Machian Gravity: A mathematical formulation for Mach's Principle

Abstract: The general theory of relativity (GR) was proposed with an aim of incorporating Mach's principle mathematically. Despite early hopes, it became evident that GR did not follow Mach's principle. Over time, multiple researchers attempted to develop gravity theories aligned with Machian idea. Although these theories successfully explained various aspects of Mach's principle, each of these theories possessed its own strengths and weaknesses. In this paper, we discuss some of these theories and then try to combine these theories into a single framework that can fully embrace Mach's principle. This new theory, termed Machian Gravity (MG) is a metric-based theory, and can be derived from the action principle, ensuring compliance with all conservation laws. The theory converges to GR at solar system scales, but at larger scales, it diverges from GR and aligns with various modified gravity models proposed to explain dark sectors of the universe. We have tested our theory against multiple observational data. It explains the galactic rotation curve without requiring additional dark matter (DM). The theory also resolves the discrepancy between dynamic mass and photometric mass in galaxy clusters without resorting to DM, but it introduces two additional parameters. It can also explain the expansion history of the universe without requiring dark components.