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 $\propto \partial_t^2 M_{\rm loc}(t)$, amplified by a Sciama-scale cosmic potential $\Phi/c^2\sim -1$. 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 $H^{\alpha\beta}\,\partial_\alpha\partial_\beta H^{\mu\nu}$, including a near-zone piece $H^{00}\,\partial_t^2 H^{\mu\nu}$. 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 $\sim (U_N/c^2)(\omega L/c)^2\ll 1$, with no enhancement by any cosmological potential. In HN theory, the conformal scalar satisfies $\nabla_a\nabla^a m + (R/6)m=\lambda N$. For a localized device of size $L$ driven at angular frequency $\omega$, $|(c^{-2}\partial_t^2 m_s)|/|\nabla^2 m_s|\sim (\omega L/c)^2\ll 1$, 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 $E_{\rm int}/(M c^2)$ and by $M_{\rm dev}/M_H$. Thus any Mach-effect thrust is far too small for practical propulsion.