Steady General Relativistic Magnetohydrodynamic Inflow/Outflow Solution along Large-Scale Magnetic Fields that Thread a Rotating Black Hole

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by  Hung-Yi Pu

Draftversion April21,2015
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
STEADY GENERAL RELATIVISTIC MAGNETOHYDRODYNAMIC INFLOW/OUTFLOW SOLUTION
ALONG LARGE-SCALE MAGNETIC FIELDS THAT THREAD A ROTATING BLACK HOLE
Hung-Yi Pu1, Masanori Nakamura1, Kouichi Hirotani1, Yosuke Mizuno2,3, Kinwah Wu4, and Keiichi Asada1
1Institute ofAstronomy&Astrophysics,AcademiaSinica,11FofAstronomy-Mathematics Building,AS/NTUNo. 1,Taipei10617,
Taiwan
2InstituteofAstronomy,NationalTsingHuaUniversity,Hsinchu30013,Taiwan
3Institute forTheoreticalPhysics,Goethe University,D-60438,FrankfurtamMain,Germanyand
5 4MullardSpaceScienceLaboratory,UniversityofCollegeLondon, HolmburyStMary,Dorking,SurreyRH56NT,UK
1
0 Draft version April 21, 2015
2
ABSTRACT
r
p Generalrelativisticmagnetohydrodynamic(GRMHD) flowsalongmagneticfields threadingablack
A hole can be divided into inflow and outflow parts, according to the result of the competition between
the black hole gravity and magneto-centrifugal forces along the field line. Here we present the first
8
self-consistent, semi-analytical solution for a cold, Poynting flux-dominated (PFD) GRMHD flow,
1
whichpassesallfourcritical(inner andouter,Alfv´enandfastmagnetosonic)pointsalongaparabolic

streamline. Byassumingthatthedominating(electromagnetic)componentoftheenergyfluxperflux
]
E tube is conserved at the surface where the inflow and outflow are separated, the outflow part of the
H solution can be constrained by the inflow part. The semi-analytical method can provide fiducial and
complementarysolutionsforGRMHDsimulationsaroundtherotatingblackhole,giventhattheblack
.
h hole spin, global streamline, and magnetizaion (i.e., a mass loading at the inflow/outflow separation)
p are prescribed. For reference, we demonstrate a self-consistent result with the work by McKinney in
- a quantitative level.
o
r Subject headings: galaxies: active—– galaxies: jets — Magnetohydrodynamics (MHD) — Black hole
t physics
s
a
[
1. INTRODUCTION field line can be propagated outward in the form
2 of torsional Alfv´en waves, thus extracting the black
Relativistic jets emerging from accreting black hole
v hole energy electromagnetically (Blandford & Znajek
systems have been observed in active galactic nuclei
2
1977, hereafter BZ77). However, because the en-
(AGN), micro-quasars (stellar mass black hole X-ray
1
vironment around an accreting black hole is not a
1 binaries), and presumably gamma-ray bursts (GRBs).
perfect vacuum (contrary to the force-free treatment
2 Observationally, the bulk Lorentz factors Γ of jets in
in BZ77), the general relativistic magnetohydrody-
0 AGNs are & 10 20 (Jorstad et al. 2005; Cohen et al.
− namics (GRMHD), which consist of electromagnetic
. 2007;Gu2009;Pushkarev et al.2009;Lister et al.2013),
1 and fluid components, provide a more general pic-
and the values could be higher for some blazars (see
0 ture for the dynamics and structures of relativistic
Gopal-Krishna et al. 2006; Hovatta et al. 2009). The
5 jets in both theoretical approaches (e.g. Camenzind
Lorentz factors of micro-quasar jets are lower, mostly
1 1986a,b, 1987; Takahashi et al. 1990; Fendt & Greiner
with Γ 2 10 (e.g. Fender et al. 2004; Corbel et al.
v: 2002),b∼utst−illthereareafew foundtohaveΓ>10(see 2001; Fendt & Camenzind 1996; Fendt & Ouyed 2004)
i Miller-Jones et al.2006). Jetsingamma-rayburstersare and numerical simulations (e.g. Koide et al. 1998,
X 2000; Mizuno et al. 2004; McKinney & Gammie
supposed to be ultra-relativistic, and their Lorentz fac-
r tors can be as high as 100 1000(see, e.g. Lyubarsky 2004; Hawley & Krolik 2006; McKinney 2006;
a ∼ − Beckwith et al. 2008; Tchekhovskoy et al. 2010, 2011;
2010; Lyutikov 2011). How jets become relativistic af-
Tchekhovskoy& McKinney 2012).
ter being launched from nearby black holes is a long-
The overall configuration of an accreting black hole
standing issue. Electromagnetic or magnetohydrody-
system is schematically illustrated in Figure 1 (see
namic (MHD) mechanisms are frequently invokedto ex-
also GRMHD simulations for magnetized accretion,
tract energy and momentum from the black hole and
e.g. McKinney 2006; McKinney & Gammie 2004;
accretion disk (e.g., Meier et al. 2001, for reviews). One
Hawley & Krolik 2006). Ordered, parabolic lines are
of the key issues to be addressed is the potential of the
developed near the funnel, which is confined by the
MHD flow acceleration up to a high bulk Lorentz factor
corona and/or accretion flow. Due to the relative ab-
of Γ>10.
sence of accreting materials, the funnel region is Poynt-
An ideal engine to power relativistic jets is a spinning
ing flux-dominated (PFD). The fluid loading onto the
black hole. Close to the black hole, the rapid winding
field is accelerated inward (or outward) if the black hole
of the azimuthal component in large-scale magnetic
gravity force is larger (or smaller) than the “magneto-
fields due to the frame-dragging inside the black
centrifugal” forces (e.g. Sa¸dowski & Sikora 2010, in the
hole ergosphere results in a counter torque (induced
caseoftheaccretiondisk). Aspointedoutinthetheoret-
by the Lorentz force) against a black hole rotation.
icalworkofTakahashi et al.(1990),blackholerotational
The energy that the black hole spent to perturb the
2 H.-Y. Pu et al.
energy can be extracted outward by a PFD GRMHD
inflow. This is a direct result of that the electromag-
netic components dominating the GRMHD flow, and
theelectromagneticcomponentisresponsibleforextract-
ing the black hole energy, similar to the BZ77 process.
