Mach's Principle and Invariance under Transformation of Units

TLDR

A recently published gravitational theory compatible with Mach's principle, the Brans–Dicke theory, features a Jordan‑type gravitational field comprising a tensor and scalar component, and its invariance under coordinate‑dependent unit transformations is discussed. The study demonstrates that a coordinate‑dependent unit transformation can recast the Brans–Dicke theory into a conventional metric form satisfying the Einstein field equations. By applying a coordinate‑dependent transformation of measurement units, the authors reformulate the theory so that the gravitational field appears as a standard metric tensor. In the transformed formulation, the scalar field behaves as a matter field.

Abstract

A gravitational theory compatible with Mach's principle was published recently by Brans and Dicke. It is characterized by a gravitational field of the Jordan type, tensor plus scalar field. It is shown here that a coordinate-dependent transformation of the units of measure can be used to throw the theory into a form for which the gravitational field appears in the conventional form, as a metric tensor, such that the Einstein field equation is satisfied. The scalar field appears then as a "matter field" in the theory. The invariance of physical laws under coordinate-dependent transformations of units is discussed.