Eight Local Couplings of Gravitational Waves from Unified Field Equations

Abstract

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 \(\mathbf{p}=(p_{+},p_{\times},p_{x},p_{y},p_{b},p_{\ell})\), 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 \(\pm1\) 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-\(\pm1\) 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, \(\betag_{\perp}=\hat{\mathbf{k}}\times\partial_{t}(p_{x},p_{y})\), and the eight couplings are \(\Pvec=(\mathbf{p},\betag_{\perp})\). Here we adopt unified field equations on \(\mathbf{p}\) to clarify the origin of each component of \(\Pvec\); the measured quantity of each coupling then isolates the polarizations in that mixture, which favors identification of distinct polarizations and model tests.