The outwardenergy flux, after being extracted from the
black hole, is expected to propagate continuously out- energy flux
ward throughout the magnetosphere from the inflow re-
gion to the outflow one. In this paper, we focus on the
PFD GRMHD flow in the funnel region, including both
the inflow and outflow parts. light surface
For comparison, let us quickly consider the case when
the GRMHD flow becomes fluid-dominated. In that
outflow
case, the energy flux is dominated by the fluid compo-
nent, and therefore it has aninwarddirection for inflow, corona
but outward for outflow (c.f., the energy flux direction
+
showninFigure1). Suchdiscontinuityoftheenergyand
momentum fluxes implies that the outflow is accretion- accretion flow
powered, which is constrained by the energy input from
the disk/corona. The switch-on and switch-off of the
separation surface
extraction of the black hole energy (inflow) may closely
inflow
relate to the launching and quenching of relativistic jets
(outflow)(e.g.Pu et al.2012;Globus & Levinson2013).
Prior to the GMRHD studies mentioned, Phinney
(1983) considered the inflow and outflow along a static limit
monopole field jointly by the conservation of the to-
tal energy flux per flux tube. In this pioneering work,
they consider energy extraction from the black hole via BBllaacckk HHoollee light surface
BZ77 process (the inflow part), and the Michel’s “mini-
mum torque solution” (Michel 1969), in which the fast(- Fig. 1.— Schematic illustration of a Poynting flux-dominated
magnetosonic) point is located at infinity (the outflow (PFD)GRMHDflowconfinedbytheaccretionflowanditscorona.
Theoutward-streaming curves indicates ordered, large-scale mag-
part). We, however, suggest that a more realistic situ-
netic fields that thread the black hole event horizon. The inflows
ation can be considered: the black hole energy extrac- and the outflows (represented by thick white arrows) are along
tionprocessinthe frameworkofGRMHD, andatype of the field lines, and they are separated by the separation surface
parabolic GRMHD flows as a result of external pressure (markedbyadashedline). Theenergyflux(representedbyagray
arrow) is outward in both the inflow and outflow regions, as the
confinements provided by the corona/accretion. Recent
blackholerotationalenergyisextractedandtransportedoutward.
observationalevidencealsosupportsthisidea;nearbyac- Thestatic limit(dashed curve)andthelightsurface(solidcurve)
tive radiogalaxy,M87,exhibits the parabolicstreamline outsidetheblackhole(blackregion)arealsoshown.
up to 105 Schwarzschild radius (Asada & Nakamura
∼
2012).
Furthermore, we are interested in the case that the flow,(ii)atthetheinflow/outflowseparationsurfacethe
fast point of the outflow is located at a finite distance. extractedenergyfluxiscarriedoutcontinuously,and(iii)
Thisconsiderationisdirectlyrelatedtheconversionfrom in the outflow region the flow passes the fast point, and
Poynting to kinetic energy fluxes of the flow and there- hence the bulk Lorentz factor increases. Although this
fore the jet acceleration. Poloidal magnetic flux is re- picture has been already recognised in the quasi-steady
quired to diverge sufficiently rapidly in order for most state in GRMHD simulations (e.g. McKinney 2006;
of the Poynting flux to be converted into the kinetic en- McKinney & Gammie 2004; Hawley & Krolik 2006), no
ergy flux beyond the fast point (also known as the mag- steady solution is available in the literature.
netic nozzle effect (e.g. Camenzind 1989; Li et al. 1992; Inthispaper,wepresentthefirstsemi-analyticalwork.
Begelman & Li 1994; Takahashi & Shibata 1998) . We consider the energy extraction from the black hole
Beskin & Nokhrina(2006)examinethe accelerationof via the GRMHD (inflow), and the perturbed force-free
the jet along a parabolic streamline by introducing a parabolic field line in Beskin & Nokhrina (2006) (out-
small perturbation into the force-free field. As a result, flow). With given black hole spin, field angular velocity,
the fast point is located at a finite distance. This indi- andmagnetizationatthe separationsurface,we areable
cates how plasma loading in the flow plays a role in ac- to to constrain the outflow solution by the inflow solu-
celeratingtheflow,aswellasaconversionfromPoynting tion. For a reference, we adopt similar parameters re-
to kinetic/particle energies. They consider the behavior ported in the GRMHD simulation of McKinney (2006,
of the outflow in the flat spacetime. However, we are ;hereafter M06). Our semi-analytical solution passes
interested in both the inflow and outflow near a black all the critical points (inner and outer, Alfv´en and fast
hole. points), and agrees with the inflow and outflow proper-
Allofthesetheoreticalworksprovideimportantpieces ties along a mid-level field line in M06.
towardapicturethatincludesthefollowingprocessalong The paper is organized as follows. In 2, we outline
§
thefieldline: (i)intheinflowregiontherotationalenergy the GRMHD formulation and the wind equation (WE).
