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WDS'08 Proceedings of Contributed Papers, Part III, 198–203, 2008.
ISBN 978-80-7378-067-8 © MATFYZPRESS
Asymptotically Uniform Electromagnetic Test Fields
Around a Drifting Kerr Black Hole
O. Kopáček
Astronomical Institute, Academy of Sciences, Prague, Czech Republic.
Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic.
Abstract. A test field solution describing the electromagnetic field around a
Kerr black hole which is drifting in an arbitrary direction with respect to the
asymptotically uniform magnetic field with the general orientation (with respect
to the rotation axis) is constructed. Proper tetrad choice of a physically relevant
observer is discussed. Force lines of electric and magnetic intensities as measured
by the orbiting (freely falling, respectively) observer are explored and plotted for
some special cases.
Introduction
Electromagnetic (EM) test field solutions of Maxwell equations in curved spacetime play an improtant role in astrophysics since we can usually suppose that astrophysically relevant EM fields are
weak enough, so that their influence upon background geometry may be neglected.
We are interested in the solutions describing an originally uniform magnetic field under the influence
of the Kerr black hole. Since the Kerr metric is asymptotically flat, this EM field reduces to the original
homogenous magnetic field in the asymptotic region. First, such a test field solution was given by Wald
[1974] for the special case of perfect alignment of the asymptotically uniform magnetic field with the
symmetry axis. More general solution for arbitrary orientation of the asymptotic field was given by Bičák
et al. [1985]. We use this solution to construct the EM field around the Kerr black hole which is drifting
through the asymptically uniform magnetic field.
As we are motivated to investigate such a fields by the opportunity of making judgements about
the motion of charged particles, we need to project acquired coordinate components onto a properly
chosen tetrad. A realistic (though still simple enough) scenario of the motion of the test particles in
the vicinity of the Kerr black hole is to bind it to the geodesic circular orbit in the equatorial plane
above marginally stable orbit, rms , and let it freely fall below rms . This is roughly how we model the
accretion process in the accretion discs. We are especially curious whether these fields would under
certain circumstances allow the particle to escape from the equatorial plane and form an outflow of the
matter from the accretion disc.
Lines of force
Kerr metric in Boyer-Lindquist coordinates t, r, θ, φ (Misner et al., 1973):
ds2 = −
sin2 θ
Σ 2
∆
[dt − a sin θ dφ]2 +
[(r2 + a2 )dφ − a dt]2 +
dr + Σdθ2 ,
Σ
Σ
∆
(1)
where
∆ ≡ r2 − 2M r + a2 , Σ ≡ r2 + a2 cos2 θ.
(2)
Natural definition of the lines of force (of magnetic and electric fields), as measured by given observer equipped with the orthonormal tetrad eµ(α) , is their identification with the lines along which
magnetic/electric charge connected to this observer would start to move due to the presence of the EM
field F µν . Coordinate components of the magnetic (electric) field intensities are determined by (coordinate components of) the Lorentz force felt by the unit magnetic (electric) charge are (Hanni et al.,
1973):
B µ =∗F µν uν , E µ = F µν uν ,
(3)
where uν represents a 4-velocity of the charge.
Tetrad components of the vector field determining desired lines of force are given as the spatial part
of the projection onto eµ(α) :
∗ µ ν
(i)
µ ν
B (i) = B(i) = e(i)
= E(i) = e(i)
µ F νu , E
µF νu ,
198
(4)
KOPÁČEK: ELECTROMAGNETIC FIELDS AROUND A DRIFTING KERR BLACK HOLE
(α)
where e µ are 1-forms dual to the tetrad vectors eµ(α) . Lowering/rising spatial tetrad indices doesn’t
matter since the tetrad is supposed to be orthonormal – g(µ)(ν) = η(µ)(ν) .
Electromagnetic field
We start with Fµν describing the test field with asymptotic form of a general (ie. not necessarily
