Gravitational Lagrangians, Mach's principle, and the equivalence principle in an expanding universe

TL;DR

This paper examines a gravitational Lagrangian derived from special relativity to test Mach's principle in an expanding universe, finding consistency at critical density with mass renormalization, and compares it to general relativity corrections.

Contribution

It applies Kennedy's Lagrangian to a cosmological model, demonstrating its consistency with Mach's principle and deriving higher order corrections related to the equivalence principle.

Findings

Lagrangian consistent with Mach's principle at critical density

Higher order corrections align gravitational and inertial masses

Comparison with Einstein-Infeld-Hoffmann corrections shows similarities and differences

Abstract

The gravitational Lagrangian based on special relativity and the assumption of a fourth rank tensor interaction, derived by Kennedy (1972), is used to check Mach's principle in a homogeneous isotropic expanding universe. The Lagrangian is found to be consistent with Mach's principle when the density is the critical density and inertial mass is suitably renormalized. The Kennedy approach only gives the Lagrangian to first order in the gravitational coupling constant. By invoking the equivalence principle higher order corrections are found which renormalize the gravitational masses to the same values as the inertial masses. It is not the same as the correction derived from general relativity by Einstein-Infeld-Hoffmann, but otherwise the Lagrangians agree.