oftheblackholeisextractedoutwardbytheGRMHDin- In 3, with the consideration of the conservation of en-
§
Steady GRMHD Inflow/Outflow Solution 3
ergyfluxininflowandoutflowregionneartheseparation F uµ=0 . (6)
µν
surface,wediscussthematchingconditiontoconnectthe (ξ Tµν) =0 , (7)
µ ;ν
inflow and outflow part of a PFD GRMHD flow. In 4,
we introduce our model setup. We adopt similar para§m- (ηµTµν);ν=0 , (8)
eters to those reported by M06 and compare the solu- where ξµ = ∂/∂t and ηµ = ∂/∂φ are the Killing vec-
tionobtainedbythematchingconditionwiththatofthe tors. These conservation equations (equations 5–8) give
time-averaged GRMHD numerical simulation results in four conserved quantities along a streamline. By denot-
M06. Finally, a summary is given in 5. ing the poloidal stream function as Ψ(r,θ), they are: (i)
§
the angularvelocityofthe fieldline, Ω (Ψ),(ii) the par-
2. STATIONARY,AXISYMMETRICMHDFLOWINKERR ticle number flux per unit magnetic fluFx (mass loading),
SPACETIME
η(Ψ),(iii)thetotalenergyoftheflowperparticle,E(Ψ),
2.1. Basic Formulae and(iv) the totalangularmomentum per particle, L(Ψ)
The theory about stationary and axisymmet- (Camenzind 1986a,b, 1987; Takahashi et al. 1990):
ric ideal GRMHD flows has been in several works
(Camenzind 1986a,b, 1987; Fendt & Camenzind F F
tr tθ
1996; Fendt & Ouyed 2004; Takahashi et al. 1990; ΩF(Ψ)= = , (9)
F F
Fendt & Greiner 2001). For completeness, in this rφ θφ
section we summary and present the necessary formulae
for the purpose of this paper. √ g nur √ g nuθ
η(Ψ)= − = −
The naturalunit systemis usedthroughoutthis work. F − F
θφ rφ
As c = G = M = 1, the gravitational radius r =
g √ g nut(Ω Ω )
GM/c2 =1, where c is the speed of light, G is the grav- = − − F , (10)
F
itational constant, and M is the mass of the black hole rθ
(conversionsfromthec.g.s.unitstothe naturalunits for
the physical variables here can be found in tables 3 and E(Ψ)=E +E
FL EM
4 in Pu et al. (2012)). The flows occur in a background
Ω
Kerr space-time, which is stationary and axisymmetric. = µu F √ g Frθ
t
For a metric signature [ + + +], the Kerr metric (in − − 4πη −
Boyer-Lindquistcoordin−ates) reads ΩF
= µu B , (11)
t φ
− − 4πη
∆ a2sin2θ 4arsin2θ
ds2= − dt2 dtdφ
− Σ − Σ
L(Ψ)=L +L
Asin2θ Σ FL EM
+ dφ2+ dr2+Σ dθ2 , (1) 1
Σ ∆ =µu √ g Frθ
φ
− 4πη −
wherea J is theangularmomentumofthe blackhole,
∆ r2 ≡ 2r+a2, Σ r2 +a2cos2θ, and A (r2 =µu Bφ , (12)
a2)≡2+ −a2sin2θ. ≡ ≡ − φ− 4πη
△
We also assume that the flow is cold. For a highly- with√ g =Σsinθ. Here,Ω=uφ/utisthefluidangular
relativistic flow, the thermal pressure p is insignificant −
velocity and µ is the relativistic specific enthalpy, which
compared with the rest-mass energy density and the ki-
becomes m in the cold limit. The covariant magnetic
neticenergydensityinthefluid,andhencethecoldlimit field observedby a distant observerwith uν =(1,0,0,0)
is justified.
is given by
The flow is magnetizedandthe stress-energytensorof 1
the fluid has two components: Bµ ǫνµαβFαβuν , (13)
≡ 2
Tµν =TFµLν +TEµMν , (2) and its toroidal component is given by
where the fluid component is given by B =√ g Frθ =(∆/Σ)sinθF , (14)
φ rθ
−
Tµν =ρuµuν , (3)
FL where ǫνµαβ = √ g [νµαβ] is the Levi-Civita tensor,
−
and the electromagnetic component by and[νµαβ] is the completely antisymmetric symbol(see
Appendix).
Tµν = 1 FµγFν 1gµνFαβF , (4) The outward energy flux in the flow is
EM 4π (cid:18) γ − 4 αβ(cid:19) r = Tr =nEur , (15)
E − t
where uµ is the 4-velocity of the fluid and ρ is the rest-
and the outward angular momentum flux is
mass energy density. The electromagnetic field tensor
F satisfies Maxwell’s equations, and the proper num- r = Tr =nLur . (16)
µν L − φ
ber density n (= ρ/m, where m is the rest-mass of the
particles) satisfies the mass continuity equation. Splittingthemintofluid(i.e.EFL andLFL)andtheelec-
Under the ideal MHD condition, a stationary and ax- tromagnetic components (i.e. EEM and LEM) gives
isymmetric flow obeys four conservation laws: r= r+ r
FL EM
E E E
(nuµ) =0 , (5) =nE ur+nE ur
;µ FL EM
4 H.-Y. Pu et al.
Ω B
= nµu ur F φ A , (17) (Takahashi et al. 1990), where
t φ,θ
− − 4π Σsinθ
K = (g Ω2 +2g Ω +g ) , (25)
and 0 − φφ F tφ F tt
2
L
Lr=LFLr +LEMr K2= 1−ΩFE , (26)
=nL ur+nL ur (cid:18) (cid:19)
FL EM
2
r 1 L L
=−nµuφur+ EΩEFM . (18) K4=gt2φ−−gttgφφ "gφφ+2gtφE +gtt(cid:18)E(cid:19) # .(27)
AsinitiallyproposedbyTakahashi et al.(1990),thecase The Alfv´en Mach number M is given by
A
of r > 0 for inflow (which requires a negative total en-
ergEy) is known as the MHD Penrose process. For later M2 4πµη2/n=4πµn up 2 =4πµη up , (28)
studies, the term is instead used to indicate a nega- A ≡ B B
tive energy orbit of the fluid component, r > 0 (e.g. (cid:18) p(cid:19) (cid:18) p(cid:19)
Hirotani et al. 1992; Koide et al. 2002; SEeFmLenov et al. where the re-scaled poloidal field is
20T04h;eKboumlkisLsaorroevnt2z0f0a5c;tKoroiodfeth&eBflaobwaf2o0r1a4)d.istinct ob- Bp2=BjBj(gtt+gtφΩF)−2
server can be defined by 1
= grrΨ2 +gθθΨ2 . (29)
Γ=√ g ut . (19) ∆sin2θ ,r ,θ
− tt Alongthestreamlinese(cid:2)veralcharacteristi(cid:3)csurfacescan
If all the energy in the Ponyting flux is converted to be defined. Their definition and properties are summa-
the fluid’s bulk (kinetic) energy at a large distance, the rized in the Appendix.
terminal Lorentz factor will be The conservedquantities E, L andη can be expressed
E r in terms of three system parameters: (i) the launching
Γ = = E . (20) point of the flow, r , (ii) the location of the Alfv´en sur-
∞ µ ρur ⋆
face, r , and (iii) the magnetization parameter at the
A
In addition, the angular velocity of the fluid at a large launching point, σ⋆. Explicitly, the relations are
distance will be
L g +g Ω
tφ φφ F
L r = , (30)
uφ∞ = µ = ρLur . (21) E −gtt+gtφΩF (cid:12)(cid:12)rA
E˜2=K0 , (cid:12)(cid:12) (31)
Equations(20)and(21)thereforeprovidetheupperlimit K
oftheterminalLorentzfactorandangularvelocityofthe 2(cid:12)r⋆
(cid:12)
fluid at large distances. σ⋆= (cid:12)(cid:12) Φ2⋆ = Φ⋆ 1 , (32)
4πm(nu √ g) 4πm η
2.2. Wind Equation p − |r⋆ | |
where E˜ = E/µ, the flux function Φ = B √ g, and Φ
The streamline of the flow is represented by the func- p − ⋆
denotes that it is evaluated at r =r . In terms of these
tion Ψ(r,θ) = const. The WE (i.e., the relativistic ⋆
parameters, the Mach number can be written as
Bernoulli equation), describing the fluid motion along
the streamlines can be obtained using the normalization µ f
condition uαuα =−1. The WE therefore has the form: MA =(cid:20)mup√−g σ⋆(cid:21)|1/2 , (33)
u2+1= E 2 U (r,θ) , (22) where f = Φ⋆/Φ. Note that up = 0 at r = r⋆ has been
p µ g assumed in deriving the relation (31). In addition, the
(cid:18) (cid:19)
relation (32) implies that knowing the mass loading η is
where the poloidal component of the 4-velocity is given equivalent to knowing the σ .
