The Astrophysical Journal, 792:88 (10pp), 2014 September 10 doi:10.1088/0004-637X/792/2/88C© 2014. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

CAUSAL EXTRACTION OF BLACK HOLE ROTATIONAL ENERGY BYVARIOUS KINDS OF ELECTROMAGNETIC FIELDS

Shinji Koide and Tamon BabaDepartment of Physics, Kumamoto University, 2-39-1, Kurokami, Kumamoto 860-8555, Japan

Received 2013 November 22; accepted 2014 June 25; published 2014 August 20

ABSTRACT

Recent general relativistic magnetohydrodynamics (MHD) simulations have suggested that relativistic jets fromactive galactic nuclei (AGNs) have been powered by the rotational energy of central black holes. Some mechanismsfor extraction of black hole rotational energy have been proposed, like the Penrose process, Blandford–Znajekmechanism, MHD Penrose process, and superradiance. The Blandford–Znajek mechanism is the most promisingmechanism for the engines of the relativistic jets from AGNs. However, an intuitive interpretation of this mechanismwith causality is not yet clarified, while the Penrose process has a clear interpretation for causal energy extractionfrom a black hole with negative energy. In this paper, we present a formula to build physical intuition so that inthe Blandford–Znajek mechanism, as well as in other electromagnetic processes, negative electromagnetic energyplays an important role in causal extraction of the rotational energy of black holes.

Key words: black hole physics – galaxies: nuclei – gamma-ray burst: general – magnetic fields –magnetohydrodynamics (MHD) – plasmas – stars: black holes

1. INTRODUCTION

A number of observations suggest that phenomena in the mostactive regions in the universe are related to black holes. Someof the most active objects in the universe—for example, activegalactic nuclei (AGNs), microquasars (black hole binaries), andgamma-ray bursts (GRBs)—emit relativistic jets (Biretta et al.1999; Pearson & Zensus 1987; Mirabel & Rodriguez 1994;Tingay et al. 1995; Kulkarni 1999). It is believed that these rel-ativistic jets are caused by the drastic phenomena around theblack holes at the centers of these objects. The possible energysources of the drastic phenomena are the gravitational energyof the matter falling toward the black hole and the rotationalenergy of the black hole itself. Recently, numerical simulationsof general relativistic magnetohydrodynamics (GRMHD) havesuggested that the relativistic jet is launched from the vicinityof the black hole, i.e., inside of the ergosphere (Koide 2004;Koide et al. 2006), and some long-term simulations showedthat the energy seems to be supplied from the rotational energyof the black hole (McKinney 2006; McKinney et al. 2012). Inthese GRMHD simulations, the black hole rotational energyseems to be extracted through the magnetic field flux tubes dueto the so-called Blandford–Znajek mechanism (Blandford &Znajek 1977). It was proposed as the mechanism in the force-free condition, by which the rotational energy of the black hole isextracted directly through the horizon along the magnetic fluxtubes. However, in principle, causality prohibits the outwardtransportation of any material, energy, and information acrossthe horizon. Thus, as pointed out by Punsly & Coroniti (1989,1990a, 1990b), the Blandford–Znajek mechanism seems to becontradictory to causality. On the other hand, in the Penrose pro-cess, the black hole spin energy is extracted causally due to thenegative energy at infinity (or just called “energy”) of a particlecaused by fission (Penrose 1969). Takahashi et al. (1990) andHirotani et al. (1992) found an axisymmetric steady-state solu-tion of ideal MHD plasma inflow with negative energy towardthe rotating black hole. When the negative energy of the inflow-ing plasma in the ergosphere is swallowed by the black hole, theblack hole rotational energy decreases, that is, the black holeenergy is extracted, as in the Penrose process. The difference

between the ideal MHD mechanism and the Penrose process isthat the negative energy is produced by the magnetic tensionforce in the ideal MHD inflow, while in the Penrose process it iscaused by the fission of a particle. This MHD energy extractionmechanism is called the “MHD Penrose process” (see Table 1).The MHD Penrose process was mimicked and confirmed by theGRMHD simulations of initially uniform, very strongly magne-tized plasma around a rapidly rotating black hole, which showedthat the negative energy of plasma is produced quickly in theergosphere (Koide et al. 2002; Koide 2003). However, becauseof the short time duration of the simulation, the numerical solu-tion is far from a stationary state. Komissarov (2005) performeda long-term GRMHD simulation with a similar initial situationto that of Koide (2003) and confirmed the MHD Penrose pro-cess in the early stage. Furthermore, he found that the MHDPenrose process is a transient phenomenon, and alternately,the outward electromagnetic energy flux through the horizoncontinues to appear almost everywhere with the exception ofa very thin equatorial belt. He remarked that the pure elec-tromagnetic mechanism with ideal MHD condition continuesto operate to extract the rotational energy of the black hole.Strictly speaking, this electromagnetic mechanism should bedistinguished from the original Blandford–Znajek mechanismbecause the original mechanism is derived with the force-freecondition, while the electromagnetic energy extraction mecha-nism was shown with the ideal MHD simulations. In this paper,we call the mechanism shown by the simulations the “MHDBlandford–Znajek mechanism,” while the original mechanismis called the “force-free Blandford–Znajek mechanism.” Con-sidering the numerical results, Komissarov (2009) discussed theelectromagnetic extraction mechanism of the black hole energy,including the force-free Blandford–Znajek mechanism, MHDPenrose mechanism, and superradiance in the wide view. How-ever, unfortunately, a convincing explanation with respect to thecausality of these mechanisms, which should also yield the con-ditions of the mechanisms, is not given, except for the MHD Pen-rose process (Komissarov 2009). Koide (2003) pointed out thatthe force-free Blandford–Znajek mechanism uses the negativeelectromagnetic energy at infinity to extract the spin energy ofthe black hole. This point of view was discussed extensively by

