chapter  6
13 Pages

In this lecture we focus on the wave amplitude itself, and how it and the polarization depend on the motions in the source. Consider an isolated source with a stress-energy tensor T Î±Î² . As in chapter 2, the Einstein equation is(

âˆ’ âˆ‚ 2

âˆ‚ t2 +âˆ‡2

) hÎ±Î² = âˆ’16Ï€T Î±Î² (6.1)

(hÎ±Î² = hÎ±Î²âˆ’ 12Î·Î±Î²h and h Î±Î²

,Î² = 0). Its general solution is the following retarded integral for the field at a position x i and time t in terms of the source at a position yi and the retarded time t âˆ’ R:

hÎ±Î²(xi , t) = 4 âˆ« 1

R T Î±Î²(t âˆ’ R, yi ) d3y, (6.2)

where we define R2 = (xi âˆ’ yi )(xi âˆ’ yi ). (6.3)

Let us suppose that the origin of coordinates is in or near the source, and the field point x i is far away. Then we define r 2 = xi xi and we have r 2 yi yi . We can, therefore, expand the term R in the dominator in terms of y i . The lowest order is r , and all higher-order terms are smaller than this by powers of r âˆ’1. Therefore, they contribute terms to the field that fall off faster than r âˆ’1, and they are negligible in the far zone. Therefore, we can simply replace R by r in the dominator, and take it out of the integral.