⋆
by In the cold limit, the WE is a polynomial equation of
u2 =uju , (23) 4-th order in u :
p j p
withthesummationoverthe poloidalindicesj = r, θ . 4
The term Ug(r,θ) in the right-hand side of eq{uatio}n Aiuip =0 . (34)
(22), which is evaluated along the magnetic field line i=0
X
in the calculation, is related to the conserved quanti-
The coefficients (r;Ψ,Φ,r ,r ,σ ,Ω ) are given by
ties, and its explicit expression depends on the assumed Ai ⋆ A ⋆ F
backgroundspace-time (see Camenzind (1986a,b, 1987); =C2 ,
Fendt & Camenzind (1996); Fendt & Ouyed (2004) for A4 2
= 2C K ,
the Minkowski and Schwarzschild spacetimes; and A3 − 2 0
Takahashi et al. (1990); Fendt & Greiner (2001) for the A2=C22+K02+E˜2K4C22
Kerr space-time). = 2C K +2E˜2K C
In a Kerr space-time, we obtain A1 − 2 0 2 2
=K2 E˜2K K ,
K K 2K M2 K M4 A0 0 − 0 2
U (r,θ)= 0 2− 2 A− 4 A , (24) where C =√ g f/σ (Fendt & Greiner 2001).
g (M2 K )2 2 − ⋆
A− 0
Steady GRMHD Inflow/Outflow Solution 5
3. MATCHINGCONDITIONOFTHEINFLOWAND Again, from dynamical point of view, the relatively
OUTFLOW strong dependence can be understood by the fact that
In the work of Phinney (1983), the matching of the the fast point of the outflow can vary from finite dis-
inflowandoutflowpartsoftheflowisconstrainedbythe tance to infinity. Because the uncertainty of the δ is
conservation of the energy flux per magnetic flux in the introducedby the uncertainty of(σ⋆)−, insteadof (σ⋆)+
inflow and outflow region (see also 4.2), the outflow can still be well constrained.
§
The matching condition then play the role to constrains
(ηE)inflow =(ηE)outflow (35) theoutflowbysinglingouttheoutflowsolutionthatsat-
isfies (B ) =(B )+.
Remember that both η and E of the inflow and outflow φ − φ
are constant.
Consider equation (35) at the separation surface, r , 4. FLOWALONGAPARABOLICFIELDLINEWITHA
s FINITE-DISTANCEFASTPOINT
for PFD flow (E E E ), we further consider
≈ EM ≫ FL 4.1. Model Setup
(η) (E ) =(η) (E )+ , (36)
inflow EM − outflow EM In general, the field configuration should be con-
to be the matching condition of the inflow and outflow. sistently determined by solving the trans-field equa-
The superscripts “ ” (or “+”), respectively, denote the tion, (i.e. the Grad-Shafranov equation). The trans-
physical value com−puted at the location very close to r field equation in cold limit involves the stream func-
s
in the inflow (or outflow) region, that is, r r (or tion Ψ, and the derivative of the conserves quanti-
rex→prerss+se)d. Aasfter some algebra, equation (36) ca→n als−so be Btieess,kidnΩ&F/PdΨar,’edvη/1d9Ψ93,)d.EH/odwψe,vdeLr,/sdoψlv(inNgittthaeettraaln.s1-fi9e9l1d;
( r ) =( r )+ , (37) equationanalyticallyis very challengingand beyond the
EEM − EEM scope of this paper.
or Ontheotherhand,weareinterestedinthe casewhere
Ω Ω
F(Bφ)− = F(Bφ)+ . (38) the fast point of the outflow is located at a finite dis-
4π 4π tance. It is therefore essential to consider an additional
Equation (37) implies that the matching condition we modification on the original force-free field line due to
adopt is equivalent to the statement: the outward Poyt- the MHD flow. We leave a better consideration of field
ing energy flux is continuous at the separation surface1. configurationforafuturework,andadoptthestreamline
Equation(38)revelsthatsuchconditionguaranteesthat function in Beskin & Nokhrina (2006) as the prescribed
the toroidal field is continuous at the separation point, parabolic field
provided that ΩF is the same constant in the inflow and Ψ=Ψ0+ǫf , (41)
outflow region.
where Ψ is the the flat spacetime parabolic force-free
It is interesting to note that the matching condition 0
field generated by the toroidal surface current distri-
does not require that η or E should be continuous
EM bution, I = C/4πr(1 + Ω r2)1/2, on equatorial plane
when crossing r =r . That is, if we define F
s
(Blandford 1976; Lee & Park 2004),
(η) (E )+ (σ )+
inflow EM ⋆
δ = = , (39) πC
≡(cid:12) (η)outflow (cid:12) (cid:12) (EEM)− (cid:12) (cid:12) (σ⋆)− (cid:12) Ψ0 = Ω sinh−1(ΩFr(1−cosθ)), (42)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) F
δ isnot(cid:12)necessaryfo(cid:12)run(cid:12)ity. Thela(cid:12)str(cid:12)elationin(cid:12)equation
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) and
(39) is obtained with the help of equation (32). Never-
ǫf =ǫπCΩ rsinθ , ǫ 1 (43)
theless, due to following reason, the outflow can still be F
≪
properly constrained by the inflow even with the uncer- is the perturbation introduced by the MHD effect. The
tainty of δ.
constant C is assumed to be unity. Note that by the
poClooindsaildefirelda lflinowe wailtohngsoameprsepscerciibfiecd,anhgoullea-rthvreelaodciintyg help of the relation sinh−1(x) = ln(x + √x2+1), Ψ0
is proportional to r(1 cosθ), which is the same as
field. Znajek (1977) showed that, due to the regular- −
the dominating term of the parabolic field2 in BZ77.
ity requirement at the event horizon, r =1+√1 a2,
+ In addition, B = 0 is guaranteed. It is shown in
−
the derivative of the stream function Ψ is finite and B ∇ ·
φ Beskin & Nokhrina (2006) that, althoughǫf/Ψ 1 on
0
satisfies ≪
the (outer) fast surface, the perturbation method is not
r2 +a2 applicable beyond the fast point. As a result, we can
Bφ(r+,θ)=−sinθr2 ++a2cos2θ(ΩH −ΩF)Ψ,θ , (40) only discuss the flow solution up to the outer fast point.
