The Scalar Mach-Sciama Theory of Gravitation

TL;DR

This paper develops a scalar-tensor gravity theory inspired by Mach's principle, linking local inertial properties to the matter distribution causally, and demonstrating consistency with cosmological and weak-field observations.

Contribution

It formulates a scalar realization of Mach's principle within scalar-tensor gravity, implementing causal selection rules for solutions and connecting scalar evolution to matter content.

Findings

Reproduces Machian scaling in an expanding universe.

Ensures vanishing Eötvös parameter at leading order.

Remains consistent with weak-field tests.

Abstract

We formulate a scalar realization of Sciama's Machian programme within the general Bergmann-Wagoner class of scalar--tensor gravity. Starting from a universally conformally coupled matter sector, we rewrite the field equations in terms of the invariant set $\{{\cal I}_1,{\cal I}_2,{\cal I}_3,\hat g_{\mu\nu}\}$, so that Machian requirements can be stated independently of conformal frame. Sciama's causal postulate is implemented not by modifying the local dynamics, but as a selection rule on the solution space of the invariant scalar equation: the admissible configuration is the retarded response to the matter distribution in the causal past, with any source-free contribution removed. In a spatially flat FLRW universe, and in the light, slowly varying regime, the prescription reduces to an explicit temporal kernel that links the background scalar evolution to the matter content within the Hubble region, reproducing the expected Machian scaling in an expanding background. Universal coupling to a single physical metric implies that structureless test bodies share the same acceleration in a given external configuration, so that the E\"otvs parameter vanishes at leading order. Nonuniversal effects can arise only when gravitational binding energy contributes appreciably to the total mass, as in strongly self-gravitating objects. Thus, the theory implements a causal Machian determination of the cosmological inertial scale while remaining consistent with standard weak-field tests.