A Topological Interpretation of Mach's Principle in General Relativity

TL;DR

This paper explores a topological perspective on Mach's principle within General Relativity, linking the curvature action to topological properties of space-time and proposing alternative formulations inspired by elliptic complexes.

Contribution

It introduces a topological interpretation of the curvature action in General Relativity based on the Lovelock and Gibbons-Hawking-York terms, connecting it to Mach's principle and suggesting new formulations.

Findings

Curvature action relates to topological properties of space-time.

Mach's principle guides alternative formulations of gravity.

Deviations from standard curvature action are discussed in modified gravity theories.

Abstract

Starting from the Lovelock action and its supplementation by the relevant Gibbons-Hawking-York boundary term, the curvature action corresponding to second-order General Relativity is stated in accordance to the topological properties of the space-time manifold $\mathcal{M}$ with metric solutions being interpreted as topological solitons. Furthermore, this is shown to arise naturally from a topological interpretation of Mach's principle, with the appropriate manifestation of general covariance. Mach's principle is again invoked to suggest formulations of the curvature action in alternative elliptic complexes. The extent of these deviations from the curvature action as constructed in the first part of this paper are remarked upon in the context of contemporary modified theories of gravity.