Mach's Principle and Spatial Scale-Invariance of Gravity

TL;DR

This paper explores how Mach's principle implies that gravity is inherently scale-invariant and describes the most general spacetime compatible with this, allowing arbitrary matter properties and emphasizing the relational nature of inertia.

Contribution

It introduces a class of spacetimes with spatial homothety that embody Mach's principle, showing they accommodate arbitrary matter and are the most general form consistent with scale-invariant gravity.

Findings

Spacetime admits three independent spatial homothetic Killing vectors.

Matter properties in such spacetimes are arbitrary and can follow any equation of state.

The spacetime is globally degenerate in anti-Machian scenarios like vacuum.

Abstract

Gravity does not provide any scale for matter properties. We argue that this is also the implication of Mach's hypothesis of the relativity of inertia. The most general spacetime compatible with this property of gravity is that admitting three, independent spatial homothetic Killing vectors generating an arbitrary function of each one of the three spatial coordinates. The matter properties for such a spacetime are (spatially) arbitrary and the matter generating the spacetime admits {\it any} equation of state. This is also the most general spacetime containing the weak gravity physics in its entirety. This spacetime is machian in that it is {\em globally} degenerate for anti-machian situations such as vacuum, a single matter particle etc. and, hence, has no meaning in the absence of matter.