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

Abstract

A frequency-dependent Machian effect previously put forward by Woodward (that for a body undergoing mass–energy fluctuations, the second time derivative of the mass–energy density is a source of a gravitational field) is discussed within Einstein’s theory and justified using Hoyle–Narlikar’s conformal gravitational theory. It is shown that Einstein’s theory has a similar term that is 3rd order post-Newtonian, but besides the issue of coordinate-dependence, the Machian significance of any field term in Einstein’s equation depends on the (universe’s) cosmological solution to the field equations. Therefore, Woodward’s theory is examined within Hoyle–Narlikar’s scalar–tensor theory of gravitation (a theory that was expressly developed with the intent to incorporate Mach’s principle) for a universe undergoing accelerating expansion (hereby accounted for by a positive cosmological constant). It is shown under gauge invariant expressions that the conformal, scalar–tensor gravitational theory of Hoyle and Narlikar has a similar term of first order when the mass–energy fluctuation is due to distant objects but that it effectively becomes a higher order effect when the mass–energy fluctuations arise from fluctuation of the (local) mass–energy (as is necessarily the case in Woodward’s experimental results, since the only mass that can be purposively fluctuated in energy, monochromatically, is the local mass, instead of the distant masses responsible for most of the inertia according to Mach’s principle). Therefore this effect appears too small for practical space travel application (unless the spaceship is near a black hole or a neutron star). Present cosmological measurements of the possible time variation of G are shown to occur at much lower frequencies and therefore cannot be used to rule out Woodward’s effect if G exhibits significant time-dependence at higher frequencies than observed in these cosmological measurements.