Electromagnetic extraction of energy from Kerr black holes

When a rotating black hole is threaded by magnetic field lines supported by external currents flowing in an equatorial disc, an electric potential difference will be induced. If the field strength is large enough, the vacuum is unstable to a cascade production of electron–positron pairs and a surrounding force-free magnetosphere will be established. Under these circumstances it is demonstrated that energy and angular momentum will be extracted electromagnetically. As a further consequence it is shown that charge can never contribute significantly to the geometry of a rotating hole. The fundamental equations describing a stationary axisymmetric magnetosphere are derived and the details of the energy and angular momentum balance are discussed. A perturbation technique is developed which can be used to provide approximate solutions for slowly rotating holes. Solutions appropriate when the field lines threading the hole lie on conical and paraboloidal surfaces at large distances are described to illustrate this mechanism. These ideas are incorporated into a discussion of a model of active galactic nuclei containing a massive black hole surrounded by a magnetized accretion disc. In this model relativistic electrons can be accelerated at large distances from the hole and therefore will not incur serious losses, which is a defect of some existing models. In addition, if the field lines have paraboloidal shape, the energy will be beamed along antiparallel directions as observations of both compact and extended radio sources seem to require

Trans-Magnetosonic Accretion in a Black Hole Magnetosphere

We present the critical conditions for hot trans-fast magnetohydrodynamical (MHD) flows in a stationary and axisymmetric black-hole magnetosphere. To accrete onto the black hole, the MHD flow injected from a plasma source with low velocity must pass through the fast magnetosonic point after passing through the ``inner'' or ``outer'' Alfven point. We find that a trans-fast MHD accretion solution related to the inner Alfven point is invalid when the hydrodynamical effects on the MHD flow dominate at the magnetosonic point, while the other accretion solution related to the outer Alfven point is invalid when the total angular momentum of the MHD flow is seriously large. When both regimes of the accretion solutions are valid in the black hole magnetosphere, we can expect the transition between the two regimes. The variety of these solutions would be important in many highly energetic astrophysical situations.Comment: 27 pages, 12 figures, accepted to Ap

Electromagnetic extraction of energy from Kerr black holes

When a rotating black hole is threaded by magnetic field lines supported by external currents flowing in an equatorial disc, an electric potential difference will be induced. If the field strength is large enough, the vacuum is unstable to a cascade production of electron–positron pairs and a surrounding force-free magnetosphere will be established. Under these circumstances it is demonstrated that energy and angular momentum will be extracted electromagnetically. As a further consequence it is shown that charge can never contribute significantly to the geometry of a rotating hole. The fundamental equations describing a stationary axisymmetric magnetosphere are derived and the details of the energy and angular momentum balance are discussed. A perturbation technique is developed which can be used to provide approximate solutions for slowly rotating holes. Solutions appropriate when the field lines threading the hole lie on conical and paraboloidal surfaces at large distances are described to illustrate this mechanism. These ideas are incorporated into a discussion of a model of active galactic nuclei containing a massive black hole surrounded by a magnetized accretion disc. In this model relativistic electrons can be accelerated at large distances from the hole and therefore will not incur serious losses, which is a defect of some existing models. In addition, if the field lines have paraboloidal shape, the energy will be beamed along antiparallel directions as observations of both compact and extended radio sources seem to require

Analytic Solutions to the Constraint Equation for a Force-Free Magnetosphere around a Kerr Black Hole

The Blandford-Znajek constraint equation for a stationary, axisymmetric black-hole force-free magnetosphere is cast in a 3+1 absolute space and time formulation, following Komissarov (2004). We derive an analytic solution for fields and currents to the constraint equation in the far-field limit that satisfies the Znajek condition at the event horizon. This solution generalizes the Blandford-Znajek monopole solution for a slowly rotating black hole to black holes with arbitrary angular momentum. Energy and angular momentum extraction through this solution occurs mostly along the equatorial plane. We also present a nonphysical, reverse jet-like solution.Comment: 6 pages, accepted for publication in Ap

Force-Free Magnetosphere of an Accretion Disk -- Black Hole System. I. Schwarzschild Geometry

In this paper I study the magnetosphere of a black hole that is connected by the magnetic field to a thin conducting Keplerian disk. I consider the case of a Schwarzschild black hole only, leaving the more interesting but difficult case of a Kerr black hole to a future study. I assume that the magnetosphere is ideal, stationary, axisymmetric, and force-free. I pay a special attention to the two singular surfaces present in the system, i.e., the event horizon and the inner light cylinder; I use the regularity condition at the light cylinder to determine the poloidal electric current as a function of poloidal magnetic flux. I solve numerically the Grad--Shafranov equation, which governs the structure of the magnetosphere, for two cases: the case of a nonrotating disk and the case of a Keplerian disk. I find that, in both cases, the poloidal flux function on the horizon matches a simple analytical expression corresponding to a radial magnetic field that is uniform on the horizon. Using this result, I express the poloidal current as an explicit function of the flux and find a perfect agreement between this analytical expression and my numerical results.Comment: 28 pages, 3 figures; submitted to the Astrophysical Journal; some minor corrections made and a reference adde