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The Astrophysical Journal, 792:88 (10pp), 2014 September 10 Koide & Baba

Table 1Classification of Various Mechanisms of Energy Extraction from a Black Hole

Mechanism Form of Negative Energy Torque for Redistribution Output Energy Referencesof Angular Momentum

Penrose process Mechanical energy ofparticle

Force of particle fission Mechanical energy ofparticle

Penrose (1969)

Magnetic Penrose process Mechanical energy ofelectrically chargedparticles

Electromagnetic force Mechanical energy ofelectrically chargedparticles

Wagh (1989)

Force-free Blandford–Znajekmechanism

Electromagnetic energy Electromagnetic tensionforce (force-free)

Electromagnetic energy Blandford & Znajek (1977)

MHD Blandford–Znajekmechanism

Electromagnetic energy Electromagnetic tensionforce (MHD)

Electromagnetic energyand kinetic energy(Alfven wave)

Takahashi et al. (1990); Koide(2003); Komissarov (2005)

MHD Penrose process Mechanical energy ofplasma

Lorentz force (magnetictension, MHD)

Electromagnetic energyand kinetic energy(Alfven wave)

Takahashi et al. (1990);Hirotani et al. (1992); Koideet al. (2002); Koide (2003)

Energy extraction withmagnetic reconnection

Mechanical energy ofplasmoid

Magnetic tension due tomagnetic reconnection

Mechanical energy ofplasmoid

Koide (2009)

Superradiance Electromagnetic energyof electromagnetic wave

“Half-mirror” effect dueto quantum tunneling

Electromagnetic energyof electromagnetic wave

Press & Teukolsky (1972);Teukolsky & Press (1974);Lightman et al. (1975)

Krolik et al. (2005) and Lasota et al. (2014) for ideal MHD andforce-free Blandford–Znajek mechanisms, respectively. How-ever, it is often difficult to build physical intuition regarding theMHD/force-free Blandford–Znajek mechanisms with causality.Here we present an intuitive formula for the electromagneticmechanism of the energy extraction from the rotating black holeto aid in building physical intuition regarding the mechanisms.The formula is also applicable to other electromagnetic mecha-nisms such as the MHD Penrose process (Takahashi et al. 1990;Hirotani et al. 1992; Koide et al. 2002; Koide 2003) and super-radiance (Press & Teukolsky 1972; Teukolsky & Press 1974;Lightman et al. 1975).

In Section 2, we review the energy and angular mo-mentum transport of the electromagnetic field around theblack holes briefly but sufficiently. In Section 3, we explainthe electromagnetic mechanisms of black hole energy ex-traction that is, the force-free Blandford–Znajek mechanism,MHD Blandford–Znajek mechanism, and superradiance withincausality. We summarize our explanation about the energyextraction mechanisms from the black hole, including bothBlandford–Znajek mechanisms, in Section 4.

2. ELECTROMAGNETIC ENERGY AND ANGULARMOMENTUM TRANSPORT NEAR

A ROTATING BLACK HOLE

We review the electromagnetic energy and angular momen-tum transport in the spacetime (x0, x1, x2, x3) around a spinningblack hole based on the so-called “3+1 formalism.” The scale ofa small element in the spacetime around the rotating black holeis given by

ds2le = gμνdxμdxν = −h2

0dt2 +3∑

i=1

[h2

i (dxi)2 − 2h2i ωidtdxi

].

(1)Here we have gij = 0 (i �= j ), g00 = −h2

0, gii = h2i , and

gi0 = g0i = −h2i ωi , where Greek indices (μ, ν) run from 0 to 3

and Roman indices (i, j ) run from 1 to 3. Throughout this paper,we use the natural unit system, where the light speed, electricpermittivity, and magnetic permeability in a vacuum are unity:c = 1, ε0 = 1, and μ0 = 1. When we define the lapse functionα and shift vector βi by

α =√√√√h2

0 +3∑

i=1

(hiωi)2, βi = hiωi

α, (2)

the line element ds is written as

ds2le = −α2dt2 +

3∑i=1

(hidxi − αβidt)2. (3)

The determinant of the matrix with elements gμν is given by√−‖g‖ = αh1h2h3, and the contravariant metric is writtenexplicitly as g00 = −(1/α2), gi0 = g0i = −(βi/αhi), andgij = (1/hihj )(δij − βiβj ), where δij is the Kronecker δsymbol.

The relativistic Maxwell’s equations are

∇μ∗Fμν = 0, (4)

∇μFμν = −J ν, (5)

where ∇ν is the covariant derivative, Fμν is the electromagneticfield-strength tensor, and ∗Fμν is the dual tensor of Fμν ,∗Fμν ≡ (1/2)εμνλσFλσ (εμνλσ is the Levi–Civita antisymmetrictensor, which is a tensor density of weight −1), and J ν =(ρe, J

1, J 2, J 3) is the electric four-current density (ρe is theelectric charge density)(Jackson 1979). The electric field Ei andthe magnetic field Bi are given by Ei = Fi0 (i = 1, 2, 3) andB1 = F23, B2 = F31, B3 = F12, or Bi = (1/2)ε0ijkFjk = ∗F 0i ,respectively. Using the four-vector potential Aμ, we have Fμν =∇μAν −∇νAμ = ∂μAν − ∂νAμ because of the symmetry of theChristoffel symbols, Γλ

μν = Γλνμ.

