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

62 Mass- and current-quadrupole radiationdipole, or momentum part of the field, which is also part of the non-wave solution.Our gauge transformation has incorporated a Lorentz transformation that has putus into the rest frame of the source.) The time-dependent part of the field is nowpurely spatial, transverse (because everything is multiplied by ⊥), and traceless(as can be verified by explicit calculation).The expression for the spatial part of the field actually does not depend onthe trace of S jk , as can be seen by constructing the trace-free part of the tensor,defined as:˜S jk = S jk − 1 3 δ jk S l l. (6.24)In fact, it is more conventional to use the mass moment here instead of the stress,so we also define˜M jk = M jk − 1 3 δ jk Ml l , ˜S jk = 1 d 2 ˜M jk2 dt 2 . (6.25)In terms of ˜M the far field is¯h TTij = 2 (⊥ ik ⊥ jl ¨˜M kl + 1 r2 ⊥ij ¨˜M )kl n l n k . (6.26)This is the usual formula for the mass-quadrupole field. In textbooks the notationis somewhat different than we have adopted here. In particular, our tensor ˜Mis what is called I– in Misner et al (1973) and Schutz (1985). It is the basis ofmost gravitational-wave source estimates. We have derived it only in the contextof linearized theory, but remarkably its form is identical if we go to the post-Newtonian approximation, where the gravitational waves are a perturbation ofthe Newtonian spacetime rather than of flat spacetime.Given this powerful formula, it is important to try to interpret it andunderstand it as fully as possible. One obvious conclusion is that the dominantsource of radiation, at least in the slow-motion limit, is the second timederivativeof the second moment of the mass density T 00 (the mass-quadrupolemoment). This is a very important difference between gravitational wavesand electromagnetism, in which the most important source is the electricdipole.In our case the mass-dipole term is not able to radiate because it isconstant, reflecting conservation of the linear momentum of the source. Inelectromagnetism, however, if the dipole term is absent for some reason (allcharges positive, for example) then the quadrupole term dominates and it looksvery similar to equation (6.26).6.2.3 Radiation patterns related to the motion of sourcesThe projection operators in equation (6.26) show that the radiative field istransverse, as we expect. However, the form of equation (6.26) hides two equallyimportant messages: