(ed.). Gravitational waves (IOP, 2001)(422s).

Application of the TT gauge to the current-quadrupole field 67−++−Figure 6.1. A simple current-quadrupole radiator. The left-hand panel shows how the twowheels are connected with blade springs to a central axis. The wheels turn in oppositedirections, each oscillating back and forth about its rest position. The right-hand panelshows the side view of the system, and the arrows indicate the motion of the near side ofthe wheels at the time of viewing. The + signs indicate where the momentum of the massof the wheel is toward the viewer and the − signs indicate where it is away from the viewer.The simplicity of these expressions is striking. There are two basic caseswhere one gets current-quadrupole radiation.• If there is an oscillating angular momentum distribution with a dipolemoment along the angular momentum axis, as projected onto the sky, thenin an appropriate coordinate system ¨J xx will be nonzero and we will have ⊗radiation. To have a non-vanishing dipole moment, the angular momentumdensity could, for example, be symmetrical under reflection through theorigin along its axis, so that it points in opposite directions on opposite sides.• If there is an oscillating angular momentum distribution with a dipolemoment along an axis perpendicular to the angular momentum axis, asprojected onto the sky, then in an appropriate coordiate system ¨J xy will benonzero and we will have ⊕ radiation.6.3.3 A model system radiating current-quadrupole radiationTo see that the first of these two leads to physically sensible results, let us considera simple model system that actually bears a close resemblance to the r-modesystem. Imagine, as in the left panel of figure 6.1, two wheels connected by anaxis, and the wheels are sprung on the axis in such a way that if a wheel is turnedby some angle and then released, it will oscillate back and forth about the axis.Set the two wheels into oscillation with opposite phases, so that when one wheelrotates clockwise, the other rotates anticlockwise, as seen along the axis.Then when viewed along the axis, the angular momentum has no componenttransverse to the line of sight, so there is no radiation along the axis. This