Electromagnetic Fields of Separable Space-Times

Carter derived the forms of the metric and the vector potentials of the space-times in which the relativistic Schrodinger equation for the motion of a charged particle separates. Here we show that on each `spheroidal' surface a rotation rate exists such that relative to those rotating axes the electric and magnetic fields are parallel and orthogonal to the spheroid which is thus an equipotential in those axes. All the finite Carter separable systems without magnetic monopoles or gravomagnetic NUT monopoles have the same gyromagnetic ratio as the Dirac electron.Comment: 9 pages; accepted for publication in Class. Quantum Gra

A Toy Model for the Magnetic Connection between a Black Hole and a Disk

A magnetic field connecting a Kerr black hole to a disk rotating around it can extract energy and angular momentum from the black hole and transfer them to the disk if the black hole rotates faster than the disk. The energy can be dissipated and radiated away by the disk, which makes the disk shine without the need of accretion. In this paper we present a toy model for the magnetic connection: a single electric current flowing around a Kerr black hole in the equatorial plane generates a poloidal magnetic field which connects the black hole to the disk. The rotation of the black hole relative to the disk generates an electromotive force which in turn generates a poloidal electric current flowing through the black hole and the disk and produces a power on the disk. We will consider two cases: (1) The toroidal current flows on the inner boundary of the disk, which generates a poloidal magnetic field connecting the horizon of the black hole to a region of the disk {\it beyond} the inner boundary; (2) The toroidal current flows on a circle inside the inner boundary of the disk but outside the horizon of the black hole, which generates a poloidal magnetic field connecting a portion of the horizon of the black hole to the {\it whole} disk. We will calculate the power produced by the magnetic connection and the resulting radiation flux of the disk in the absence of accretion, and compare them with that produced by accretion.Comment: 9 pages, including 9 figure

Issues in the Blandford-Znajek Process for GRB Inner Engine

Several issues regarding the Blandford-Znajek process are discussed to demonstrate that it can be an effective mechanism for powering the gamma ray bursts. Using a simple circuit analysis it is argued that the disk power increases the effective power of the black hole-accretion disk system, although a part of disk power can be dissipated into black hole entropy. Within the framework of the force-free magnetosphere with the strong magnetic field, the magnetically dominated MHD flow is found to support the Blandford-Znajek process and it is demonstrated that the possible magnetic repulsion by the rotating black hole will not affect the efficiency substantially.Comment: 10 pages, 1 figure, 3 references added, more discussions on the magnetic field on the black hole, accepted for publication in Ap

Hot Settling Accretion Flow onto a Spinning Black Hole

We study the structure and properties of hot MHD accretion onto a Kerr black hole. In such a system, the hole is magnetically coupled to the inflowing gas and exerts a torque onto the accretion flow. A hot settling flow can form around the hole and transport the angular momentum outward, to the outer edge of the flow. Unlike other hot flows, such as advection- and convection-dominated flows and inflow-outflow solutions (ADAFs, CDAFs, and ADIOS), the properties of the hot settling flow are determined by the spin of the central black hole, but are insensitive to the mass accretion rate. Therefore, it may be possible to identify rapidly spinning BHs simply from their broad-band spectra. Observationally, the hot settling flow around a Kerr hole is somewhat similar to other hot flows in that they all have hard, power-law spectra and relatively low luminosities. Thus, most black hole candidates in the low/hard and, perhaps, intermediate X-ray state may potentially accrete via the hot settling flow. However, a settling flow will be somewhat more luminous than ADAFs/CDAFs/ADIOS, will exhibit high variability in X-rays, and may have relativistic jets. This suggests that galactic microquasars and active galactic nuclei may be powered by hot settling flows. We identify several galactic X-ray sources as the best candidates.Comment: 7 pages, 1 figure. Submitted to Ap

Force-Free Magnetosphere of an Accretion Disk -- Black Hole System. II. Kerr Geometry

We consider a stationary axisymmetric force-free degenerate magnetosphere of a rotating Kerr black hole surrounded by a thin Keplerian infinitely-conducting accretion disk. We focus on the closed-field geometry characterize by a direct magnetic coupling between the disk and the hole's event horizon. We first argue that the hole's rotation necessarily limits the radial extent of the force-free link on the disk surface: the faster the hole rotates, the smaller the magnetically-connected inner part of the disk has to be. We then show that this is indeed the case by solving numerically the Grad--Shafranov equation--the main differential equation describing the structure of the magnetosphere. An important element in our approach is the use of the regularity condition at the inner light cylinder to fix the poloidal current as a function of the poloidal magnetic flux. As an outcome of our computations, we are able to chart out the maximum allowable size of the portion of the disk that is magnetically connected to the hole as a function of the black hole spin. We also calculate the angular momentum and energy transfer between the hole and the disk that takes place via the direct magnetic link. We find that both of these quantities grow rapidly and that their deposition becomes highly concentrated near the inner edge of the disk as the black hole spin is increased.Comment: 45 pages, 12 figures. Accepted for publication in the Astrophysical Journa