2. MHD description of black hole magnetospheres
The basic equations describing a magnetohydrodynamic (MHD) configuration under the assumptions of axisymmetric, stationary and ideal MHD in the context of Kerr metric were first derived by Blandford & Znajek (1977) and Znajek (1977). In this paper, we apply the MHD formulation in the 3+1 formalism introduced by Thorne & Macdonald (1982), Macdonald & Thorne (1982), or Thorne et al. (1986) (hereafter TPM). In the notation, we follow TPM and Okamoto (1992).
2.1. Space-time around rotating black holes
In the 3+1 split the space-time around rotating black holes with a mass M and angular momentum per unit mass, is described using Boyer-Lindquist coordinates with the line element
t denotes a global time in which the system is stationary, is the angle around the axis of symmetry, and are similar to their flat space counterpart spherical coordinates. The parameters of the metric tensor are defined as usual,
is the angular velocity of the differentially rotating space, or the angular velocity of an observer moving with zero angular momentum (ZAMO), , respectively. is the red shift function, or lapse function, describing the lapse of the proper time in the ZAMO system to the global time t, .
The electromagnetic field , the current density , and the electric charge density are measured by the ZAMOs according to the local flat Minkowski space. These local experiments then have to be put together by a global observer for a certain global time using the lapse and shift function for the transformation from the local to the global frame. In spite of this transformation, Maxwell's equations in the 3+1 split look very similar to those in Minkowski space. There is just an additional source term from the differential rotation of space (see below).
2.2. The cross-field force-balance
The magnetospheric structure follows from the force-balance across the flux surfaces. The projection of the equation of motion perpendicular to the field lines provides the stream equation. Here, in the force-free case, only Lorentz forces (perpendicular to the flux surfaces) are considered.
Under the assumption of axisymmetry a magnetic flux function (or stream function) can be defined measuring the magnetic flux through a loop of the Killing vector ,
Equivalently, the poloidal current is defined by integration of the poloidal current density through the same loop
The indices P and T denote the poloidal and toroidal components of a vector. The force-free assumption,
implies that the poloidal current flows parallel to the poloidal magnetic field . Thus, .
With the assumption of a degenerated magnetosphere, i.e.
an angular velocity of field lines can be derived from the derivative of the time component of the vector potential
With the additional assumption of stationarity, Ampère's law can be expressed as
(TPM). The differential rotation of space provides an additional source term with the dimension of a current density. The toroidal current density follows from a projection of Eq. (4), the equation of motion in the force-free limit, perpendicular to ,
The toroidal component of Ampère's law (7) eventually leads to the stream equation, a non linear partial differential equation of second order for the flux function ,
and denotes the positions of the two light surfaces,
The indicates the derivative . The sign holds for the outer light surface with , while the - sign stands for the inner light surface, where . Throughout this paper we will assume a constant angular velocity of the field lines, const. This assumption will be discussed below (Sect. 3.5).
The stream equation was first derived by Blandford & Znajek (1977) and further evaluated in the 3+1 formalism by TPM. A general version of the stream equation including inertial terms and entropy was obtained by Beskin & Pariev (1993).
In the special relativistic limit, , , and the stream eq. (9) becomes identical to the pulsar equation (Scharlemann & Wagoner 1973).
For large r, and the metric reduces to Minkowski. We define an asymptotic light cylinder, (here and in the following denote the cylindrical coordinates). Using the normalization
the stream equation can be written dimensionless
The coupling constant measures the strength of the (poloidal current) source term,
(A similar normalization for the term in Eq. (9) would reveal a coupling constant .)
Two typical length scales enter the problem. (1) The scale of the horizon radius determines the influence of gravitation on the metric. (2) The asymptotic light cylinder describes the influence of rotational effects on the electrodynamics. The interrelation between these two scaling parameters follows from the definition of the rotation law for the field, , in terms of the rotation of the black hole, .
2.4. The regularity condition
At the light surfaces, the stream equation becomes singular and reduces to a non-linear, partial differential equation of first order,
This regularity condition for is equivalent to an inhomogeneous Neumann-type boundary condition on the poloidal magnetic field component parallel to the surface (the light surface),
where denotes the unit vector normal to the light surface.
Note that the regularity condition depends on the strength of the poloidal current as well as the current distribution. This has far reaching consequences for the global field topology. For the special relativistic case, we have shown that the shape of the jet boundary is determined by the regularity requirement (Fendt 1994, Fendt et al. 1995). As it will discussed below, the same applies in the general relativistic case.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998