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

Chapter 6Mass- and current-quadrupole radiationIn this lecture we focus on the wave amplitude itself, and how it and thepolarization depend on the motions in the source. Consider an isolated sourcewith a stress-energy tensor T αβ . As in chapter 2, the Einstein equation is( )− ∂2∂t 2 +∇2 h αβ =−16πT αβ (6.1)(h αβ = h αβ − 1 2 ηαβ h and h αβ ,β = 0). Its general solution is the following retardedintegral for the field at a position x i and time t in terms of the source at a positiony i and the retarded time t − R:h αβ (x i , t) = 4∫ 1R T αβ (t − R, y i ) d 3 y, (6.2)where we defineR 2 = (x i − y i )(x i − y i ). (6.3)6.1 Expansion for the far field of a slow-motion sourceLet us suppose that the origin of coordinates is in or near the source, and the fieldpoint x i is far away. Then we define r 2 = x i x i and we have r 2 ≫ y i y i . Wecan, therefore, expand the term R in the dominator in terms of y i . The lowestorder is r, and all higher-order terms are smaller than this by powers of r −1 .Therefore, they contribute terms to the field that fall off faster than r −1 , and theyare negligible in the far zone. Therefore, we can simply replace R by r in thedominator, and take it out of the integral.The R inside the time-argument of the source term is not so simple. If wesuppose that T αβ does not change very fast we can substitute t − R by t − r (theretarded time to the origin of coordinates) and expand( ) 1t − R = t − r + n i y i + O , with n i = x ir r , ni n i = 1. (6.4)58