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The Astrophysical Journal, 792:88 (10pp), 2014 September 10 Koide & Baba

The electromagnetic energy–momentum tensor Tμν

EM isgiven by

Tμν

EM = FμσF νσ − 1

4gμνF λκFλκ . (6)

The total energy–momentum tensor T μν is

T μν = pgμν + hUμUν + Tμν

EM, (7)

where p, h, and Uμ are the proper pressure, the proper enthalpydensity, and the four-velocity of the plasma, respectively. Theenergy–momentum conservation law is given by

∇μT μν = 0. (8)

The force-free condition is

JμFμν = 0, (9)

and the general relativistic Ohm’s law is

FμνUν = η[Jμ + (UνJν)Uμ], (10)

where η is the resistivity of the plasma. The ideal MHD conditionis given by setting η = 0, FμνU

ν = 0.Here we introduce a local coordinate frame, the so-called

“fiducial observer (FIDO) frame,” (x0, x1, x2, x3). Using thelocal coordinates of the frame xμ, the line element becomes

ds2le = ημνdxμdxν = −dt2 +

3∑i=1

(dxi)2,

where ημν is the metric of Minkowski spacetime. Comparingthis metric with Equation (3), we get

dt = αdt, dxi = hidxi − αβidt, (11)

and we have partial derivative relations,

∂

∂t= ∂t

∂t

∂

∂t+

∑i

∂xi

∂ t

∂

∂xi= 1

α

∂

∂t+

∑i

βi

hi

∂

∂xi,

∂

∂xi= 1

hi

∂

∂xi. (12)

Then a contravariant vector aμ in the FIDO frame of anarbitrary contravariant vector aμ in the global coordinates xμ iswritten as

a0 = αa0, ai = hiai − αβia0 (13)

and the covariant vector aμ is

a0 = 1

αa0 +

∑i

βi

hi

ai, ai = 1

hi

ai . (14)

We use the quantities observed by the FIDO frame because theycan be treated intuitively and yield formulae more easily. This isbecause the relations between the variables in the FIDO frameare the same as those in the special theory of relativity andsimilar to the Newtonian relation.

Using the quantities of the electromagnetic field in the FIDOframe, Maxwell’s equations are written using the following 3+1formalism,

∂Bi

∂t= −

∑j,k

hi

h1h2h3εijk ∂

∂xj

[αhk

(Ek −

∑l,m

εklmβlBm

)],

(15)

α(J i + ρeβi) +

∂Ei

∂t=

∑j,k

hi

h1h2h3εijk ∂

∂xj

[αhk

(Bk +

∑l,m

εklmβlEm

)],

(16)

∑i

1

h1h2h3

∂

∂xi

(h1h2h3

hi

Bi

)= 0, (17)

ρe =∑

i

1

h1h2h3

∂

∂xi

(h1h2h3

hi

Ei

), (18)

where εijk = ε0ijk .For convenience, we introduce the derivatives of an arbitrary

three-vector field a and an arbitrary scalar field φ measured bythe FIDO frame as

∇ · a =∑

i

1

h1h2h3

∂

∂xi

(h1h2h3

hi

ai

), (19)

(∇φ)i = 1

hi

∂φ

∂xi, (20)

(∇ × a)i =∑j,k

hi

h1h2h3εijk ∂

∂xj(hka

k). (21)

We express Maxwell’s equations in vector form as

∂B∂t

= −∇ × [α(E − β × B)], (22)

α(J + ρeβ) +∂E∂t

= ∇ × [α(B + β × E)], (23)

∇ · B = 0, (24)

ρe = ∇ · E, (25)

where β = (β1, β2, β3), E = (E1, E2, E3), B = (B1, B2, B3),and J = (J 1, J 2, J 3).

The 3+1 form of the force-free condition is

J · E = 0, ρe E + J × B = 0, (26)

and Ohm’s law is written as

E + v × B = 1

γη[ J − ρ ′

eγ v], (27)

where γ = U 0 is the Lorentz factor, v = (U 1/γ , U 2/γ , U 3/γ )is the three-velocity, and ρ ′

e = −J νUν is the electric chargedensity observed by the plasma rest frame (the proper electriccharge density). The conservation equation of the electric chargeis derived using Equations (23) and (25) as

∂ρe

∂t+ ∇ · [α(J + ρeβ)] = 0. (28)

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The Astrophysical Journal, 792:88 (10pp), 2014 September 10 Koide & Baba

We present the equations of energy and angular momentumconservation around a spinning black hole. When ξμ is a Killingvector, we have a conservation law associated with Equation (8)

1√−‖g‖∂

∂xμ(√

−‖g‖T μνξν) = 0. (29)

Because ‖g‖ = −(αh1h2h3)2, this equation yields

∂

∂t(αT 0νξν) = − 1

h1h2h3

∑i

∂

∂xi(αh1h2h3T

iνξν). (30)

Using the Killing vector χν = (−1, 0, 0, 0), we have the law ofenergy conservation

∂e∞

∂t= −∇ · S, (31)

where e∞ ≡ αT 0νχν is called energy at infinity (or just“energy”) density and Si ≡ αhiT

iνχν is the ith componentof energy flux density. Here we also express these quantities inthe FIDO frame as

e∞ = α(ε + ργ ) +∑

i

αβiQi, (32)

Si = α

⎡⎣αQi + e∞βi +

∑j

αβj T ij

⎤⎦ , (33)

where ε + ργ = T 00, γ = U 0, and Qi = T 0i .We separate these quantities into the hydrodynamic and

electromagnetic components:

e∞ = e∞hyd + e∞

EM, (34)

Si = Sihyd + Si

EM, (35)

where

e∞hyd = α(hγ 2 − p) +

∑i

αβihγ 2vi , (36)

e∞EM = α

((B)2

2+

ˆ(E)2

2

)+

∑i

αβi(E × B)i , (37)

Sihyd = α2hγ 2

⎛⎝1 +

∑j

βj vj

⎞⎠ (vi + βi), (38)

SiEM = α2[(E − β × B) × (B + β × E)]i , (39)

where the subscripts “hyd” and “EM” indicate hydrodynamicand electromagnetic components, respectively, and (E)2 =(E1)2 + (E2)2 + (E3)2, (B)2 = (B1)2 + (B2)2 + (B3)2. Here,SEM = (S1

EM, S2EM, S3

EM) can be regarded as the Poynting vector.The general relativistic Maxwell’s Equations (22)–(25) yield

∂e∞EM

∂t= −∇ · SEM − α(v + β) · fL, (40)

where fL = ρeE + J × B is the Lorentz force density.

