Osaka City University Yousuke Takamori

Collaborators : Hideki.Ishihara(OCU), Ken-ichi.Nakao(OCU), Masashi.Kimura(OCU),and Chul-Moon Yoo

(APCTP) 117/July/2009 12th MG meeting , Paris

・ Introduction

・ Force-free magnetic fields around an extreme Reissner-Nordstrom

black hole

・ Exapmles ① Vacuum ②Non-rotating magnetic fields ③Rotating magnetic fields

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Active Galactic Nuclei(AGN)

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Black holes would be central engine.

If the Blandford-Znajek mechanism

works to extract energy of the black

hole, magnetic fields play an important role.

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Electromagnetically energy extraction from a Kerr black hole.

rotation of the magnetic

field line

magnetic field line

rotation of the black hole

ＢＨ

The rotational energy of a Kerr black hole

can be extracted electromagnetically.

The Poynting flux at the event horizon

： radial component of the magnetic flux density 　　： angular velocity of the black

hole 　　4： angular velocity of magnetic

field lines 　　

(Blandford and Znajek, 1977)

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The efficiency of energy extraction would decrease by the

Meissner effect of the black holes in the vacuum case.

Does the Meissner effect appear in the cases that some

current exist?

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The energy flux depends on magnetic field configurations at the horizon. 　　

Stationary and axisymmetric magnetic fields in a vacuum

are expelled from the event horizon of extremely rotating

black holes. (Bicak, 1976) 　　

( at the horizon.)

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The Meissner effect appears in extreme Kerr black holes with

degenerate horizon. Instead of complicated Kerr metric, we consider a

static and spherical black hole with degenerate horizon.The metric of static and spherical black hole with

degenerate horizon.(extrme Reissner-Nordstrom black holes)

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The black hole has degenerate horizon.

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Maxwell equations

・ Stationary, axisymmetric and force-free electromagnetic fields.

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The force-free conditions

:field strength tensor

:4-current density

Stationary and axisymmetric

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Stationary, axisymmetric and force-free magnetic fields

are represented by these quantities.

： magnetic flux 　　

： current 　　

： angular velocity of

magnetic field lines 　　

Magnetic fields or Current densities

ＢＨ

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The G-S equation has two kinds of singular surfaces.

・ event horizons ：

・ light surfaces ：　　

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We impose a boundary condition at the horizon(Znajek, 1977):

Because the horizon is degenerate, there is an additional boundary

condition at the horizon:

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①

②

(vacuum)

(non-rotating magnetic fields)

③ (rigid-rotating magnetic fields )

(i) Rotating split monopole(ii) Non-radial magnetic field

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Magnetic field configurations at the horizon are monopole ones.

Because no magnetic monopole exists in the nature, the black hole

with degenerate horizon exhibits the Meissner effect.

The G-S equation(extreme ceases)

The horizon regularity condition comes from the G-S equation:

(※The horizon boundary conditions are trivial in the vacuum case.)

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From the G-S equation:

The G-S equation

From the horizon boundary conditions:

These conditions are consistent!17/July/2009 12th MG meeting , Paris

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From the G-S equation:

The Meissner effect exhibits!In this case, there is not a current at

the horizon.

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① ②

①

②to the next page

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In this case, we can construct a split monopole configuration

at the horizon.

There is a current sheet on the equatorial plane.

The Meissner effect dissapears! But the magnetic field

configurations at the horizon are only split monopole ones.

The magnetic fields have radial shape

at the horizon.

ＢＨ

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The G-S equation

We have to be mindful of the light surfaces.

Inner Light Surface

Outer Light Surface

ＢＨ

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Rotating split monopole is an exact solution.(Michel, 1973)

Blandford, 1977

The solution has a toroidal component.

There is a current sheet on the

equatorial plane.

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Inner Light Surface

ＢＨ

We consider slow-rotating magnetic fields. In this case, the inner

light surface is close to the event horizon.

We consider near the horizon.

Outer Light Surface

We do not consider far region from

the horizon.

All quantities can be represented by Taylor expansion from the

horizon. 17/July/2009 12th MG meeting , Paris

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We define these quantities.

We write quantities at the horizon as follows:

The horizon locates at

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The G-S equations at the horizon

The light surface regularity condition

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We can check that the horizon

boundary conditions are consistent with these

equations.

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We can expand all quantities in

Current and quantities at the horizon

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Quantities at the inner light surface

The location of the inner light surface

We can check the G-S equation and the inner light surface regularity

condition each order 17/July/2009 12th MG meeting , Paris

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We can construct to satisfy these equations.

From the G-S equation:

From the inner light surface regularity condition:

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ＢＨ

magnetic field lines

There is a current sheet on the equatorial plane.

Non-zero exists.

Non-zero exists.

inner light surface

North hemisphere

South hemisphere

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In a stationary axisymmetric and force-free electromagnetic system

in black holes with degenerate horizon, the Meissner effect

disappears if there exist some current.

Non-rotating split monopole

Rotating split monopole

Rotating cylindrical magnetic field

And we show some magnetic field configurations threading

the horizon with a current sheet on the equatorial plane.

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・・・ .

・ Magnetic fields without a current sheet.・ Differential rotating magnetic fields・ The Meissner effect in extreme Kerr black holes

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