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

TL;DR

The paper tests Woodward's proposed Machian gravitational thrust in covariant general relativity and in Hoyle–Narlikar conformal scalar–tensor gravity. In GR, a second-time-derivative structure appears in the relaxed, harmonic-form equations but is a high-order, locally suppressed quasilinear effect with coefficient set by the small local potential $U/c^2$, not a cosmological $\

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 $Φ/c^2\sim -1$. 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 $H^{αβ}\,\partial_α\partial_βH^{μν}$, including a near-zone piece $H^{00}\,\partial_t^2 H^{μν}$. 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 $\sim (U_N/c^2)(ωL/c)^2\ll 1$, with no enhancement by any cosmological potential. In HN theory, the conformal scalar satisfies $\nabla_a\nabla^a m + (R/6)m=λN$. For a localized device of size $L$ driven at angular frequency $ω$, $|(c^{-2}\partial_t^2 m_s)|/|\nabla^2 m_s|\sim (ωL/c)^2\ll 1$, 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.

Figures (5)

• Figure 1: Schematic Machian hierarchy: a cosmic-scale background potential versus a local, weak-field device-driven perturbation. Potential perturbation delta phi G delta M/r. Any effective Mddot coupling in Einstein's general relativity is suppressed by the local potential U/c^2 << 1 instead of the cosmic potential -Phi/c^2 1.
• Figure 2: Visualizing the interface condition error. (A) The incorrect ansatz assumes the field fluctuation $\delta m$ tracks the density fluctuation $\delta \rho$ at the boundary, implying equal logarithmic derivatives. (B) In a physical piezoelectric device, density changes are driven by particle number kinematics ($\delta N$), while the scalar field $\delta m$ (sourced by the tiny mass ratio $M_{\text{dev}}/M_H$) remains effectively flat. The gradient of the field is negligible compared to the gradient of the material density.
• Figure 3: Two logically distinct moves. A: the inconsistent "hybrid" step criticized in Sect. 3.4, where $m$ is treated as a field in derivative terms while $6/m^2$ is simultaneously replaced by a constant. B: the controlled alternative: decompose $m=m_0+m_s$, expand all occurrences consistently, and remember that the inverse wave operator is nonlocal (Green-operator form), so pointwise identifications such as $m\propto N$ are not available.
• Figure 4: Weak-field reduction logic used in Sect. \ref{['sec:HN-weakfield-waves']}. The key steps are: (i) linearize about Minkowski on the scales of interest; (ii) use a harmonic-type gauge to remove the double-divergence term in $R$; (iii) obtain a driven Minkowski wave equation for the trace scalar $h$; and (iv) note that converting this trace-sector result into a Newtonian-potential equation requires the full linearized field equations (e.g. the $00$ component) and is not pursued here.
• Figure 5: Logical steps and notation needed to compare the weak-field wave equation derived here with the Woodward/Sciama-motivated form. The key technical inputs are (i) the approximation applied specifically to the $m_s$ term and (ii) distinguishing the universe’s potential $\Phi$ from the local potential $\phi$.