If ημ = (0, 0, 0, 1) is the Killing vector for the azimuthaldirection, we have the equation of angular momentum conser-vation

∂l

∂t= −∇ · M, (41)

where l ≡ αT 0νην and Mi ≡ αhiTiνην are the total angular

momentum density and the angular momentum flux density,respectively. Using the quantities measured in the FIDO frame,we have

l = h3Q3, (42)

Mi = αh3(T i3 + βiQ3). (43)

These variables also can be divided into the hydrodynamic andelectromagnetic components, denoted by the subscripts “hyd”and ‘EM,” as follows:

l = lhyd + lEM, (44)

Mi = Mihyd + Mi

EM, (45)

wherelhyd = h3hγ 2v3, (46)

lEM = h3(E × B)3, (47)

Mihyd = αh3[pδi3 + hγ 2vi v3 + cβihγ 2v3], (48)

MiEM = αh3

[((B)2

2+

(E)2

2

)δi3 − BiB3

− EiE3 + βi(E × B)3

]. (49)

In this case, from Equations (32) and (42), we have a relation ofthe energy and the angular momentum,

e∞ = α(ε + ργ ) + ω3l = α

[ε + ργ +

β3

h3l

], (50)

when ω1 = ω2 = 0. Furthermore, Equations (36), (37), (46),and (47) yield

e∞hyd = α(hγ 2 − p) + ω3lhyd, (51)

e∞EM = α

((B)2

2+

ˆ(E)2

2

)+ ω3lEM. (52)

The general relativistic Maxwell’s Equations (22)–(25) read

∂lEM

∂t= −∇ · MEM − h3f

3L . (53)

Hereafter, we consider the electromagnetic energy transportwhen we have the relation between the electric field andmagnetic field as

E = −vF × B. (54)

Here, vF is a certain vector field and does not always meanthe real velocity, while in the ideal MHD case, it is identifiedby the plasma velocity v. It is noted that the drift velocity due to

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The Astrophysical Journal, 792:88 (10pp), 2014 September 10 Koide & Baba

the electric field E, vE = ((E × B)/B2), can be used as one ofthe vectors of vF. Intuitively, vF is regarded as the velocity of themagnetic field lines, while this intuition is not rigorous becausewe cannot identify the magnetic field lines at the different times.However, we do not have such a serious contradiction with theinterpretation, and we often recognize vF as the velocity of thefield line implicitly. Using Equation (54), we have

e∞EM = α

((B)2

2+

(E)2

2

)+ αβ · (E × B)

= α

[1

2

(1 + v2

F⊥)

+ β · vF⊥

](B)2, (55)

where vF⊥ is the component of vF perpendicular to the magneticfield B, vF = vF‖ + vF⊥, vF‖ ‖ B, vF⊥ ⊥ B. Here we used therelations (E)2 = (B)2v2

F⊥, E × B = (B)2vF⊥. With respect tothe energy transport flux density, we have

SEM = α2

[ {1

2

(1 + v2

F⊥)

+ β · vF⊥

}(B)2(vF⊥ + β) +

(1 − v2

F⊥)

×{

(B)2

2(vF⊥ + β) − (β · B)B

}]. (56)

Using Equations (55) and (56), we obtain

SEM = αe∞EM(vF⊥ + β) + α2(1 − v2

F⊥)

×{

B2

2(vF⊥ + β) − (β · B)B

}. (57)

With respect to the angular momentum of the electromagneticfield, assuming Equation (54), we have

lEM = h3(B)2v3F⊥, (58)

MiEM = αh3

[1

2

(1 + v2

F⊥)(B)2δi3 + βi(B)2v3

F⊥

− BiB3 − EiE3

]. (59)

In this case, we also have

e∞EM = α

(B)2

2

(1 + v2

F⊥)

+ ω3lEM. (60)

3. CAUSAL ENERGY EXTRACTION FROM BLACKHOLES WITH SEVERAL KINDS OF

ELECTROMAGNETIC FIELDS

3.1. Force-free Electromagnetic Field Case:Blandford–Znajek Mechanism

In this subsection, we consider the energy transport near thehorizon in the force-free limit case, which is assumed in theoriginal work on the Blandford–Znajek mechanism (Blandford& Znajek 1977). Here we use the Kerr metric for spacetime(x0, x1, x2, x3) = (t, r, θ, φ) with ωφ � 0 in this section. Theforce-free condition, JμFμν = 0, reads

J · E = 0, (61)

Figure 1. Geometric relation between the vectors B, BP, Bφ

, vF, vF‖, vF⊥,

and vφF⊥.

ρe E + J × B = 0. (62)

This means there are no energy and momentum transformsbetween the electromagnetic field and plasma. In such a case,we can write the electromagnetic field by Equation (54). Thisis because when ρe �= 0, we have E = −( J/ρe) × B, andconfirm Equation (54) with vF = ( J/ρe). When ρe = 0, we haveJ × B = 0, i.e., J ‖ B. Furthermore, because of Equation (61),we have J ⊥ E, and then E ⊥ B. We confirm Equation (54)with vF = (1/B2)E × B.

