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

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,soDefine the mass quadrupole momentTwice using ${\partial }_{\mu }{T}^{\mu \nu }=0$, discarding boundary terms, givesHence

In Lorenz gauge, ${\partial }_{\mu }{\bar{h}}^{\mu \nu }=0$. A leading outgoing far-zone field depends on retarded time $u=t-r$, so ${\partial }_j=-{\hat{x}}_j{\partial }_u+O(r^{-1})$. Taking $\nu =i$ and using ${\bar{h}}^{0i}=-{\bar{h}}_{0i}$ givesDropping a nonradiative integration constant and using part i,

With $u=t-r$, direct integration givesTherefore the time-dependent spatial field isThe gravitational-wave frequency is $2\omega$. On the positive $z$-axis this matrix is transverse and traceless. It is a rotating combination of the plus polarization and cross polarization, with amplitude proportional to $1/z$, exactly as for a plane wave propagating in the $z$ direction.

The quadrupole formula in units $G=c=1$ isHere $I_{kk}=m{a}^2/3$ is constant, so its trace subtraction has no third derivative. If $B=4m{a}^2{\omega }^3/3$, thenThus ${\stackrel{...}{I}}_{ij}{\stackrel{...}{I}}_{ij}=2B^2$ andRestoring units multiplies this by $G/c^5$.