4. VARIATIONS ON THE THEME

4.1. Special Relativity

Special relativistic gas dynamics (and MHD) is, in many respects, similar to its subrelativistic counterpart and many hydrodynamical effects like shocks, nozzle flow etc, carry over. In particular, it has recently been possible to demonstrate special relativistic, self-similar solutions of disk-driven, collimated outflow (Li, Chiueh & Begelman 1992, cf. Camenzind 1989, Lovelace et al. 1986). These solutions, typically, exhibit the property of equipartition between magnetic and mechanical energy asymptotically and arbitrarily large speed.

However, the surprisingly large effects of relativistic beaming imply that our observations of the powerful VLBI jets may be quite misleading although it is these that promise the finest probe of the jet collimation region. As we have already discussed, radiative drag becomes increasingly important in a particle-dominated flow with increasing bulk Lorentz factor. However, much larger bulk Lorentz factors are possible if most of the momentum flux, in the region where the radiation energy density is high, is in an electromagnetic form. Provided that the energy can be transported at high speed to the radio-emitting region there is no difficulty of principle in acquiring electrons from the surrounding gas and accelerating them to energies ~ 100 - 1000 MeV perhaps through shock waves. (Radio source models relying on this principle can have a beaming angle much larger than ~ c/v_ob and this may alleviate problems in accounting for the unbeamed sources.) Arguably, the most attractive way to produce an electromagnetically-dominated outflow is to extract spin energy from a Kerr black hole.

4.2. Black Hole MHD

As has been reviewed elsewhere (e.g., Thorne, Price & MacDonald 1986), it is also possible to extract the rotational kinetic energy of a spinning black hole as well as that of an accretion disk. A spinning black hole, described by the Kerr metric can be described by two parameters (e.g., Misner, Thorne & Wheeler 1973), that can be chosen to be the irreducible mass m_i and the angular velocity . The irreducible mass is analogous to the rest mass in ordinary relativistic dynamics and the minimum mass to which a black hole can be reduced through classical processes. The actual gravitational mass of the hole m is related to the irreducible mass through the familiar-looking formula

where = 2m_i < 2^-1/2 in gravitational units. The mass difference, m - m_i, can be extracted in principle. In practice, no more than half of this difference is likely to be extracted, the balance going into the irreducible mass.

Magnetic fields that thread the hole's event horizon and are supported by external currents will induce an electric potential difference between the poles and the equator of the hole. This will cause currents to flow. Fiducial numbers for ~ 10⁸ M black holes in AGN are magnetic fields of strength ~ 10⁴ G, potential differences V ~ 10²⁰ V and currents I ~ 10¹⁸ A. At these large potentials, there is ample electric field strength to break down the vacuum and to create electron-positron pairs within a black hole magnetosphere. This allows current to flow continuously from the horizon to infinity, while particles move inward across the horizon, but outward at large radius in a relativistic MHD wind. Remember that only a negligible kinetic energy flux need be associated with those particles that are carrying the electrical current; the overwhelming majority would then be transported in the form of electromagnetic Pointing flux. However, if there is significant dissipation in the flow (and this is not an unreasonable supposition), then the mechanical energy flux may eventually dominate the outflowing wind.

This picture of electromagnetic energy extraction is reminiscent of (and derived from) theories of pulsar electrodynamics. However, there are some crucial differences. Firstly, there is the need for charged particle creation in the magnetosphere and the attendant dissipation. Provided that ions are extractable from neutron star surfaces, there is no necessity for this to occur, at least in simple models. However, it is observed that rapidly spinning, strong field pulsars, notably the Crab pulsar, are quite dissipative with at least several percent of the energy flux appearing as -rays. It is reasonable to guess that an even higher percentage of the extracted power might be dissipated in this manner in the case of black holes. (It is unlikely that the 3 GeV -rays detected by the EGRET instrument on the Compton Gamma Ray Observatory originate directly from a black hole magnetosphere as the opacity to pair production on ~ keV X-rays is much greater than unity. It is far more likely that these -rays are formed at greater distances from the hole, in an outflowing relativistic jet.)