In the steady-state and axisymmetric case, Equations (15),(17), (61), and (62) yield

vF = hφ

α(ΩF − ωφ)eφ = R

α(ΩF − ωφ)eφ, (63)

where ΩF is a constant along the magnetic flux surface, R ≡hφ = h3 corresponds to the distance from the z axis, and eφ

is the unit vector for azimuthal direction (Blandford & Znajek1977). Because the triangle of vF⊥ and vF and the triangle of BP

and B are similar (Figure 1), we found the following relation,

vF⊥vF

= BP

B. (64)

Here we define BP and Bφ as the poloidal and azimuthalcomponents of the magnetic field B, respectively. We then have

vF⊥ = vF√1 + (Bφ/BP)2

. (65)

The Znajek boundary condition at the horizon (Znajek 1977) isexpressed as

Bφ

BP

= vφ

F . (66)

Then, very near the horizon, we also have

vF⊥ ≈ vF√1 + v2

F

, (67)

where “≈” means asymptotic equivalence. In the limit towardthe horizon (r → rH, rH is the radius of the black hole), wehave vF → ∞ when ΩF �= ΩH; we then find vF⊥ → 1. Here wewrite the value of ωφ at the horizon by ΩH. Eventually, usingEquation (57) we obtain, very near the horizon,

SEM = αe∞EM(vF⊥ + β). (68)

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The Astrophysical Journal, 792:88 (10pp), 2014 September 10 Koide & Baba

Figure 2. Geometry of vectors vF, vF⊥, β, and the magnetic field lines in the case of ΩF < ΩH (left panel) and ΩF > ΩH (right panel) near the horizon.

The directions of vF of the cases of ΩF < ΩH and ΩF > ΩHare opposite because of Equation (63), and the slopes of themagnetic field lines in the two cases are also opposite (Figure 2).Then, the direction of vF⊥ is always directed toward the innerregion of the black hole when ΩF �= ΩH. Then, when e∞

EM < 0,the electromagnetic energy flux is directed outward, and theenergy of the black hole is extracted through the horizon.

Next, we determine the condition of the negative energye∞

EM < 0 at the horizon. When ΩF �= ΩH, vF⊥ is directed towardthe black hole horizon in both cases of ΩF < ΩH and ΩF > ΩH.Because the triangle of vF⊥, vF and the triangle of BP, B aresimilar (Figure 1), we found (vφ

F⊥/vF⊥) = (BP/B), and we thenobtain

vφ

F⊥ =(

BP

B

)2

vF = vF

1 + v2F

. (69)

Finally, using the second equation of Equation (2) andEquation (63), we get

e∞EM =

[1

2

(1 +

v2F

1 + v2F

)+

βφvφ

F

1 + v2F

]αB2

= 1

2

α2 + 2R2ΩF(ΩF − ωφ)

α2 + R2(ΩF − ωφ)2αB2. (70)

At the horizon (α −→ 0, ωφ −→ ΩH), Equations (63) and (66)

yield B =√

(Bφ)2 + (BP)2 ≈ |Bφ| = |(R/α)(ΩF − ΩH)|BPH

because |Bφ| � BP, where BPH is the value of BP at the horizon.Eventually, at the horizon we found

e∞EM ≈ R2

H

αΩF(ΩF − ΩH)(BPH)2, (71)

SEM = R2HΩF(ΩF − ΩH)(BPH)2(vF⊥ + β), (72)

where RH is the value of R = hφ at the horizon. It is noted that theradial component of the electromagnetic energy flux is identicalto the simple equation given by McKinney & Gammie (2004,Equation (34) in the paper), if we set the force-free condition atthe horizon, v

⊥F = er . Then, when 0 < ΩF < ΩH, the negative

energy of the electromagnetic field is realized (e∞EM < 0), and the

rotational energy of the black hole is extracted. This is exactlythe same condition of the Blandford–Znajek mechanism. Thissuggests even in the Blandford–Znajek mechanism, to extract

Figure 3. Magnetic field surfaces in the steady-state ideal MHD case and thespatially orthogonal coordinates (s, Ψ, φ).

the black hole rotational energy, the negative energy of theelectromagnetic field is utilized as a mediator. In conclusion,putting the negative electromagnetic energy into the black hole,the black hole rotational energy is extracted causally in theBlandford–Znajek mechanism.

Sometimes the energy extraction of the rotating black holeis intuitively explained by the torque of the magnetic field atthe horizon. This intuitive explanation is not appropriate withrespect to causality because at the horizon, no torque affects thematter and field at the horizon outward. Equations (71) and (72)suggest that the falling of the negative (electromagnetic) energyinto the black hole could decrease the black hole energy toextract the black hole energy.

3.2. Ideal MHD Case: MHD Blandford–ZnajekMechanism/MHD Penrose Process

We consider the ideal MHD case in the spacetime aroundthe spinning black hole. We assume the situation is stationaryand axisymmetric, the same as the force-free case in theprevious section. In such a case, the magnetic flux surfacesare stationary and axisymmetric and are expressed as a constantazimuthal component of the vector potential, Aφ . We introducethe new coordinate system (t, s, Ψ, φ), where t is the timeof Kerr spacetime, φ is the azimuthal coordinate, Ψ = Aφ ,and the coordinate s is set outward along the intersectionline of a magnetic surface and the meridian plane (φ =const.) (Figure 3). Here we set the coordinates s so that itis perpendicular to the coordinate Ψ. The s coordinate at the

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horizon is sH. Essentially, this coordinate system correspondsto the Boyer–Linquist coordinate (t, r, θ, φ) where t = t ,s = s(r, θ ), Ψ = Ψ(r, θ ), and φ = φ. Then, the length ofa line element in the spacetime of the rotating black hole isgiven by

ds2le = −h2

t dt2 + h2s ds2 + h2

ΨdΨ2 + h2φdφ2 − 2h2

φωφdtdφ.

