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

Expansion for the far field of a slow-motion source 59The two conditions r ≫ y i y i and the slow-motion source, can be expressedquantitatively as:r ≫ ¯λR ≪ ¯λwhere ¯λ is the reduced wavelength ¯λ = λ/2π and R is the size of source.The terms of order r −1 are negligible for the same reason as above, but thefirst term in this expansion must be taken into account. It depends on the directionto the field point, given by the unit vector n i . We use this by making a Taylorexpansion in time on the time-argument of the source. The combined effect ofthese approximations ish αβ = 4 r∫[T αβ (t ′ , y i ) + T αβ ,0(t ′ , y i )n j y j + 1 2 T αβ ,00(t ′ , y i )n j n k y j y k+ 1 6 T αβ ,000(t ′ , y i )n j n k n l y j y k y l +···]d 3 y. (6.5)We will need all the terms of this Taylor expansion out to this order.The integrals in expression (5.5) contain moments of the components of thestress-energy. It is useful to give these names. Use M for moments of the densityT 00 , P for moments of the momentum T 0i and S for the moments of the stressT ij . Here is our notation:∫∫M(t ′ ) = T 00 (t ′ , y i ) d 3 y, M j (t ′ ) = T 00 (t ′ , y i )y j d 3 y,∫∫M jk (t ′ ) = T 00 (t ′ , y i )y j y k d 3 y, M jkl (t ′ ) = T 00 (t ′ , y i )y j y k y l d 3 y,∫∫P l (t ′ ) = T 0l (t ′ , y i ) d 3 y, P lj (t ′ ) = T 0l (t ′ , y i )y j d 3 y,∫P ljk (t ′ ) = T 0l (t ′ , y i )y j y k d 3 y,∫∫S lm (t ′ ) = T lm (t ′ , y i ) d 3 y, S lmj (t ′ ) = T lm (t ′ , y i )y j d 3 y.These are the moments we will need.Among these moments there are some identities that follow from theconservation law in linearized theory, T αβ ,β = 0, which we use to replace timederivatives of components of T by divergences of other components and thenintegrate by parts. The identities we will need areṀ = 0, Ṁ k = P k , Ṁ jk = P jk + P kj , Ṁ jkl = P jkl + P klj + P ljk ,(6.6)Ṗ j = 0, Ṗ jk = S jk , Ṗ jkl = S jkl + S jlk . (6.7)