A Machian wave effect in conformal, scalar-tensor gravitational theory

Abstract

Woodward proposed that driven mass-energy fluctuations could yield a frequency-dependent "Machian" gravitational response $\propto {\partial }_t^2{M}_{\mathrm{loc}}(t)$, amplified by a Sciama-scale cosmic potential $\hat{I}¦/c^2\sim -1$. We test this claim covariantly in (i) Einstein gravity and (ii) Hoyle-Narlikar (HN) conformal scalar-tensor gravity. In GR, the Landau-Lifshitz relaxed equations in harmonic gauge contain nonlinear terms of the form $H^{\hat{I}\pm {\hat{I}}^2}{\partial }_{\hat{I}}\pm {\partial }_{\hat{I}}^2{H}^{\hat{I}¼\hat{I}½}$, including a near-zone piece $H^{00}{\partial }_t^2{H}^{\hat{I}¼\hat{I}½}$. These terms are not independent matter sources; they arise from expanding the curved wave operator about a flat background. Moving them back to the left-hand side restores the quasilinear principal part, and for laboratory devices their size is suppressed by $\sim (U_N/c^2)(\ddot{I}‰L/c{)}^2\ll 1$, with no enhancement by any cosmological potential. In HN theory, the conformal scalar satisfies ${\nabla }_a{\nabla }^{a}m+(R/6)m=\hat{I}»N$. For a localized device of size $L$ driven at angular frequency $\ddot{I}‰$, $|(c^{-2}{\partial }_t^2{m}_s)|/|{\nabla }^2{m}_s|\sim (\ddot{I}‰L/c{)}^2\ll 1$, so the response is effectively instantaneous (Poisson-like), not wave-amplified. Baryon-number conservation fixes the scalar charge, so the rest-mass monopole cannot oscillate; any radiating monopole requires nonconservative internal-energy variations, further suppressed by $E_{\mathrm{int}}/(M{c}^2)$ and by $M_{\mathrm{dev}}/M_H$. Thus any Mach-effect thrust is far too small for practical propulsion.

26 pages, 5 figures