We assume the ideal MHD condition, UμFμν = 0, which yields

E + v × B = 0. (73)

Using the coordinates (s, Ψ, φ), Equations (15)–(18), (31), (41),and (73) yield the following conservation variables along themagnetic surface:

M(Ψ) = hφhΨραU s = αρU s

Bs, (74)

Bs(Ψ) = hΨhφBs = 1, (75)

ΩF(Ψ) = α

hφ

[vφ + βφ − Bφ

Bsvs

], (76)

L(Ψ) = hφ

[h

ρUφ − α

MBφ

], (77)

H (Ψ) = h

ρ[αγ − hφ(ΩF − ωφ)] = h

ρα(γ − v

φ

F Uφ). (78)

It is noted that quantities with hats are variables observed by theFIDO frame. It is also noted that the distribution of Ψ is deter-mined by the transverse equation called the “Grad–Shafranovequation” (Beskin & Kuznetsova 2000). Recently, numericalsimulations of GRMHD provide a more complete feature ofthe mechanism such as the distribution of Poynting flux overthe event horizon, the relative importance of negative energyat infinity fluid and electromagnetic field, the energy flux fromthe black hole to the disk through the magnetic field lines, etc.(McKinney et al. 2012; Hawley & Krolik 2006). It is noted thatthe numerical, time-dependent simulations showed that magne-torotational instability (MRI) always causes fluctuations, and nosteady state of plasma and the magnetic field is found.

At the black hole horizon, the lapse function α becomes 0, hsbecomes infinite, while ωφ = (αβφ/hφ) −→ ΩH, hΨ −→ hΨH,hφ −→ RH are finite except on the z axis. Hereafter, we discussthe quantities along a certain fixed magnetic flux surface Ψ = Ψ.

Because the horizon is not a real singular surface, and thedensity ρ and pressure p are measured by the plasma rest frame,ρ and p should be finite at the horizon. Then, from Equations (74)and (75), αU s and Bs must be finite at the horizon, where wewrite Bs at the horizon by Bs

H. At the horizon, the plasma fallsvertically to the horizon at the light velocity,1vφ = vΨ = 0,

1 This is also derived as follows. Extremely near the horizon, αBφ is finite,because vs is finite. Then, from Equation (77), Uφ is finite. At the horizon,

because αUs is finite and α −→ 0, U s and γ =√

1 + (U s )2 + (Uφ )2 are

infinite. Then, vφ = (Uφ/γ ) becomes zero at the horizon, and γ = |U s |.Finally, at the horizon, vs = (U s/γ ) = −1.

vs = −1, and the second equation of Equations (2) and (76)yield

αBφ

Bs≈ RH(ΩF − ΩH). (79)

Using Equations (73) and (76), we have

E = −v × B = Bs

(−vφ + vs Bφ

Bs

)eφ × es

= −R

α(ΩF − ωφ)eφ × (Bses) = −vF × B, (80)

where we put vF = (R/α)(ΩF − ωφ)eφ and eφ , es are the unitbase vectors along the φ and s coordinates, respectively. Verynear the horizon, we have

vF ≈ RH

α(ΩF − ΩH)eφ. (81)

Equations (79) and (81) present the geometrical disposition ofvectors B and vF, as shown in Figure 2. When ΩF �= ΩH, wefound that the vector of vF is always directed toward the blackhole inner region.

Intuitively, at the horizon of the rotating black hole, the plasmafalls into the black hole radially with the speed of light (vs = −1,vφ = 0 at s = sH). When the azimuthal component of themagnetic field is finite outside of the horizon and stationary, themagnetic field lines are twisted extremely strongly in appearancenear the horizon because of Equation (76) where αBφ is uniformalong the magnetic surface and α vanishes at the horizon. Thisis due to the difference in the lapse of time near the black holeand is an apparent feature in the Kerr metric. In such a case, theperpendicular component of the velocity to the magnetic fieldis identical to the plasma velocity; we then have

v⊥F = v⊥ = 1 (82)

at the horizon.2 Then, the electromagnetic energy flux densityat the horizon is given by

SEM = αe∞EM(vF⊥ + β), (83)

from Equation (57). When e∞EM becomes negative at the horizon,

the electromagnetic energy is transported outward through thehorizon when ΩF �= ΩH, because vF⊥ is always directed inwardtoward the black hole’s inner region (see Figure 2). Here,because vF⊥ vanishes if ΩF = ΩH, no electromagnetic output isexpected; we then consider only the case of ΩF �= ΩH.

As shown in Equation (55), the density of electromagneticenergy at infinity is given by

e∞EM = α

[1

2

(1 + v2

F⊥)

+ βφvφ

F⊥

](B)2. (84)

2 This equation is also derived as follows. Using the similarity of the triangleof vF⊥ and vF and the triangle of BP = B

sand B (Figure 1), we found

(vF⊥/vF) = (BP/B) = (Bs/B), where B =√

(Bs )2 + (Bφ )2 and BP = Bs .

Very near the horizon, because Bφ is much larger than Bs , we have(Bs/B) ≈ (Bs/|Bφ |), and then (vF⊥/vF) ≈ (Bs/|Bφ |). Using Equations (79)and (81), at the horizon we confirm vF⊥ = (Bs/|Bφ |)vF = 1.

