ABSTRACT
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 αβ . As in chapter 2, the Einstein equation is(
− ∂ 2
∂ t2 +∇2
) hαβ = −16πT αβ (6.1)
(hαβ = hαβ− 12ηαβh and h αβ
,β = 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 − R:
hαβ(xi , t) = 4 ∫ 1
R T αβ(t − R, yi ) d3y, (6.2)
where we define R2 = (xi − yi )(xi − 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 −1. Therefore, they contribute terms to the field that fall off faster than r −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.
