3.3. Kinematics of superluminal expansion
As described in the contribution by Kellermann, there is now good evidence for superluminal expansion in at least three compact sources, if we adopt the cosmological interpretation of large redshifts. A variety of physical models have been proposed to account for this phenomenon, which we now describe using a loose kinematic classification developed in the review by Blandford, McKee and Rees  (to which the reader is referred for further details and references to the literature).
When a radio source of size R changes in a time ~ R/c, then light travel time effects in the emergent radiation must be taken properly into account. The main requirements on a physical model, if the sources observed so far are typical, is to produce a double or aligned triple expanding with a speed 2c - 8c, successive expansions occurring in the same direction. Although "acausal" models have been proposed in which there are two or more independent outbursts, the observed predominance of expansion over contraction raises problems for this interpretation. That superluminal expansion is indeed possible can be simply demonstrated by considering a "ballistic" model in which a shell of radio emitting plasma moves radially outwards from an origin `o' with speed . At a fixed observed time, the locus of the shell is a prolate ellipsoid with one focus at 0 and eccentricity /c. The observer therefore sees a circle of radius equal to the semiminor axis of the ellipsoid expanding with speed 2V(1 - V2 / c2)-1/2, which exceeds 2c if > 2-1/2 c. In a straight-forward variation designed to produce an expanding double, two radio-emitting plasmoids or beams are ejected along antiparallel directions with the nearer making an angle ~ (1 - 2 / c2)1/2 with the observer direction. An expansion speed 0 (1 - 2 / c2)1/2 will be observed. However the probability of the observer having this orientation is only ~ 2c2 / 02, which is typically very small when 0 >> c. Extreme superluminal expansion would then have to be a comparatively rare phenomenon amongst a complete sample of compact radio sources.
In a simple example of a "screen" model a relativistic signal (e.g., a shock) illuminates a thin circular ring, (e.g., the inner edge of a disc of accreting matter), radius R. An observer in the plane containing the ring will see a double, of separation x, separating with a speed 0 ~ (R / x)c;x << R. Screen models are usually characterised by deceleration of the observed expansion, although more complex observations can be interpreted using suitably tailored screens. The signal motion need not be rectilinear. One ingenious suggestion  is that the emitting electrons are channelled at relativistic speeds along the north and south polar field lines of a magnetic dipole. They would then only be observed when the field line pointed towards the observer, and an outburst originating near the centre of the dipole would create an expanding double. (For an equatorial observer 0 = 4.4c.)
In most simple types of screen model superluminal effects can usually be achieved without the observer having a privileged orientation (in contrast to ballistic models). Apparent velocities 0 >> c will however only be observed for a fraction ~ (c / 0)2 of the total emission time, and the actual size of the screen will exceed the observed size by a factor typically ~ (0 / c).
There are many examples in physics of cases where the space velocity of some feature (although not the velocity of energy transport) can exceed c. For example the line of intersection of two almost parallel surfaces (e.g. shock waves), rapidly approaching one another, can be arbitrarily large. In "phase" models (e.g., Epstein and Geller ) the observed expansion is caused directly by such a mechanism rather than light travel time effects. Proper allowance for these effects will however in general result in a subrelativistic apparent expansion, unless the observer lies in a direction roughly perpendicular to the phase velocity, again with a rather low probability ~ (c/v0).
As well as the observed rate of expansion, we must also consider the relative surface brightness of different parts of the source. Of course this depends in detail on a variety of astrophysical effects particle acceleration, magnetic field strength, energy and isotropy of the central outbursts etc.- of which our knowledge must be limited. One important effect that can be considered is the Doppler shift of the emitting plasma parametrised by the factor D = (1 - 2 / c2)1/2(1 - em cos / c)-1 where is the angle between the velocity em of the emitting plasma (i.e., of the frame in which the emission is more or less isotropic) and the line of sight. em need not be the same as the velocity of the signal. It can be argued that em must be neither too small nor too large, and is optimally mildly relativlstic. If em << c, D 1 for all and so all regions accessible to the signal should radiate towards the observer, not just those displaying superluminal expansion. Unless the observer has a special orientation, the flux from the most rapidly moving components would be dominated by that from the slower moving background, contrary to what is observed. If em is ultrarelativistic then only material moving with an angle (1 - 2 / c2)1/2 will be seen and then it is difficult to explain how an observer in an arbitrary direction will see doubles expanding along a preferred axis.
A type of model that seems able to explain the existing data was proposed by Blandford et al. . An outburst associated with the primary energy source initiates a relativistic signal (for instance, a burst of electromagnetic energy or a strong shock) which propagates outwards illuminating a suitably shaped screen (such as the walls of a channel) similar to that invoked in Section 2 to be responsible for the supply of energy to extended double radio sources. The burst gives the radio emitting plasma a mildly relativistic velocity, thus ensuring that only the front of the screen is highlighted. (Note that this mildly relativistic expansion velocity is possibly required to avoid excessive Compton losses, and is perhaps implied by the inferred dominance of particle pressure.) With this sort of geometry, separation speeds ~ 5c are achievable without the observer having a particularly privileged orientation. It remains to be seen if this type of explanation is adequate to account for observed sources.
These "intrinsic" kinematical models are not the only ones capable of reproducing superluminal expansion. Radio waves produced in a single spherically symmetric outburst can also be refracted or scattered so as to produce an apparently separating double. If the radiation is scattered through a small angle whilst propagating towards the observer, then the unscattered photons will arrive before the scattered photons and a roughly circular source of expansion speed ~ 2c / will be produced. Unfortunately, in a model like this, the primary source size must be smaller than the observed size (implying coherent emission), in contrast to most intrinsic models, and this increases the ratio of the Compton scattered flux to the synchrotron flux. One such model, proposed by Wilson , involves a very compact variable source at the centre of a non-variable line source (which may be regarded as the inner part of the "beams" of an extended source) embedded in a cloud of thermal plasma. If induced scattering dominates spontaneous scattering in the cloud, then an outburst in the central component will be followed by an apparent "expanding double" of enhanced surface brightness moving out along the line source. This type of model has the virtue of being able to explain superliminous expansion along a fixed axis which is not close to the line of sight; but has the possible disadvantage that the apparent brightness of the expanding components would be strongly frequency-dependent.
The VLBI and variability data can generally be explained in terms of relativistic effects where the (phase or physical) velocities corresponding to Lorentz factors 5. The main exceptions to this are spectacular variables such as AO 02354+164 and those sources - e.g., 3C 454.3 - which appear to vary at frequencies below 1 GHz.