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

68 Mass- and current-quadrupole radiationis sensible, because when projected onto the plane of the sky the two wheelsare performing exactly opposite motions, so the net effect is that there is zeroprojected momentum density.When viewed from a direction perpendicular to the axis, with the axis alongthe x-direction, then the angular momentum is transverse, and it has oppositedirection for the two wheels. There is therefore an x-moment of the x-componentof angular momentum, and the radiation field will have the ⊗ orientation.To see that this has a physically sensible interpretation, look back again at thepolarization diagram, figure 2.1, and look at the bottom row of figures illustratingthe ⊗ polarization. See what the particles on the x-axis are doing. They aremoving up and down in the y-direction. What motions in the source could beproducing this?At first one might guess that it is the up-and-down motion of the mass in thewheels as they oscillate, because in fact the near side of each wheel does exactlywhat the test particles at the observer are doing. However, this cannot be theexplanation, because the far side of each wheel is doing the opposite, and whenthey both project onto the sky they cancel. What in fact gives the effect is that atthe top of the wheel the momentum density is first positive (towards the observer)and then negative, while at the bottom of the wheel it is first negative and thenpositive. On the other wheel, the signs are reversed.Current-quadrupole radiation is produced, at least in simple situations likethe one we illustrate here, by (the second time-derivative derivative of) thecomponent of source momentum along the line of sight. If this is positive in thesense that it is towards the observer, then the momentum density acts as a positivegravitational ‘charge’. If negative, then it is a negative ‘charge’. The wheelshave an array of positive and negative spots that oscillates with time, and thetest particles in the polarization diagram are drawn toward the positive ones andpushed away from the negative ones. Interestingly, in electromagnetism, magneticdipole and magnetic quadrupole radiation are also generated by the component ofthe electric current along the line of sight.This is a rather simple physical interpretation of some rather more complexequations. It is possible to re-write equation (6.38) to show explicitly thecontribution of the line-of-sight momentum, but the expressions become evenmore complicated. Instead of dwelling on this, I will turn to the question ofcalculating the total energy radiated by the source.6.4 Energy radiated in gravitational wavesWe have calculated the energy flux in equation (5.14), and we now have the TTwave amplitudes. We need only integrate the flux over a distant sphere to getthe total luminosity. We do this for the mass and current quadrupoles in separatesections.