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

Application of the TT gauge to the current-quadrupole field 65The TT projection of the equation for the metric ish TTij = 4 r ṠTTijk n k . (6.32)6.3.2 Separating the current quadrupole from the mass octupoleThe last equation is compact, but it does not have the ready interpretation thatwe have at quadrupole order. This is because the moment of the stress, S ijk ,does not have such a clear physical interpretation. We see from equations (6.6)–(6.8) that S ijk is a complicated mixture of moments of momentum and density.To gain more physical insight into radiation at this order, we need to separatethese different contributions. It is straightforward algebra to see that the followingidentity follows from the earlier ones:Ṡ ijk = 1 ...6 M ijk + 2 ¨P [ jk]i 3+ 2 ¨P [ik] j 3, (6.33)where square brackets around indices mean antisymmetrization:A [ik] := 1 2 (Aik − A ki ).This is a complete separation of the mass terms (in M) from the momentum terms(in P) because the only identities relating the momentum moments to the massmoments involve the symmetric part of P ijk on its first two indices, and this isabsent from equation (6.33).The first term in equation (6.33) is the third moment of the density, and thisis the source of the mass-octupole field. It produces radiation through the thirdtime-derivative. Since we are in a slow-motion approximation, this is smaller thanthe mass-quadrupole radiation by typically a factor v/c. Unless there were somevery special symmetry conditions, one would not expect the mass octupole to beanything more than a small correction to the mass quadrupole. For this reason wewill not treat it here.The second and third terms in equation (6.33) involve the second momentof the momentum, and together they are the source of the current-quadrupolefield. It involves two time-derivatives, just as the mass quadrupole does, but theseare time-derivatives of the momentum moment, not the mass moment, so theseterms produce a field that is also v/c smaller than the typical mass-quadrupolefield. However, it requires less of an accident for the mass quadrupole to beabsent and the current quadrupole present. It just requires motions that leave thedensity unchanged to lowest order. This happens in the r-modes. Therefore, thecurrent quadrupole deserves more attention, and we will work exclusively withthese terms from now on.The terms in equation (6.33) that we need are the ones involving ¨P ijk .These are antisymmetrized on the first two indices, which involves effectivelya vector product between the momentum density (first index) and one of themoment indices. This is essentially the angular momentum density. To make