Paper 7: Vector-Mode Gravitational Wave Amplitude from a Rotating Binary in the Four-Dimensional Scale Space Framework

The four-dimensional scale space framework predicts five gravitational wave polarisation classes, including two vector modes hxs, hys coupling spatial and scale degrees of freedom. Paper 2 established the existence of these modes but could not evaluate their amplitude for lack of the rotating metric; Paper 6 derived the linearised scale-Kerr exterior rotating perturbation; Paper 8 performed the interior matching, fixing the mode spectrum and correcting the amplitude normalisation by the factor 5/7 (first-principles Hartle–Thorne matching). We now combine these results to compute the vector-mode amplitude and LIGO detector response for a binary pulsar system. We derive the vector-mode polarisation tensors e^Vx ij= ˆpi ˆnj + ˆni ˆpj and compute the antenna pattern functions FVx, FVy for an L-shaped interferometer. The sky-averaged RMS antenna pattern is ⟨F^2 Vx⟩^(1/2) ≈0.44, comparable to the tensor-mode value ⟨F^2 +⟩^(1/2) ≈0.45 — the vector modes are geometrically well-coupled to L-shaped detectors. However, the scale-space vector mode hxs couples spatial displacement dx to the scale coordinate s; a freely falling scale-stationary mirror produces no spatial strain at leading order. The effective coupling requires a mirror with nonzero scale velocity ˙s mirror, introducing a suppression factor ˙s/c∼1.4 ×10^−9 for Earth-surface mirrors (derived from the linearised geodesic equation in Appendix D, using t as the external time parameter). For PSR J0737−3039 (PA = 22.7 ms, d = 1.15 kpc), the Paper 8-corrected radiated sideband amplitude is A^sb xs ≈3.2 ×10^−26 (Paper 8: ×5/7) at f ≈44.05 Hz. Combined with the scale-coupling suppression, the effective strain at a LIGO mirror is ≈ 2.2 ×10^−35, giving SNR ≲ 10^−8 — far below detection by any planned strain detector. For a hypothetical scale-velocity-sensitive detector with LIGO-comparable noise, the unsuppressed SNR for J0737 would be ≈2 after one year; the nearby millisecond pulsar PSR J0437−4715 (d = 157 pc, P = 5.76 ms) gives unsuppressed SNR∼6×10^3 and is the most promising target for future scale-sensitive instrumentation.