Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 36D (General Relativity)

Use $G=c=1$ and the physical real part of the printed plane wave. Its null wavevector obeys $k^{\alpha }{k}_{\alpha }=0$, so each component satisfies $\square h_{\alpha \beta }=0$. Direct differentiation of the defined stress tensor givesThe trace is proportional to $k^{\nu }{k}_{\nu }$ for a single plane wave and hence vanishes. ThereforeThis is local conservation of energy and momentum conservation; spatial integration gives conserved total charges when the boundary flux vanishes. For a finite-amplitude complex phasor, one takes the real field before quadratic products and may time-average: $⟨{\tau }_{\mu \nu }⟩=k_{\mu }{k}_{\nu }{H}^{\alpha \beta }{\overline{H}}_{\alpha \beta }/(64\pi )$ in transverse-traceless gauge. Its energy density is nonnegative and its flux travels in the positive $z$ direction.

Using the stipulated point-mass quadrupole, take $z(t)=L\cos \omega t$ and $Q_{ij}=m(x_i{x}_j-r^2{\delta }_{ij}/3)$. The only nonzero components areSince $z^2=L^2(1+\cos 2\omega t)/2$, its third derivative is $4L^2{\omega }^3\sin 2\omega t$. Thereforeand the total outward radiated power isQuadrupole radiation oscillates at twice the motion frequency. The approximation assumes slow motion $L\omega \ll c$ and a prescribed motion relative to the rest of the source. A lone freely moving point mass cannot execute this oscillation; a physical conserved source includes the accelerating apparatus or other matter, whose quadrupole would also need specification. The displayed value is the point-mass contribution requested by the supplied formula.