The Scalar Mach-Sciama Theory of Gravitation

TL;DR

The paper presents a covariant scalar–tensor realization of Mach's principle within the Bergmann–Wagoner framework by enforcing a universal conformal coupling to matter and reframing the dynamics in frame-invariant quantities $\mathcal{I}_1,\mathcal{I}_2,\mathcal{I}_3$ and $\hat{g}_{\mu\nu}$. Machian causality is realized through a retarded selection rule on the invariant scalar equation, so the physically relevant scalar configuration is entirely generated by past matter, with no free homogeneous part. In a spatially flat FLRW background and the light, slowly varying regime, this yields a simple temporal kernel $K(t,t')$ that links the background scalar evolution to matter inside the Hubble region, thereby fixing the inertial calibration via $m_{IA}=m_{0}\sqrt{\mathcal{I}_1}$ and relating it to $M_H/R_H$. The leading-order weak equivalence principle holds for structureless test bodies, with potential departures only from Nordtvedt-type self-gravity, encoded by body-specific sensitivities $s_A$. Overall, the work provides a covariant, causal mechanism by which cosmic matter history determines local inertial properties, compatible with standard weak-field tests and offering a platform for further cosmological and strong-field explorations.

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 $\{{\cal I}_1,{\cal I}_2,{\cal I}_3,\hat g_{μν}\}$, 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 Eötvs 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.