Eight Local Couplings of Gravitational Waves from Unified Field Equations

A gravitational-wave (GW) detector records a linear mixture of local couplings under the assumption that extra polarizations enter geodesic deviation. Vacuum general relativity (GR) admits two transverse-traceless (TT) amplitudes. It remains to determine the largest set of couplings that can sit in that mixture, and whether a tensor-only arm-length result selects GR uniquely. The little group $E(2)$ of a null four-momentum classifies the strain amplitudes $\mathbf{p}=(p_{+},p_{\times },p_x,p_y,p_b,p_{\mathrm{\ell }})$, commonly written $h_P$, which determine the electric tidal tensor along a ray. Geodesic deviation, recorded as differential arm length, therefore contains only those $p_P$ that enter that tensor. Lorentz mixing at helicity $\pm 1$ supplies a gravito-magnetic (GEM) field that does not enter the tidal tensor. if GEM field is static, that does not propagate as a wave. A time-varying helicity-$\pm 1$ current sources a GEM wave that enters the mixture as a coupling, to be isolated by its measured quantity. For a radiation-zone wave that depends only on retarded time, ${\text{betag}}_{\perp }=\widehat{\mathbf{k}}\times {\mathrm{\partial }}_t(p_x,p_y)$, and the eight couplings are $\text{Pvec}=(\mathbf{p},{\text{betag}}_{\perp })$. Here we adopt unified field equations on $\mathbf{p}$ to clarify the origin of each component of $\text{Pvec}$; the measured quantity of each coupling then isolates the polarizations in that mixture, which favors identification of distinct polarizations and model tests.