyo U) Kerr BHs as accelerators 28/2-1/3/2012 57 / 37


Geodesic motion

The geodesic equation→ “the equation of motion”

d2 rdτ2= −1

2∂V∂r

The normalisation (drdτ

)2

= E2 − V

The effective potential

V(r, E, L) = 1 − 2r+

L2 + a2 − E2 a2

r2−

2(L − Ea)2

r3,

where r = r/M, t = t/M, a = a/M, τ = τ/M, E = E/µ, andL = L/(µM).



Trasition formalism: Ori & Thorne (2000)

Expand V(r, E, L) around the ISCO in terms of R = r − rISCO,χ = Ω−1

ISCO(E − EISCO), and ξ = L − LISCO. Then, the “EOM” implies

d2Rdτ2= −αR2 + βξ − 1

2

(Ω∂2V

∂E∂r

)ISCO

(χ − ξ) + · · · ,

where

α =14

(∂3V∂r3

)ISCO

, β = −12

(∂2V

∂L∂r+ Ω

∂2V

∂E∂r

)ISCO

.

The energy and angular momentum losses

χ = ξ = −ηκτ, where κ =325

(Ω7/3 dt

dτE)

ISCO.

Suggested from δE = ΩδL for circular orbits.Redefinition of the variables

R = η2/5R0X, τ = η−1/5τ0T, R0 = (βκ)2/5α−3/5, τ0 = (αβκ)−1/5.



Ori-Thorne solution

The Ori-Thorne equation and the numerical solution

X = −X2 − T.

The asymptotic behaviours

X ≈√−T as T → −∞, X ≈ − 6

(Tdiv − T)2as T → Tdiv ' 3.412.



Transition formalism: Kesden (2011)

To restore the consistency with the normalisation, we introduce

χ − ξ = η6/5(χ − ξ)0Y, (χ − ξ)0 = α−4/5(βκ)6/5

(∂V

∂L

)−1

ISCO.

The EOM and the normalisation imply

X = −X2 − T + εY, Y = 2X + 2εYX,

where

ε = η2/5C, C = −12α−3/5(βκ)2/5

Ω ∂2V

∂E∂r

(∂V

∂L

)−1ISCO

.

The asymptotic behaviours:

Y ≈ −43

(−T)3/2 for T → −∞, Y ≈ − 12Tdiv − T

for T → Tdiv.

The lowest-order formalism will be justified only if ε . 1.



Energy and angular momentum losses

Applying the transition formalism, we obtain

E = EISCO + ∆Etr + ∆Enorm, L = LISCO + ∆Ltr,

where

∆Etr = ΩISCO∆Ltr = −ΩISCOη4/5κτ0T,

∆Enorm = ΩISCOη6/5(χ − ξ)0Y.

The ISCO crossing time T0 ' 0.72: X(T0) = 0.

The horizon crossing time TH: X(TH) = XH



Maximal rotation limit

For δ = 1 − a∗ → 0, gravitational radiation power is vanishing. In fact,the numerical results by the GREMLIN code can be fit by

E = Aδm,

where A ' 1.80 and m ' 0.317 (Hughes, Kesden 2011).

After a straightforward calculation, we obtain ε ' η2/5δ2m/5−1/3 and

∆Etr ' −28/33−1/2δ1/3ε2T,∆Enorm ' 24/331/2δ2/3ε3Y.



Radiative effects are subdominant

Horizon crossing: Since XH ' −1/(2ε), Tdiv − TH ' 2√

3ε andY(TH) ' −2

√3ε−1/2. Then, we find

∆Etr ' −28/33−1/2δ1/3ε2TH,

∆Enorm ' −27/33δ2/3ε5/2.

When the object plunges into the horizon, to estimate

E − ΩH L ' µ[∆Enorm − (ΩH − ΩISCO)∆Ltr + (EISCO − ΩH LISCO)

],

we find

∆Enorm = O(ε5/2δ2/3),−(ΩH − ΩISCO)∆Ltr = O(ε2δ2/3),(EISCO − ΩH LISCO) = O(δ1/2).

Therefore, the last term is dominant in the limit δ → 0. It is justified toneglect GWs as the first approximation.



Maybe bounded but significantly high

The near-horizon collision (assuming m = 1/3)

Ecm ' 2.6 × 1030GeV( µ2

100GeV

)1/2 (M

10M

)1/2

ε5/4

' 4.6 × 1058erg(µ2

M

)1/2 (M

108 M

)1/2

ε5/4,

The on-ISCO collision (assuming m = 1/3)

Ecm ' 1.3 × 1021GeV( µ1

100GeV

)1/6 ( µ2

100GeV

)1/2 (M

10M

)1/3

ε5/6

' 2.3 × 1057erg(µ1

M

)1/6 (µ2

M

)1/2 (M

108 M

)1/3

ε5/6.

Both are bounded for ε . 1, but still high enough of physical interest.


Summary (details)

Table of contents








Summary (details)

as particle accelerators

(Nearly) extremal BHsas neutral particle accelerators

Kerr BHs (Banados, Silk & West 2009)general rotating BHs (Zaslavskii 2010, 2012)Kerr-Newmann family of BHs (Wei et al. 2010)Sen BHs (Wei et al. 2010)accelerating and rotating BHs (Yao et al. 2011), ...

as charged particle acceleratorReissner-Nordstrom BHs (Zaslavskii 2010)general stationary charged BHs (Zhu et al. 2011), ...

BHs with strong test fieldsWeakly magnetised BHs as charged particle accelerators (Frolov 2011)

Naked singularities (NSs)Kerr NSs as neutral particle accelerators (Patil & Joshi 2011)Reissner-Nordstrom NSs as charged particle accelerators (Patil et al.2011)


Summary (details)

Towards physics and astrophysics

On the other hand, the physical quantity is still diverging. This is notyet astrophysics. Bounding the CM energy is very important.

Observational effects? That is the question! Be patient! RememberBlandford-Znajek effect appeared in 1977, which was 8 years afterthe discovery of Penrose process in 1969. We are working in thisdirection.


粒子加速器としての回転 bh high-velocity collision...

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