Geodesic deviation and strain response

Retain the previous signature, x0=ct and Riemann sign convention. In the TT representation of a weak plane wave, the relevant tidal curvature is minus one half the second time derivative of the spatial strain tensor divided by c squared. Geodesic deviation then gives a relative acceleration equal to positive one half the strain's second derivative acting on the unperturbed separation. The sign of acceleration at a positive strain maximum is consequently restoring for a sinusoid.

This is an expansion for neighboring freely falling bodies and weak perturbations. A local detector-frame description neglects curvature variation across the apparatus when its size is much smaller than a wavelength. Integrating the acceleration gives a leading displacement equal to one half the strain tensor acting on the initial separation, after choosing the initial conditions that remove unrelated free drift. An arbitrary integration constant is not a new wave polarization.

For an arm along a unit vector u, the fractional length perturbation in the long-wavelength limit is one half of u contracted twice with the TT tensor. Orthogonal arms along x and y respond oppositely to a plus wave arriving perpendicular to their plane. Their differential length change is therefore twice the single-arm change, while division by the common arm length gives the usual optimal differential strain. Missing this distinction creates a common factor-of-two error.

The actual optical instrument compares light phases accumulated along round trips and through its optical configuration. Treating that output as a calibrated strain invokes a response model. Suspensions, control loops and finite light travel time matter outside the ideal free-mass long-wavelength regime. The tensor contraction provides the geometric leading response, while a frequency-dependent transfer function handles the full instrument rather than redefining the incoming wave amplitude.

In TT coordinates initially resting freely falling masses remain at fixed spatial coordinates, while the spatial metric changes their proper separation. In a local inertial description the leading tidal displacement appears as motion. Neither picture should be used halfway through the calculation with the other's light-propagation rules. The final observable is a comparison performed by a physical clock and light path, which is why changing gauge cannot remove a genuine measured tidal effect.

A laboratory-sized detector can often use a locally planar incoming wave even when the source is cosmological. Propagation then determines the observer-frame frequency and amplitude before the local response is applied. If the arm length is no longer small compared with wavelength, the wave phase varies along the light path and the simple instantaneous length formula fails. This limitation is especially relevant for long space-based baselines; a correct response includes time delays rather than assuming a uniform strain across every arm.