Osaka City University Yousuke Takamori Collaborators : Hideki.Ishihara(OCU), Ken-ichi.Nakao(OCU), Masashi.Kimura(OCU),and Chul-Moon Yoo (APCTP) 117/July/ th 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 217/July/200912th MG meeting, Paris

Active Galactic Nuclei(AGN) 3 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. 17/July/200912th MG meeting, Paris

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) 17/July/200912th MG meeting, Paris

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? 5 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.) 17/July/200912th MG meeting, Paris

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) 6 The black hole has degenerate horizon. 17/July/200912th MG meeting, Paris

Maxwell equations ・ Stationary, axisymmetric and force-free electromagnetic fields. 7 The force-free conditions :field strength tensor :4-current density Stationary and axisymmetric 17/July/200912th MG meeting, Paris

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 817/July/200912th MG meeting, Paris

The G-S equation has two kinds of singular surfaces. ・ event horizons ： ・ light surfaces ： 917/July/200912th MG meeting, Paris

10 We impose a boundary condition at the horizon(Znajek, 1977): Because the horizon is degenerate, there is an additional boundary condition at the horizon: 17/July/200912th MG meeting, Paris

① ② (vacuum) (non-rotating magnetic fields) ③ (rigid-rotating magnetic fields ) (i) Rotating split monopole (ii) Non-radial magnetic field 1117/July/200912th MG meeting, Paris

12 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.) 17/July/200912th MG meeting, Paris

13 From the G-S equation: The G-S equation From the horizon boundary conditions: These conditions are consistent! 17/July/200912th MG meeting, Paris

14 From the G-S equation: The Meissner effect exhibits! In this case, there is not a current at the horizon. 17/July/200912th MG meeting, Paris ①② ① ② to the next page

15 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 17/July/200912th MG meeting, Paris

16 The G-S equation We have to be mindful of the light surfaces. Inner Light Surface Outer Light Surface ＢＨ 17/July/200912th MG meeting, Paris

17 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. 17/July/200912th MG meeting, Paris

18 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/200912th MG meeting, Paris

19 We define these quantities. We write quantities at the horizon as follows: The horizon locates at 17/July/200912th MG meeting, Paris

The G-S equations at the horizon The light surface regularity condition 20 We can check that the horizon boundary conditions are consistent with these equations. 17/July/200912th MG meeting, Paris

21 We can expand all quantities in Current and quantities at the horizon 17/July/200912th MG meeting, Paris

22 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/200912th MG meeting, Paris

23 We can construct to satisfy these equations. From the G-S equation: From the inner light surface regularity condition: 17/July/200912th MG meeting, Paris

24 ＢＨ magnetic field lines There is a current sheet on the equatorial plane. Non-zero exists. inner light surface North hemisphere South hemisphere 17/July/200912th MG meeting, Paris

25 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. 17/July/200912th MG meeting, Paris

26 ・・・. ・ Magnetic fields without a current sheet. ・ Differential rotating magnetic fields ・ The Meissner effect in extreme Kerr black holes 17/July/200912th MG meeting, Paris