+ The following assumptions for a PFD GRMHD flow
where Ω is the angular velocity of the hole. As a along a hole-threading field line are considered. First,
H
result, (B ) is insensitive to different value of (σ )
φ − ⋆ −
((B ) const.). From dynamical point of view, this 2 Due to a similar toroidal surface current distribu-
φ − ≈ tion, I = C/4πr, on the equatorial plane, the parabolic
can be understood by the fact that the fast point of a
force-free field around a black hole considered in BZ77:
PFDGRMHDinflowisalwayslocatedclosetotheblack Ψ = C r(1 cosθ)+2(1+cosθ)[1 ln(1+cosθ)] , also fol-
hole event horizon (Appendix). lowsΨ2 {r(1 −cosθ)atlargedistance.−Thisisoneofth}esolutions
Foroutflow,however,thereisno constraintatinfinity, of the s∝ource−-free Maxwell equation in Schwarzschild spacetime:
and therefore (Bφ)+ depends on (σ⋆)+ more strongly. ((1s−in2/θr)Ψ,r),r+(r2s1inθΨ,θ),θ =0;whileEquation(42)isoneof
thesolutionsofthesource-freeMaxwellequationinflatspacetime:
1 cf. equation(35)gives(Er)−=(Er)+. (sin1θΨ,r),r+(r2s1inθΨ,θ),θ =0.
6 H.-Y. Pu et al.
inflow outflow
0.630
-| +|
B)φ 0.625 B)φ
|( 0.620 |(
0.615
9000 11000 13000 15000 17000 19000 9000 11000 13000 15000 17000 19000
(σ )- (σ )+
* *
Fig. 2.— The toroidal field, Bφ, as a function of magnetization parameter, σ⋆, near the separation surface, rs for the inflow (left) and
mo4ua.t2tfl)co.hwiWng(hrciegonhntδd)i=tpioa1nrt(c(ooσfn⋆ts)ht−rea=isno(ltuσht⋆ei)o+onu.tflT1o3hw7e0bs0uy)p,tethrhesecirnmiflpaottwc”h;−itn”hgaatcnoidsn,d”si+itni”ognldineisngoaotluewtattyhhseesvaoatuliutsflefioecwdo.msopluutteiodnatthart→satris−sfyan(Bdφr)+→=rs+(B,φre)s−pe(csteieve§l3y..2Tanhde
§ ≃
we assume r = r . The assumption of r = r and Consistentsolutions for PFD inflow andoutflow along
⋆ s ⋆ s
u =0ensuresu hasasmoothtransitionfromu <0 a field line are obtained iteratively until the matching
p|r⋆ p p
(inflow) to u > 0 (outflow). Second, to guarantee conditionissatisfied. Foreachσ ,becauseΩ (=Ω /2)
p ⋆ F H
the flow is PFD, we require σ 1 (see also 3.4.1 of and r (determined by where K = 0 is along the field
⋆ ≫ § ⋆ 0′
Fendt & Greiner (2001) for an estimation) and ǫ 1. line; see Appendix) are known, we can solve WE (equa-
≪
Furthermore, we assume ǫ is constant along field lines. tion (34)) for the flow solution by requiring the physical
The higher the value of σ , the more magnetically dom- solution to pass through the fast surface. For example,
⋆
inated the flows are. for the case ǫ = 0.065, the profiles of (B ) and (B )+
φ − φ
Amongalltheparameterspaceweseekfortheparame- asafunctionof(σ ) and(σ )+,respectively,areshown
⋆ − ⋆
terset ΩF,η,E,L ,whichgivesasimilartime-averaged in Figure 2.
{ }
GRMHD simulation result in M06 for comparison. We Aconsistentinflow/outflowsolutionexistswhenasuit-
therefore focus on a spinning black hole with its dimen- able set (ǫ,(σ ) ,(σ )+) is applied. As mentioned in 3,
⋆ − ⋆
sionless spin a = 0.9 and a field line that threads the the tendency is for (B ) const. to result in mul§ti-
φ −
event horizon at mid-latitude, θ = 60◦. As mentioned ple choices of (σ)− such th≈at (Bφ)− = (Bφ)+ is satis-
in 2.2, the set ΩF,η,E,L can be equivalently deter- fied. This leads to certain amount of freedom for choos-
§ { }
mined by ΩF,σ⋆,r⋆,rA . We adopt r⋆ =rs (assumed), ing the value for δ. For simplicity, δ = 1 is assumed,
{ }
and ΩF =1/2ΩH (similar to the result of M06) in both so (σ⋆)− = σ⋆ + = σ⋆. We can read from Figure 2
inflowandoutflowregion. Thenwedetermineσ⋆ andrA that(Bφ)− =(B|φ)+ is satisfiedwhen (ǫ,σ⋆)=(0.065,
(notethatoncerA isdetermined,thelocationofthefast 13700). ≃
surface is determined accordingly) by the constraint of: Bythesamemethod,foranyspecificvalueofǫ(orσ ),
⋆
(i) (EEM/EFL) ∼ 102 near the separation surface3 simi- thereisacorrespondingσ⋆(orǫ)thatsatisfiesthematch-
lar to the case in M06, and (ii) the matching condition. ing condition. The quantitative relation shows that, as
We note that ΩF ≈ 1/2 ΩH is self-consistently ob- σ⋆ increases, ǫ also increases. This implies that, because
tained in a steady PFD GRMHD flow solution for a there is more mass loading onto the field, the field pro-
monopole field geometry (Beskin & Kuznetsova 2000), gressivelybunchesuptowardtotherotationalaxisofthe
while it may not be relevant for a parabolic field geom- black hole. Finally, after σ is chosen by the matching
⋆
etry. BZ77 examined the parabolic streamline in which condition, the parameter set Ω ,σ ,r ,r of the in-
F ⋆ ⋆ A
ΩF decreases when shifting the angle from close to the flow/outflowpartofthe solutio{nisuniquely}determined.
