Gravitational waves and detector measurements

Write $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with small perturbations and keep only first-order terms. Coordinate freedom changes parts of h, so a wiggling component by itself is not an observable wave. In a source-free region, an appropriate gauge isolates two independent radiative tensor polarizations. Their measurable content appears through tidal effects, such as relative motion of nearby free test masses, and through optical phase along a detector’s light paths.

Define the transverse spatial strain matrix H operationally by $d\ell^2=(\delta_{ij}+H_{ij})dx^i dx^j$ in a local transverse-traceless wave description. With our mostly-minus spacetime signature, this H is the negative of the corresponding covariant spatial spacetime perturbation h. For propagation along z, plus polarization stretches x while squeezing y; cross polarization has the same pattern rotated by forty-five degrees. The explicit definition avoids a silent sign change between conventions.

For an optimally oriented plus wave and a detector small compared with its wavelength, the leading fractional responses are $\delta L_x/L=h_+/2$ and $\delta L_y/L=-h_+/2$. For equal nominal arms, the differential displacement is $\delta L_x-\delta L_y=Lh_+$. Stating whether a number refers to one arm or the differential response matters: the two differ by a factor of two under this convention.

A real interferometer measures a light phase response with finite light travel time, optical cavities, control systems, and a directional antenna pattern. The simple arm-displacement estimate is a useful long-wavelength interpretation, not a complete calibration pipeline. Freely suspended mirrors can have nearly fixed coordinates in one useful wave gauge while their measured proper separation changes. There is no contradiction because coordinate separation and the metric together determine physical distances.

An isolated system cannot radiate a changing total mass monopole while conserving mass-energy, and its conserved center-of-mass motion rules out leading gravitational dipole radiation. The leading slow-motion radiation is quadrupolar. A compact binary’s changing quadrupole creates an evolving waveform; as it loses orbital energy, its orbital frequency increases and its signal chirps. The final strongly relativistic merger requires methods beyond a lowest-order quadrupole estimate.

Detection is an inference problem. Calibration connects instrument output to strain, data-quality checks identify disturbances, and waveform comparisons use a noise-weighted statistic. A compatible arrival-time delay and morphology across separated instruments strengthen the interpretation, while background estimation quantifies how noise can imitate it. The original binary-black-hole discovery paper provides a concrete example of this chain. A visually attractive sinusoid alone supplies none of these statistical or instrumental guarantees.