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

6.4.1 Mass-quadrupole radiationEnergy radiated in gravitational waves 69The mass-quadrupole radiation field in equation (6.26) must be put into the energyflux formula, and the dependence on the direction n i can then be integrated overa sphere. It is not a difficult calculation, but it does require some angular integralsover multiple products of the vector n i , which depends on the angular directionon the sphere. By symmetry, integrals of odd numbers of factors of n i vanish. Foreven numbers of factors, the result is essentially determined by the requirementthat after integration the result must be fully symmetric under interchange of anytwo indices and it cannot have any special directions (so it must depend only onthe Kronecker delta δ i j ). The identities we need are∫n i n j d = 4π 3 δij , (6.41)∫n i n j n k n l d = 4π15 (δij δ kl + δ ik δ jl + δ il δ jk ). (6.42)Using these, one gets the following simple formula for the total luminosity ofmass-quadrupole radiationL massgw = 1 ... ...5 〈 ˜M jk ˜M jk 〉 . (6.43)Here we still preserve the angle brackets of equation (5.14), because this formulaonly makes sense in general if we average in time over one cycle of the radiation.6.4.2 Current-quadrupole radiationThe energy radiated in the current quadrupole is nearly as simple to obtain as themass-quadrupole formula. The extra factor of n i in the radiation field makes theangular integrals longer, and requires two further identities:∫n i n j n k n l n p n q d = 4π 7 δ(ij δ kl δ pq) , (6.44)ɛ ijk ɛ i′ j ′ k ′ = δ ii′ δ jj′ δ kk′ + δ ij′ δ jk′ δ ki′ + δ ik′ δ ji′ δ kj′− δ ii′ δ jk′ δ kj′ − δ ij′ δ ji′ δ kk′ − δ ik′ δ jj′ δ ki′ , (6.45)where the round brackets indicate full symmetrization on all indices. Theexpression is simplest if we define˜J jk := ɛ jlm ˜P k lm + ɛ klm ˜P j lm ,where˜P kij := P kij − 1 3 δij P kl l.The result of the integration of the flux formula over a distant sphere is[18, 25], in our notation,L currentgw = 4 5 〈 ...˜J jk ...˜J jk 〉. (6.46)