pole to equatorial plan. In M06, the field geometry be- The relaxation of the assumption δ = 1 is discussed at
comes almost monopolar in the vicinity of the horizon, the end of 4.3.2.
so that Ω 1/2Ω is observedalongthe field line (see §
F H
≈
also Beskin 2009). In the present paper, although the 4.3. Self-Consistent Inflow/Outflow Solution
parabolic field is prescribed as a global field geometry,
4.3.1. Flow Properties
we nevertheless adopt the constant value of 1/2 Ω in
H
our fiducial solution for convenience. We adopt the above parameter set (ǫ,σ ) = (0.065,
⋆
≃
13700) as the fiducial model parameters, because
4.2. Matching the Inflow and Outflow the resulting flow solution satisfies our requirement
(E /E ) 102 ( 4.1). The conserved quantities,
EM FL |r⋆ ∼ §
Ω ,η,E,L , of our fiducial flow solution are shown in
F
{ }
3 In top panel of Figure 7 of M06 Γ(∞EM)/Γ(∞MA) is 103 (see Table 1. The mass loading η changes sign according to
oMf0t6hefodredfienfiintiiotinonins),thwishipcahpeeqru.ivNaoletentthtoatEoEuMr/dEeFfiLnit∼io1n0∼2ofinEtEeMrmiss guironasn.dBΨe,cθau(esequtahteionsig(n10h)a)sinnoinsflpoewcifiacndmeoauntiflnogw, trhee-
4π smallerthanΓ(∞EM) usedinM06. Thefactorof4π isabsorbed absolute value η is shown. By the assumption δ = 1,
intothedefinitionofFµν inM06. (η) = (η)| | (equation (39)).
inflow outflow
| | | |
Steady GRMHD Inflow/Outflow Solution 7
2
TABLE 1
Conserved Quantities of the Fiducial FlowSolutions
a=0.9
1.5
(ǫ=0.065,σ⋆ 13700)† withδ=1
≃
Inflow Outflow
η(Ψ)m 7 10−5 M)
|EL(|(ΨΨ))//µµ ≃−171220≃ × ≃171244 2z (c/G 1
≃− ≃
ΩF(Ψ) ≃0.157
Notes.
† aconsistent inflow/outflow solutionis obtained when a suitable 0.5
set(ǫ,σ⋆)isapplied,suchthat thematchingconditionissatisfied Black Hole
(see 3and 4.2).
§ §
0
0 0.5 1 1.5 2 2.5
R (c2/GM)
TABLE 2
Propertiesof PFD GRMHDFlow Along the Same
Fig.3.— Characteristic points of a fiducial PFD GRMHD in-
Hole-Threading Field Line
flow. Towardtheblackhole: Alfv´ensurfaces(filledcyantriangles),
light surfaces (empty green circles), and fast surfaces (filled blue
InflowSolution Outflow Solution
squares). The event horizon and static limit(the outer boundary
ur <0 >0 oftheergosphere)areshownbythethinsolidanddashedlines,re-
uθ >0 <0 spectively. Fieldlinesarerepresentedbythethicksolidline. The
uφ >0 >0 Alfv´en surfaces are located inside the the negative energy region
ut >0 >0 (shaded region), implying that the black hole energy is extracted
outward.
E=EFL+EEM <0 >0
g +g Ω =0. Thus,theregionmustbeinsidetheergo-
EFL >0 >0 tt tφ F
EEM <0 >0 sphere, where gtt > 0. As the flow becomes increasingly
PFD, the location of the Alfv´en surface moves toward
L=LFL+LEM <0 >0 the light surface, finally entering the negative energy re-
LFL >0 >0 gion (see the Appendix). For PFD GRMHD inflow, the
LEM <0 >0 fastsurfaceislocatedverycloseto,andalmostcoincides
with, the blackhole eventhorizon. This is whythe PFD
r = r + r >0 >0 inflow solutions are all similar, as mentioned in 3. In
E E†EFEErLFrML EEM <>00 >>00 Ffaisgturseur3fawcee poflotthteheflolowc,awtiohnicshosfhtahreeAthlfev´esnamsuerffae§caetuarneds
Notes. mentioned previously.
† ForastationaryGRMHDinflowsolutionalongahole-threading
fineolndhlionlee,-tEhFrreLad<in0gifisesladtilsifineedd.uIrnincgontrtarnasstie,nftorphanasien,florw a>lon0gias 4.3.2. Radial Structure
EFL
possible(e.g.Koideetal.2002;Komissarov2005). Let us now show the fiducial flow solution up to the
fast surface in Figure 4 and compared the result (espe-
In the inflow region (ur < 0), both E < 0 and L < 0 cially Figures 7 and 8) in M06. The top panel of Figure
indicates that the energy and angular momentum of the 4 shows the opening angle of the prescribed field, which
black hole is extracted outward ( r > 0 and r > 0). roughly follows a single power law θ r 0.52, which is
−
E L ∝
E/µ of the outflow gives the maximum possible value of ingeneralmorecollimatedcomparetotheresultofM06.
the terminal Lorentz factor (equation (20)). Although The locations of the characteristic surfaces are overlap
(E ) = (E )+ under the assumption δ = 1, the ontothe profile. TheAlfv´ensurfacesarelocatedcloseto
EM − EM
absolution value of E = E + E for the inflow is the light surfaces, and the inner fast surface is located
EM FL
slightly smaller than the value for the outflow. This is closetothehorizon. NotethatinM06theopeningangle
because, in the inflow region, the fluid component E has different slope at a different radial range (see Fig-
FL
(or L ) has an opposite sign with the electromagnetic ure 10 of M06). Instead, our prescribed field line follows
FL
component E (or L ), partly canceling the electro- a single power law. Nevertheless, with similar require-
EM EM
magnetically extracted energy (or angular momentum); ments at the separation surface ((E /E ) 102),
EM FL |r⋆ ∼
whereas in the outflow region, the fluid and the electro- the fast surface of the outflow is located at several hun-
magnetic components have the same sign, both carrying dredr fromtheblackhole,whichissimilartotheresult
g
the energy and angular momentum outward. The gen- of M06.
eralproperties ofdifferent physicalcomponents for PFD The second panel of Figure 4 shows the profiles of the
inflows and outflows are provided in Table 2. electromagnetic energy component E and the fluid
EM
The extraction of black hole rotation energy by the energy component E (both normalized by µ), and
FL
GRMHDinflowisalsoindicatedbythelocationofthein- the Lorentz factor Γ. Inside the ergosphere, g < 0,
tt
flowAlfv´ensurface. Aremarkablefeature inGRMHD is Γ is ill-defined. Therefore, only the profile segments
theexistenceofanegativeenergyregion: oncetheAlfv´en outside the ergosphere are plotted. At large distances,
surfaceofaninflowresidesinsidesucharegion,theblack g 1, Γ u = E /µ. In addition, for a
tt t FL
→ − → −
holeenergyisextractedoutward(Takahashi et al.1990). PFD flow, E E E when launching, so the
EM FL
≈ ≫
The inner boundary of the negative energy region is the maximum possible value of the terminal Lorentz factor,
inner lightsurface,andthe outer boundaryis definedby Γ = E/µ E /µ near the separation surface. As
EM
∞ ≈
8 H.-Y. Pu et al.
a result, the profile of Γ along the streamline is there- beyond the fast point, where we are not able discuss in
fore related to the conversion from E to E . In the current prescribed field configuration.
EM FL
acceleration region (& 50r ), Γ roughly follows r0.6, Theorthonormalcomponentsofthe magneticfields at
g
∝
which is similar to the result of M06, and the analyt- large distance can be defined by
ical result of Γ r0.5 obtained in (Beskin & Nokhrina
2006). It is exp∝ected that a further acceleration is ex- B¯r=√grr Br , (47)
pected to be take place beyond the fast surface due to B¯φ=√gφφ Bφ . (48)
themagneticnozzleeffect(e.g.Camenzind1989;Li et al.
1992). TheconversionefficiencyfromPoyntingtokinetic NotethatB¯r isgiveninitiallywhensolvingtheWE,and
energy, which can be approximated by Γ/Γ , is closely B¯ , which is not initially known, can be determined af-
φ
related to the location of the fast surface. F∞or example, tersolvingthe WE.The bottompanelofFigure4shows
when the fast surface is located at infinity, Γ/Γ 0. the profile of the pitch angle, tan 1(B¯r/B¯φ ). Because
For the outflow solution, Γ/Γ . 0.1 up to t∞he≈fast Br andBφ areboth functions ofg− (|see the|Appendix),
surface, which is located at ∞300rg. It is also inter- theyquicklydecreaseandchangestitgnwhenenteringthe
∼
esting to note that the flow has already reached modest ergosphere (g > 0). As a result, B¯r and B¯φ are
tt
Lorentz factors (Γ 5) at the fast surface, and most ill-defined close to the black hole, and| w|e only|plot| the
∼
ofthe Poyntingenergyhas not yetbeen convertedto ki- profile in the region where g < 0. The reason why the
tt
neticenergy. Notethat,despitethefinalvalueofΓatthe pitch angle profile in M06 does not have this problem
fast surface is similar to the result in M06, the Poynting should be related to the definition of the field. The ex-
energy in M06 at fast surface has already experienced a plicit form of the magnetic field we adopt is provided
significantdecay(morethanone orderofmagnitude)up in the Appendix. Nevertheless, at far region (e.g. the
to the fast surface. The reason why the fluid energy is outflow region), spacetime becomes more flat and the
notcorrespondinglyincreasingmaybeduetodissipative differences of the definition are less important, our re-
processes. In the inner region beneath the separation sult agrees with the result of M06. The locations where
surface, ut =EFL/µ.1,asexpectedbecausethefluid B¯r = B¯φ are close to the light surface. At a large
iowsuhtsietcrrhonrimegg−lpyiolynbieobsuentyhdoaentddtbhtyeheflthuseiedpbailsaracuktniobhnooulsenu’sdrfgaarncaedv,iat−ny.uotIunt>fltoh1we, wd|ihste|arencRe|,L|B=¯|r1|/iΩs wF,elal-sdaelsscoriobbetdabinyed|B¯irn|M≈0|B6¯.φ|RL/√gφφ,
At the end of this section, we discuss how would the
occurs.
flow solution would change if we adopt a δ, which also
Similartothe energyconversionbetweenthefluidand
satisfies the matching condition, but does not equal to
theelectromagneticcomponents,theincreaseofthefluid
unity. Keep in mind that the outflow solution is well
component of the angular momentum L is at the ex-
FL constrained by the matching condition, and the uncer-
pense of the electromagnetic component of the angular
tainty of δ is due to the degeneracy of the inflow solu-
momentum L . The profiles of L and L (nor-
EM FL EM tions ( 3). As a result, the outflow solution will remain
malized by µ) are shown in the third panel of Figure 4. §
thesameifadifferentvalueofδ isadopted. ForthePFD
Again, the profile of the fluid component L /µ=u is
FL φ GRMHDflow,becausethelocationoftheAlfv´ensurface
consistent with result of M06, but the decreases of the
isalwayslocatedneartheinnerlightsurfaceandthefast
Poynting component in the simulation are much larger
surfaceisalwayslocatedclosetothehorizon,theflowdy-
than our semi-analytical solution.
namics will therefore be similar. That is, ur,uθ,uφ,ut,
The radial and polar components of the four-velocity
and therefore E /µ = u and L /µ = u will re-
of the flow, ur and uθ, can be calculated from equations main almost uncFhLanged.−Int additioFnL, Br (prφescribed)
(10)and(23),withu determinedbytheWE.Theother
p and Bφ (constrained by the Znajek’s condition on hori-
two components of the four-velocity, ut and uφ, can be
zon described in 3) will also remain similar. The elec-
obtained by solving §
tromagnetic component, E and L , due to the de-
EM EM
µ(u +u Ω )=E Ω L , (44) pendence of 1/η, follow (E ) /(E ) =
− t φ F − F (L ) /∝(L ) =| (Eη)M inflow/(η)EM out=flo1w/|δ.
subject to the normalisation uαu = 1. The velocity | EM inflow EM outflow| | outflow inflow|
α
components ur and uθ change signs ac−ross r =rs, while 5. SUMMARY
the velocity components uφ and ut remain positive in
A semi-analytical scheme is presented to investigate
boththeinflowandoutflowregions. Theangularvelocity
of the fluid, Ω = uφ/ut, which follows the black hole’s the cold,PFDGRMHD flow solutionalongaKerrblack
hole-threading field. The continuity of the outward
rotation, is however always positive along the magnetic
field line. At the separation surface, ur = uθ = 0, and Poyntingenergyfluxacrosstheseparationsurfaceisused
asthematchingconditiontoconnecttheinflowandout-
hence Ω=Ω .
