Past exam of the relativity course of the University of Cambridge / 2021 / Section B / 5 / c

For the supplied plus polarization, the perturbation is traceless, so $h_{\mu \nu }={\varphi }_{\mu \nu }$. At a fixed longitudinal coordinate $z_0$, put $H(t)=h_{\oplus }\cos [\omega (t-z_0)]$. With the $+---$ convention chosen above, the transverse spatial metric measuring positive squared lengths isThe initial coordinate separation is $(\Delta \cos \theta ,\Delta \sin \theta )$ and remains constant in transverse-traceless gauge: $h_{00}=h_{0i}=0$ implies ${\Gamma }^i{}_{00}=0$, so initially stationary particles stay at fixed spatial coordinates. Their measured proper length is insteadThe first expression is understood within the linearized metric; only its first-order expansion is controlled. Separation along the two transverse axes varies oppositely; at $45$ degrees there is no first-order length modulation. This is the angular response of a gravitational-wave separation, despite constant coordinate separation.

If the background is instead chosen as $-+++$ while keeping the same numerical spatial amplitude, the displayed length modulation has the opposite sign. The question does not state a signature here; the convention above agrees with its earlier explicitly specified Minkowski metric. The physical alternating stretch-compress pattern is unchanged by consistently redefining the signed amplitude.