parallel) uniform magnetic field (Bičák et al., 1985). Due to the axial symetry of Kerr space-time only two
components of asymptotic field were considered in that paper without any loss of generality (asymptotic
components B0 (parallel) and B1 (equatorial) to be specific). We rewrite components of EM tensor (eq.
(A3) of Bičák et al., 1985) denoting Bx ≡ B1 , Bz ≡ B0 and splitting the result into two parts according
to the asymptotic component. We obtain the asymptotically perpendicular part of the field:
Bx
Ftr
=Bx aM rΣ−2 ∆−1 sin θ cos θ[(r3 − 2M r2 + ra2 (1 + sin2 θ) + 2M a2 cos2 θ) cos ψ
− a(r2 − 4M r + a2 (1 + sin2 θ)) sin ψ],
Bx
Ftθ
=Bx aM Σ−2 (r2 cos 2θ + a2 cos2 θ)(a sin ψ − r cos ψ),
Bx
Ftφ
=Bx aM Σ−1 sin θ cos θ(a cos ψ + r sin ψ),
Bx
Frθ
= − Bx (a cos ψ + r sin ψ)
− Bx a∆−1 (M r − a2 sin2 θ) cos ψ − a(r sin2 θ + M cos2 θ) sin ψ ,
Bx
Frφ
= − Bx sin θ cos θ (r − M a2 ∆−1 ) cos ψ − a(1 + rM ∆−1 ) sin ψ
Bx
Fθφ
(5)
Bx
+ a sin2 θFtr
,
2 2
=Bx (r sin θ + M r cos 2θ) cos ψ − a(r sin2 θ + M cos2 θ) sin ψ
+ (r2 + a2 )Bx M Σ−2 (r2 cos 2θ + a2 cos2 θ)(a sin ψ − r cos ψ)
and the part which approaches uniform field aligned along the axis:
Bz
Ftr
=Bz aM Σ−2 (r2 − a2 cos2 θ)(1 + cos2 θ),
Bz
Ftθ
=2Bz aM rΣ−2 sin θ cos θ(r2 − a2 ),
Bz
Frφ
Bz
Fθφ
(6)
=Bz r sin2 θ + Bz a2 sin2 θM Σ−2 (r2 − a2 cos2 θ)(1 + cos2 θ),
=Bz ∆ sin θ cos θ + 2(r4 − a4 )Bz M rΣ−2 sin θ cos θ,
where we use the azimuthal coordinate ψ of Kerr ingoing coordinates, which is related to Boyer–Lindquist
coordinates as follows:
r − r+
a
ln
,
(7)
ψ =φ+
r+ − r− r − r−
√
with r± ≡ M ± M 2 − a2 denoting the outer and the inner horizon. We notice that limr→∞ ψ = φ.
As we shall introduce a drift of the black hole in the general direction we lose axial symmetry
B
and need to consider all spatial components of the asymptotic magnetic field. We obtain Fµνy (which
B
Bx
may only appear due to nonzero drift) by rotating Fµν
along the z-axis by angle π2 - i.e. Fµνy =
Bx
Fµν
φ → φ − π2 , Bx → By which only causes sin ψ → − cos ψ and cos ψ → sin ψ.
Since the drift shall induce uniform electric field in the asymptotic region we need to have appropriate
E
B
Fµνx,y,z handy. We get them easily by performing dual transformation of Fµνx,y,z . Dual transformation
is carried out as follows:
1
∗
Fαβ = F µν εµναβ ,
(8)
2
where εµναβ is the Levi-Civita tensor whose components are given as (Misner et al., 1973):
p
√
(9)
εµναβ = −det||gσω ||[µναβ] ≡ −g[µναβ],
with [µναβ] denoting completely antisymmetric symbol. Determinant of the Kerr metric is g =
2
gtt grr gθθ gφφ − gφt
grr gθθ = − sin2 θ Σ2 .
Performing the dual transformation we immediately obtain EM tensors with desired asymptotics of
uniform electric field:
Ex,y,z
Bx,y,z
Fµν
=∗ Fµν
(Bx,y,z → −Ex,y,z ).
(10)
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KOPÁČEK: ELECTROMAGNETIC FIELDS AROUND A DRIFTING KERR BLACK HOLE
In the explicit form we get:
E
B
B
Ftr x,y,z = sin θ Σ Fθt x,y,z (Bx,y,z → −Ex,y,z )g φt + Fθφx,y,z (Bx,y,z → −Ex,y,z )g φφ g θθ ,
E
B
B
Ftθ x,y,z = sin θ Σ Ftr x,y,z (Bx,y,z → −Ex,y,z )g φt + Fφrx,y,z (Bx,y,z → −Ex,y,z )g φφ g rr ,
E
B
Ftφx,y,z = sin θ Σ Frθx,y,z (Bx,y,z → −Ex,y,z )g rr g θθ ,
(11)
Frθx,y,z = sin θ Σ Fφtx,y,z (Bx,y,z → −Ex,y,z ) (g φt )2 − g φφ g ,
E
B
B
Frφx,y,z = sin θ Σ Fθφx,y,z (Bx,y,z → −Ex,y,z )g φt + Fθt x,y,z (Bx,y,z → −Ex,y,z )g tt g θθ ,
E
B
B
Fθφx,y,z = sin θ Σ Fφrx,y,z (Bx,y,z → −Ex,y,z )g φt + Ftr x,y,z (Bx,y,z → −Ex,y,z )g tt g rr .
E
B
tt
Now we are fully equipped to construct any asymptotically uniform test field on the Kerr background just by linear superposing of the above EM tensors. As we are concerned in constructing Fµν
which describes the test field around the black hole drifting through asymptotically uniform magnetic
field in general direction, we shall employ Lorentz transformation to find the correct asymptotic components of such a field. Once obtained we just use them to replace the original “nondrifting” quantities
Ex , Ey , Ez , Bx , By and Bz . Matrix of general Lorentz transformation is (Jackson, 1998):
γ