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With Figure 1, we found (vφ

F⊥/vF⊥) = (vF⊥/vφ

F ) ; and we thenhave v

φ

F⊥ = ((vF⊥)2/vφ

F ). At the horizon, using Equations (81)and (82), we have

vφ

F⊥ ≈ α

RH(ΩF − ΩH). (85)

Using Equation (79), we also have B ≈ |Bφ| ≈ |(RH/α)(ΩF −ΩH)||BH

s | at the horizon because |Bφ| � |Bs |. Eventually, weobtain

e∞EM ≈ R2

H

αΩF(ΩF − ΩH)

(Bs

H

)2, (86)

SEM = R2HΩF(ΩF − ΩH)

(Bs

H

)2(vF⊥ + β). (87)

This clearly shows that when 0 < ΩF < ΩH, e∞EM becomes

negative and the electromagnetic energy flux is directed outwardthrough the horizon. It is surprising that not only the conditionof the electromagnetic energy extraction from the black hole butalso the expression of energy density and the energy flux densityat the horizon are the same as those of the Blandford–Znajekmechanism (force-free case).

In the above two cases of electromagnetic extraction of theblack hole rotational energy, the negative electromagnetic en-ergy at infinity is required as a mediator to extract the blackhole rotational energy through the horizon causally. As shownin Equation (60), we have e∞

EM = αuEM + ωφlEM, whereuEM = ((E)2/2) + ((B)2/2) is the electromagnetic energy den-sity in the FIDO frame. To realize the negative electromagneticenergy, the angular momentum of the electromagnetic field lEMshould become less than −αuEM/ωφ . Locally, the angular mo-mentum should be conserved because of Equation (41), andthen redistribution of the angular momentum is required. Inthe Penrose process, fission of a particle is utilized for redistri-bution of the angular momentum and production of a particlewith negative energy at infinity. Equation (53) indicates that dy-namically only the magnetic force (the magnetic tension in theaxisymmetric case) and the Lorentz force can redistribute theelectromagnetic angular momentum. In the ideal MHD case,magnetic tension plays an important role to redistribute theelectromagnetic angular momentum and realize the negativeelectromagnetic energy. This mechanism of energy extractionwith negative electromagnetic energy is often confused with the(original) Blandford–Znajek mechanism, where the force-freecondition is used, as we did in Section 1. However, strictlyspeaking, they should be distinguished. Hereafter, in this paper,we call the ideal MHD process with negative electromagneticenergy the “MHD Blandford–Znajek mechanism.” In the MHDBlandford–Znajek mechanism, we have to take the hydrody-namic energy flux of the plasma flow into account to discuss thenet energy flux from/into the black hole.

In fact, in the ideal MHD case, the black hole rotational energycan be also extracted with the negative hydrodynamic energy ofthe plasma. The hydrodynamic energy flux density is

Shyd = α(e∞

hyd + αp)(v + β).

Then, near the horizon if the plasma with αe∞hyd < 0 falls into

the black hole, the energy is transported outward through thehorizon because α −→ 0 at the horizon. If SEM is directed out-ward, αe∞

hyd must be smaller than zero to extract the black hole

rotational energy. This extraction mechanism of black hole rota-tional energy is called the “MHD Penrose process” (Takahashiet al. 1990; Hirotani et al. 1992; Koide et al. 2002; Koide 2003).The hydrodynamic energy is given by e∞

hyd = α(hγ −p)+ω · lhyd

where ω = (ω1, ω2, ω3) and lhyd = hγ 2h3v3 is the hydro-

dynamic angular momentum density. To realize the negativehydrodynamic energy, l3

hyd < −(α(hγ 2 − p)/ω3). Angular mo-mentum is conserved, and redistribution of the hydrodynamicangular momentum is also required. Redistribution of the hy-drodynamic angular momentum is caused by the Lorentz forceshown in Equation (53).

To distinguish between the MHD Blandford–Znajek mech-anism and MHD Penrose process, we should observe the den-sity of the electromagnetic and hydrodynamic energy at infinity(e∞

EM and e∞hyd). If the electromagnetic energy plays a main role

in extracting the black hole energy, we recognize the processas the MHD Blandford–Znajek mechanism. On the other hand,hydrodynamic or plasma energy has an important role in theextraction; this is recognized as the MHD Penrose process. Inactual cases, both are possible, while some long-term simula-tions indicate that the MHD Penrose process is transient, andthe MHD Blandford–Znajek mechanism is dominant in the laterphase of the simulations (Komissarov 2005; McKinney 2006).The electromagnetic extraction mechanisms of black hole ro-tational energy outlined in this paper are restricted to those inthe steady-state, axisymmetric cases. Recently, the long-termGRMHD simulations showed 3D dynamics of plasma inter-acting with the magnetic field around the rotating black hole(McKinney et al. 2012). Strictly speaking, the results of thispaper are not applicable to the time-dependent, axi-asymmetricnumerical results. Generalization of the results of this paperfor such time-dependent, axi-asymmetric numerical results isrequired.

3.3. Electromagnetic Wave Case: Superradiance

We briefly mention the electromagnetic wave energy transportthrough the horizon. We use the Kerr metric for the spacetime(x0, x1, x2, x3) = (t, r, θ, φ) around the spinning black hole,where we set ωφ � 0. We consider the stationary solution ofthe electromagnetic wave in a vacuum, where each componentof the electromagnetic field is proportional to f (r, θ )e−iωt+imφ

(f is a function of r and θ ). We use the short-wavelengthlimit of the electromagnetic wave, |k| � |(1/hi)∂gμν/∂xi |(i = 1, 2, 3, μ, ν = 0, 1, 2, 3), where k is the wave numberof the electromagnetic wave in a local region, which is fixedat the global coordinates. In a vacuum ( J = 0, ρe = 0),Equations (22)–(25) in the FIDO frame yield

E = − kω

× B, B = kω

× E, (88)

where k and ω are the wave number and angular frequencyof the electromagnetic wave in the FIDO frame, respectively.These equations give us the dispersion relation, ω = ±k andthe relation k ⊥ B. In this case, we identify

vF⊥ = − kω

. (89)

Because vF⊥ = |k/ω| = 1, using Equation (57), we have

SEM = αe∞EM(n + β). (90)

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When an electromagnetic wave passes through the horizonand enters into the black hole, if αe∞

EM is negative, the ro-tational energy of the black hole decreases. In this case,Equations (58), (60), and (89) read

e∞EM = α(B)2 + ω3lEM = α

(1 + ω3h3

kφ

αω

)(B)2. (91)

Very near the horizon, we have e∞EM ≈ ω3R(kφ/ω)(B)2. Because

the four-wave-number kμ = (−ω, k1, k2, k3) is the covariantvector, using Equation (14), we have

− ω = 1

α(−ω) +

β3

h3k3 = − 1

α(ω − ω3k3), k3 = 1

h3k3 = m

h3.