A second important difference is that, unlike a neutron star, which is an excellent conductor, a black hole acts as if it possesses an electrical resistance ~ 100 . Indeed, it is possible to develop a simple circuit analogy for a stationary, axisymmetric magnetosphere (e.g., Thorne et al. 1986). Consider two magnetic surfaces threading the horizon and suppose that a current I flows out along one surface and returns along the other. The current will complete within the hole across an internal resistance R_H and at large distance from the hole by crossing a load impedance R_L. Let the magnetic flux enclosed by the two surfaces be . There will be more magnetic field outside these two surfaces along which current will also flow. It is possible to ascribe an angular velocity to the magnetic field lines (essentially the angular velocity of a local observer who measures no electric field). This will be a weighted average of the hole (_H) and load (_L) angular velocities.

We might expect that _L = 0, but if the magnetic surface intersects the accretion disk then it will be non-zero. It can then be shown that the torque that is transmitted from the hole to the load by the magnetosphere is

Rearranging this equation, we obtain an energy conservation equation.

In other words, the work that it is done by the hole on the magnetosphere equals the work done by the magnetosphere on the load plus the ohmic dissipation in both the hole and load. The ohmic dissipation in the load can be large enough to satisfy the non-thermal and dynamical appetites of relativistic jets and double radio sources.

Therefore, in this model, the power derives hydromagnetically, apparently from within the horizon. This has led Punsley & Coroniti (1990) to suspect that causality is violated and that it is not possible for a black hole to evolve from generic initial conditions to this steady state. (In the case of a pulsar, there is no such problem because the neutron star is definitely causally connected to its magnetosphere.) However, an alternative view (e.g., Blandford 1990) is that the causal connection between the hole and its magnetosphere is provided gravitationally by the frame-dragging effect of the g₀ components of the metric tensor. It has not yet been possible to resolve this dispute satisfactorily by solving a suitable characteristic initial value problem completely.

4.3. Boundary Layers

At first sight, the differences between black holes and protostars seem so great that it is surprising that they could play a common role in jet formation. However, I want to argue that there is an analogy that can be exploited. Specifically, if a protostar is orbited by an accretion disk which extends down to its surface (and this requires that the stellar magnetic field not be too great), then a boundary layer should form (e.g., Torbett 1984, Pringle 1989). As long as the star is not rotating with close to the break up angular velocity, (and observations of T Tauri stars suggest that this may be the case, though see Shu et al. 1988 for an alternative view), then half of the total gravitational binding energy released by infall onto the star will come from the boundary layer.

The structure of a boundary layer is still a matter for vigorous debate (e.g., Tylenda 1981). However, provided that much of the power is released as heat (or as radiation which can be absorbed by the surrounding gas), then it will function rather like a spinning black hole in an AGN. Specifically, I propose that most of the power carried by jets is concentrated in the central core and is associated with the central object. In the absence of a confining disk wind, this energy release would probably escape quasispherically. Similar boundary layers can form in CV and XRB, and these may also influence outflows in these objects.

4.4. Powers of Ten

There may be one further key ingredient for the production of a highly collimated jet. It may be necessary that the magnetic flux thread the disk over several decades of radius. As can be seen from Table 1 (cf. Ray, Marscher, these proceedings), there is no example of a jet that has to be finely collimated within a distance 10³ the putative size of the central object. Indeed, if a jet originates from a disk extending over several decades of radius, its velocity structure is probably far more complex than popularly assumed and its degree of collimation at fixed altitude will decrease with increasing cylindrical radius.

4.5. Precession

Another important clue comes from precessing jets. The most prominent example is SS433 (Vermeulen, these proceedings, Peter & Eichler 1992, preprint) although there is some doubt in this case that the physical environment is typical of jets. Less clear-cut examples of inversion symmetric sources can be found in the extragalactic radio astronomers' œuvre, although it is not clear whether or not this morphology is attributable to precession, alignment or chance in most cases. If either precessional or aligning torques are at work, then this provides additional evidence that jet collimation occurs over many decades of radius.

The simplest explanation for jet precession, e.g., in SS433, invokes the "hotplate" model and posits that the companion star has a driven precession with a 164d period and that the accretion disk is slaved to this, perhaps in antiphase. A modulation at half of the orbital synodic period (~ 6.3 d) supports this interpretation. One objection that has been raised, though, is that the inflow timescale from the outer disk is not short compared with the precession period and so we might expect that the inner disk has a substantial phase lag with respect to the outer disk. There may, however, be a resolution of this difficulty if, as outlined above, it is poloidal magnetic field, attached to the outer disk, that is responsible for the collimation, while most of the power and thrust (and consequently the jet speed) is controlled by the inner disk.