′
−γvx
||Λνµ || = 
−γvy

−γvz
1
γvy
−γvx
(γ−1)v 2
1 + v2 x
−γvz
(γ−1)vx vy
v2
(γ−1)vy2
1 + v2
(γ−1)vy vz
v2
(γ−1)vx vy
v2
(γ−1)vx vz
v2

(γ−1)vx vz 

v2
(γ−1)vy vz  ,

v2
(γ−1)v 2
1 + v2 z
(12)
1
where v = (vx2 + vy2 + vz2 ) 2 and γ = (1 − v 2 )− 2 .
Bx
Bz
Our original field, Fµν = Fµν
+ Fµν
, has a simple asymptotic form in Minkowskian coordinates,

0
0
asymptotic

||Fµν
|| = 
0
0
0
0
−Bz
0

0
0
.
Bx 
0
0
Bz
0
−Bx
(13)
′
To transform the covariant tensor Fµν , the inverse Lorentz transformation Λµν ′ = (Λνµ )−1 would be used.
But we realize that the Boyer–Lindquist coordinate system which we use to perform all the calculations
(and also to express the EM tensor of the final field) is centered around black hole and the rest frame of
the black hole is thus our “laboratory” reference frame. As we consider a drift of the black hole against
the field eq. (13), we need to perform inverse Lorentz transformation. Quantities Bx and Bz appearing
therein would be primed in standard notation. For inverse transformation of covariant tensors we use
′
original Λνµ . Thus for “drifting” Fµν we have (denoting Fµ′ ν ′ that of eq. (13)):
asymptotic
Fµν
= Fµasymptotic
Λµµ Λνν ,
′ν′
′
′
(14)
which may be written in the matrix formalism as follows:
asymptotic
|| = ||Λµµ ||t ||Fµasymptotic
|| ||Λνν || = ||Λµµ || ||Fµasymptotic
|| ||Λνν ||,
||Fµν
′ν′
′ν′
′
′
′
′
(15)
and results in:

0
vy γBz

−v
γB
0
y
z
asymptotic
||Fµν
|| = 
γ(vx Bz − vz Bx ) −γBz + vz M
vy γBx
−vy M
2

−γ(vx Bz − vz Bx )
−vy γBx

γBz − vz M
vy M
,
0
γBx − vx M 
−γBx + vx M
0
(16)
γ
(vz Bz + vx Bx ).
with M ≡ γ+1
Final step of the derivation is thus substitution of “nondrifting” quantities Ex,y,z and Bx,y,z in the
E
B
tensors Fµνx,y,z and Fµνx,y,z by Lorentz transformed values from eq. (16) and superposing the components to acquire general EM tensor describing the field around the Kerr black hole drifting through
200
KOPÁČEK: ELECTROMAGNETIC FIELDS AROUND A DRIFTING KERR BLACK HOLE
asymptotically uniform magnetic field of general orientation:
Ex
Ey
Fµν =Fµν
(Ex → −vy γBz ) + Fµν
(Ey → γ(vx Bz − vz Bx ))+
Ez
Bx
+ Fµν
(Ez → vy γBx ) + Fµν
(Bx → γBx − vx M )+
(17)
By
Bz
+ Fµν
(By → −vy M ) + Fµν
(Bz → γBz − vz M ).
Choice of the tetrad
As we are primarily interested in astrophysically relevant situations we choose the orthonormal
tetrad carried by the observer on the circular Keplerian orbit around the black hole. Such an orbit is
specified by the values of constants of motion – by specific angular momentum L̃ ≡ uφ and specific
energy Ẽ ≡ −ut which are expressed as follows (Bardeen et al., 1972):
√
√
√
± M (r2 + a2 ∓ 2a M r)
r2 − 2M r ± a M r
p
q
, L̃(r) =
,
(18)
Ẽ(r) =
√
√
r r2 − 3M r ± 2a M r
r(r2 − 3M r ± 2a M r)
where the upper signs are valid for the prograde (direct) orbits and the lower ones for the retrograde
(counter–revolving) orbits. Such a tetrad is physical only above marginally stable orbit rms which
represents a radial boundary for the stationary geodesic motion in the equatorial plane:
p
(19)
rms = M 3 + Z2 ∓ (3 − Z1 )(3 + Z1 + 2Z2 ) ,
where Z1 ≡ 1 + 1 −
a2
M2
1/3 h
1+
a 1/3
M
+ 1−
a 1/3
M
i
and Z2 ≡
orthonormal tetrad of the orbiting frame Yokosawa et al. [2005]:
eµ(t)
= u = γ̃
µ
A
∆Σ
1/2
q
3a2
M2
+ Z12 . Above r = rms we use
[1, 0, 0, ΩKep],
1/2
∆
[0, 1, 0, 0],
=
Σ
1
eµ(θ) = √ [0, 0, 1, 0],
Σ
"
√
1/2
1/2 #
A
Σ
A
µ
√ + v(φ) Ω
e(φ) = γ̃ v(φ)
, 0, 0,
,
∆Σ
∆Σ
sin θ A
eµ(r)
(20)
where we define A ≡ (r2 + a2 )2 − a2 ∆ sin2 θ to express the coordinate angular velocity of LNRF (i.e.
angular velocity of the frame dragging) Ω = 2a
A M r. Angular velocity of a circular orbit is ΩKep =
±1
where
the
upper
signs
are
for
prograde
orbits and the lower ones for the retrograde orbits.
−1/2
3/2
M
r
±a
A sin θ
(Ω
v(φ) = Σ∆
−
Ω)
stands
for
the
linear
velocity
of the orbiting tetrad as measured by ZAMO
Kep
1/2
2
−1/2
observer in LNRF. Finally γ̃ = (1 − v(φ) )
is the relevant Lorentz factor of this motion.
For r < rms there are no more circular orbits. Thus we suppose that the orbiting observer who
reaches this limit performs a free fall to the black hole keeping the values of the constants of motion
corresponding to the marginally stable orbit at rms . He is falling with Ẽms ≡ Ẽ(rms ) and L̃ms ≡ L̃(rms )
given by eqs. (18) – (19). Having fixed ut (r < rms ) = −Ẽms , uφ (r < rms ) = L̃ms and uθ = 0 we get
radial component ur easily from the normalisation condition uµ uµ = −1. Contravariant components of
the 4-velocity are then:
1
[(r[r2 + a2 ] + 2M a2 )Ẽms − 2M aL̃ms ],
r∆ q
1
2 − 4M aẼ L̃
2
ur = − 3/2 [r(r2 + a2 ) + 2M a2 ]Ẽms
ms ms − (r − 2M )L̃ms − r∆,
r
uθ = 0,
1
[2M 2 aẼms + M (r − 2M )L̃ms ].
uφ =
r∆
ut =
201
(21)
KOPÁČEK: ELECTROMAGNETIC FIELDS AROUND A DRIFTING KERR BLACK HOLE
Spatial 1-forms of the tetrad of this falling observer may be expressed as follows Dovčiak [2004]:
