Assume a spatially compact source of size $d\ll r=|\mathbf x|$, internal speeds much smaller than one, and wavelength much larger than $d$. In the <radiation zone>,
$$
|\mathbf x-\mathbf x'|=r+O(d),
\qquad
t-|\mathbf x-\mathbf x'|=t-r+O(d),
$$
so
$$
\bar h_{ij}(t,\mathbf x)\simeq\frac4r\int d^3x'\,T_{ij}(t-r,\mathbf x').
$$
Define the mass quadrupole moment
$$
I_{ij}(t)=\int d^3x\,T_{00}(t,\mathbf x)x_ix_j.
$$
Twice using $\partial_\mu T^{\mu\nu}=0$, discarding boundary terms, gives
$$
\ddot I_{ij}=2\int d^3x\,T_{ij}.
$$
Hence
$$
\boxed{\bar h_{ij}(t,\mathbf x)\simeq\frac2r\ddot I_{ij}(t-r)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 309 / 3 / a / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-309/3/a/i/solution