(92)Then, the energy density of the electromagnetic wave very nearthe horizon is approximately given by

e∞EM ≈ ΩHα

m

ω − mΩH(B)2. (93)

When ω < mΩH, negative energy appears at the horizon, and therotational energy of the black hole is extracted. This extractionmechanism corresponds to the “superradiance.” To produce thenegative energy of the electromagnetic wave, redistributionof the angular momentum is required. To understand theredistribution process, we have to consider the structure of thesolution of the electromagnetic wave in the ergosphere.

4. DISCUSSION

In this paper, we showed simple formulae (Equations (57)and (60)) to aid in building physical intuition for the causalextraction mechanism of the energy from black holes byelectromagnetic fields with negative electromagnetic energyproduced in the ergosphere. In three cases of force-free, idealMHD conditions and electromagnetic wave in a vacuum, atthe horizon we found that vF⊥ = 1; we then have SEM =αe∞

EM(vF⊥ + β). To extract the black hole rotational energycausally, we have to put the negative electromagnetic energydown into the black hole through the horizon. To producethe negative electromagnetic energy, because of the angularmomentum conservation (60), we should redistribute the angularmomentum of the electromagnetic field, where we require thenegative electromagnetic angular momentum density,

lEM < −α(B)2

ωφ= −R(B)2

βφ< 0, (94)

at the horizon (see Equation (60)). To realize the negative an-gular momentum azimuthal component, the angular momentumshould be redistributed because the total angular momentum isconserved. Redistribution of the angular momentum of the elec-tromagnetic field is caused by the electromagnetic torque (Koide2003; Gammie et al. 2004; Hawley & Krolik 2006; Krolik et al.2005).

This point of view originates with the Penrose process(Penrose 1969), which uses the negative mechanical energyof a particle. In fact, equations of the energies of matterand the electromagnetic field have similar forms as shown inEquations (51) and (52). With this viewpoint, in general, we clas-sify the known mechanisms of energy extraction from the blackhole as shown in Table 1. The Penrose process is well known and

is briefly mentioned in Section 1. The Blandford–Znajek mech-anism, MHD Blandford–Znajek mechanism, and MHD Penroseprocess were explained in the previous sections. We showed thatin all electromagnetic mechanisms of energy extraction from thespinning black hole, the negative electromagnetic energy is uti-lized as a mediator for the causal energy extraction through thehorizon. We confirmed that the condition of energy extractionis given by the realization condition of the negative energy atthe horizon. The magnetic Penrose process was not discussedin this paper. In the magnetic Penrose process, a particle in-teracts with the electromagnetic field and falls to the negativeenergy orbit. The negative energy of the particle is used to ex-tract the black hole rotational energy. This is just the Penroseprocess with electromagnetic interaction instead of fission. Su-perradiance was mentioned in Section 3.3. We found that theelectromagnetic wave with negative energy is used to extractthe black hole rotational energy. We also add the energy extrac-tion mechanism by magnetic reconnection in the ergosphere inTable 1 (Koide 2009).

We discuss the coincidence of the formulae of the energydensity and the energy flux density of the electromagnetic fieldat the horizon for the force-free and MHD Blandford–Znajekmechanisms as shown by Equations (71), (72), (86), and (87)in Sections 3.1 and 3.2, although the conditions of the twomechanisms are different. Reasoning a posteriori, we havecoincident expressions of the electric field E = −vF × B,vF = (RH/α)(ΩF − ΩH)eφ in the assumption of stationary,axisymmetric conditions for both cases. Furthermore, we havethe coincident boundary condition at the horizon vF⊥ −→ 1 andB = BP(vF/vF⊥) for both cases. These leading equations forboth cases are the same; we then have the coincident formulaefor both mechanisms.

Here we remark on the overlap of the ideal MHD and force-free conditions. The ideal MHD (Equation (73)) and force-free(Equation (62)) conditions can both be satisfied if J = ρev + J‖and ρe �= 0, where J‖ is a vector parallel to the magnetic fieldB. The vector J‖ corresponds to the net current density alongthe magnetic field lines at the plasma rest frame. Alternatively,in ideal MHD simulations, the “force-free” condition is oftendefined by B2/(2ρh) � 1 even if J − ρev is not parallel to B.

In an astrophysical situation such as in AGNs, which mech-anism is most expected to extract the black hole rotational en-ergy and activate the region near the black hole? We think theMHD Blandford–Znajek mechanism, rather than the originalBlandford–Znajek mechanism, is the most promising process.Because the plasma near the black hole is expected to be rela-tivistically hot, the plasma beta βp = 2p/B2 never vanishes. Ofcourse, the original Blandford–Znajek mechanism is applicableas an approximation with respect to the very strong magneticfield case. Such very low plasma beta is expected at the higherlatitude of the black hole magnetosphere and the fast componentof a relativistic jet.

We are grateful to Mika Koide for her helpful comments onthis paper.

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