r
(ur [ut , ur , 0, uφ ] + [0, 1, 0, 0]),
e(r)
µ = p
∆(1 + ur ur )
eµ(θ) = [0, 0, r, 0],
r
∆
(φ)
[−uφ , 0, 0, ut].
eµ =
1 + ur ur
(22)
Structure of the electromagnetic field
We shall make just a brief qualitative overview of the possible structures of the lines of force of
magnetic and electric intensities felt by the above specified observer which may be either co-rotating or
counter-rotating with the background geometry. In all figures we show a portion of the equatorial plane.
The circle signifies the upper event horizon of the black hole.
Original asymptotically uniform magnetic field is restricted to be perpendicular to the symmetry
axis (i.e. it lies in the equatorial plane). Without any loss of generality we align it with the horizontal
axis of presented figures (only nonzero asymptotical component is thus Bx ). Since the impact of the
rotation of the black hole upon the structure of the field lines is most apparent for extreme Kerr black
hole (a = M ) we confine ourselves to this case from now on. Marginally stable orbit rms thus coincides
with the horizon at r = M for prograde orbits while it lies at rms = 9M for retrograde orbits.
First we present the field lines around a nondrifting black hole and then the drift is also introduced
rsin φ
M
3
3
2
2
1
1
rsin φ
M
0
0
−1
−1
−2
−2
−3
−3
−2
−1
0
rcos φ
M
1
2
−3
−3
3
−2
−1
0
rcos φ
M
1
2
3
Figure 1. Magnetic field as felt by co-rotating (on the left) and counter-rotating (on the right) observers
around a nondrifting extreme Kerr black hole.
rsin φ
M
3
3
2
2
1
1
rsin φ
M
0
0
−1
−1
−2
−2
−3
−3
−2
−1
0
rcos φ
M
1
2
−3
−3
3
−2
−1
0
rcos φ
M
1
2
3
Figure 2. Projection of the electric field as felt by co-rotating (on the left) and counter-rotating (on
the right) around a nondrifting extreme Kerr black hole.
202
KOPÁČEK: ELECTROMAGNETIC FIELDS AROUND A DRIFTING KERR BLACK HOLE
1.75
1
1.7
1.65
0.95
1.6
0.9
rsin φ
1.55
M
rsin φ
M
0.85
1.5
1.45
0.8
1.4
0.75
1.35
−1.8 −1.75 −1.7 −1.65 −1.6 −1.55 −1.5 −1.45
rcos φ
M
0.65
0.7
0.75
0.8
rcos φ
M
0.85
0.9
Figure 3. Projection of the electric field as felt by counter-rotating observer around a nondrifting
extreme Kerr black hole. In the left panel we zoomed the area around a null point of the electric field,
the right panel shows the structure of the electric field near the horizon.
rsin φ
M
3
1.7
2
1.6
1
1.5
0
rsin φ
M
1.4
−1
1.3
1.2
−2
−3
−3
1.1
−2
−1
0
rcos φ
M
1
2
3
−1.5
−1.4
−1.3
−1.2
rcos φ
M
−1.1
−1
−0.9
Figure 4. Magnetic field as felt by counter-rotating observer around a drifting (vx = 0.5 c and vy =
−0.7 c) Kerr black hole. Null point of the magnetic field is noticed and zoomed.
Conclusion
In this paper we presented initial steps of our investigation of the asymptotically uniform electromagnetic test fields around a drifting Kerr black hole. Components of the electromagnetic tensor Fµν
describing such a field were given explicitly (in a symbolic way regarding its length) in the terms of
the former nondrifting solution. Structure of the resulting fields has been briefly overviewed in the special case of the original uniform magnetic field (in which the Kerr black hole is then immersed) being
perpendicular to the symmetry axis.
Acknowledgments.
The author thanks Dr. Vladimı́r Karas for his kind guidance and helpful advices.
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