A Machian Approach to Relativistic Cosmology
Enrico Massa, Davide Astesiano
TL;DR
The paper embeds Mach's principle in a relativistic framework by tying spacetime geometry to the global mass–energy content, yielding Einstein‑like field equations with a Universe dependent coupling $κ$ and a cosmological rule that fixes the background structure. In cosmology, this leads to a unique FRW solution with negative spatial curvature ($k=-1$) and a scale factor $R(t)=pT\sinh(t/T)$, with a positive cosmological constant $Λ$ compatible with observations and a decomposition of matter into inhomogeneities driving geometry. Dynamically, $κ$ evolves with cosmic time, making inertial mass scale as $m ∝ κ^{-1}$ and forcing related constants to adjust (while preserving atomic structure), yet reducing locally to General Relativity. The framework offers a coherent Machian reinterpretation of Einstein’s equations in cosmology, connects global matter distribution to inertial structure, and hints at novel insights for dark matter and cosmological evolution without abandoning Solar System tests.
Abstract
We present a consistent relativistic formulation of Mach principle within a geometric theory of gravitation. In this approach, neither inertia nor free fall is assumed a priori. Instead, the motion of any local system arises from its dynamical interdependence, direct or indirect, with the rest of the Universe. This viewpoint provides a sharp criterion for what qualifies as a genuinely Machian gravitational theory. A theory is Machian only if the global mass energy distribution underwrites the very existence of the pseudo Riemannian structure of spacetime, rather than merely determining local features such as curvature on a pre existing background. We analyze the resulting cosmological model and discuss its phenomenological implications. In the appropriate local limit, the theory reduces to General Relativity, thereby preserving agreement with all Solar System tests.