F
flowpartsofaPFDGRMHDflowsolution. We consider
Theradialandtoroidalcomponentsoftheorthonormal
theparabolicfieldlineofBeskin & Nokhrina(2006),and
velocity at large distance are given by
thereforetheresultingflowpassesthroughallthecritical
u¯r=√grr ur , (45) points at a finite distance.
u¯φ=√gφφ uφ (46) fieWld,itahndsimmailganrebtilzaactkiohnoalettshpeins,epaanrgautliaornvseulrofcaictey,wofetahree
as shown in the fourth panel of Figure 4. The profile able to obtain a specific parameter set Ω ,σ ,r ,r
F ⋆ ⋆ A
of u¯r is quite similar to the result in M06, but u¯φ has thatgivesinflowandoutflowsolutionsina{greementwith}
a relativelysteeper profile compare to the simulation re- the time-averagedflow properties along a mid-level field
sult. Wesupposethisisrelatedtothefieldconfiguration line reported in the GRMHD simulation of M06.
Steady GRMHD Inflow/Outflow Solution 9
inflow outflow
)
e
e
gr 101
e
d
(

θ
100
µ| 102
/
M
E
E
|

µ 101
/
L
F
E

Γ 100
103
µ|
/
M
E 102
L
|

µ
/ 101
L
F
L
100
101

φ
-u

-ru| 100
|
10-1
)
|
φ
-B 100
/
r
-B
|
(
1
-
n
a
t
10-1
100 101 102
2
r (c /GM)
Fig.4.—FiducialPFDGRMHDflowsolutionpropertiesalongafieldline. Toppanel: Jetopeningangleoftheprescribedfield. Location
ofcharacteristic surfacesarealsoshown: separationpoint(plus sign),lightsurfaces(empty circles),Alfv´ensurfaces(filledtriangles), and
fastsurfaces(filledsquares). Thethinverticallineindicatestheangularprofileofthestaticlimit(gtt=0). Secondpanel: Electromagnetic
energycomponent(uppersolidline),EEM,andfluidenergycomponent(lowersolidline),EFL,ofthetotalenergyE=const.=EEM+EFL,
inunitoffluidrestmassenergy. TheprofileofΓisshownonlywhengtt<0(dashedline). Thirdpanel: Electromagnetic(uppersolidline),
LEM,andfluid(lowersolidline),LFL componentsoftotalangularmomentum,L=const.=LEM+LFL. Fourth panel: Theorthonormal
velocities u¯r and u¯φ. Bottom panel: The pitch angle of the orthonormal field, tan−1(B¯r/B¯φ ) (solid line), which is well-described by
tan−1|RL/√gφφ|(dashedline)atlargedistance,whereRL=1/ΩF. Becausetheorthono|malfiel|disrelatedtogttandbecomesill-defined
neartheblackhole,thepitchangleisonlyshownwhengtt<0. Alongthefieldline,thelocationoftheeventhorizon,thestaticlimit,and
theseparationpointareindicatedbytheverticalsolid,dotted-dashed, anddashedlines,respectively.
10 H.-Y. Pu et al.
In this current work, due to the limitation of the pre- We thank the anonymous referee for helpful sugges-
scribed field configuration, we can only discuss the flow tions, which significantly improved the paper. We also
solutionuptothe outerfastsurface. Asafuturework,a thank C. Fendt for helpful information about GRMHD
better considerationof the field configurationcould help outflow solutions under cold limit, and Z. Younsi for
to explore of the flow acceleration beyond the fast sur- proofreading. K.H. is supported by the Formosa Pro-
face, where the major jet acceleration takes place. gram between National Science Council in Taiwan and
Compared to the GRMHD and the general rel- ConsejoSuperiorde InvestigacionesCientificas inSpain,
ativistic force-free electrodynamics (GRFFE) (e.g. administered through grant number NSC100-2923-M-
McKinney & Narayan 2007) numerical simulation ap- 007-001-MY3. Y.M. is supported by the Ministry of
proaches, the semi-analytical approach provides a com- Science and Technology of Taiwan under the grant
plementary understanding of the relativistic jets, in the NSC 100-2112-M-007-022-MY3 and MOST103-2112-M-
sense that the numerical dissipative process is absent, 007-023-MY3, and by ERC Synergy Grant “BlackHole-
andthatthefluidcomponentisincluded. Thestationary Cam: Imaging the Event Horizon of Black Holes”.
solution obtained by the scheme can also be provided as
a reference of the time-averaged GRMHD jet behaviour
in numerical simulations.
APPENDIX
NOTES ON THEMAGNETICFIELD
Here we present the explicit form of the magnetic field. The covariant magnetic field defined in Equation (13)
1
B ǫ Fαβξν , (A1)
µ νµαβ
≡ 2
can be alternatively written as
1
Bµ ǫνµαβF ξ , (A2)
αβ ν
≡ 2
where ǫ = √ g [νµαβ], and ǫνµαβ = 1 [νµαβ], with √ g = Σsinθ. Since ξν = (1,0,0,0) and ξ =
νµαβ − −√ g − ν
(g ,0,0,g ), we can quickly read from the above−definitions that B =0 but Bt =0.
tt tφ t
6
The components of the magnetic field are therefore given by
B =√ gFθφ, (A3)
r
−
g +Ω g
Br = tt F tφF , (A4)
θφ
− √ g
−
B = √ gFrφ , (A5)
θ
− −
g +Ω g
Bθ = tt F tφF , (A6)
rφ
√ g
−
and
B =√ gFrθ , (A7)
φ
−
g
Bφ = tt F . (A8)
rθ
−√ g
−
With the relations
g +Ω g
Fθφ = gθθ tt F tφ F , (A9)
− ∆sin2θ θφ
g +Ω g
Frφ = grr tt F tφ F , (A10)
− ∆sin2θ rφ
Frθ =grrgθθ F , (A11)
rθ
one can check Br = grrB , Bθ = gθθB , and Bφ = gφφB . Note that, despite F is finite at all region, Bµ is
r θ φ µν
ill-defined near a Kerr black hole because g changes sign when entering the ergosphere.
tt
At large distance, the metric become Minkowski spacetime in spherical coordinates,
ds2 = dt2+dr2+r2dθ2+r2sin2θdφ2 , (A12)
−
and √ g =r2sinθ. In this limit, the orthonormal field has the form
−
1
B¯r √g Br = F , (A13)
≡ rr r2sinθ θφ