On Mach's principle: Inertia as gravitation

TL;DR

This paper investigates Mach's principle by calculating the universe's influence on a test mass, finding results that support the idea that inertia may be a gravitational effect related to the universe's mass distribution.

Contribution

The study provides a novel calculation of the universe's action on a test mass using linearized gravitation and retarded potentials, supporting Mach's principle.

Findings

Reactive acceleration approximately equals negative of the test mass's acceleration.

Results are consistent across different cosmological parameters.

Supports the idea that inertia arises from gravitational interaction with the universe.

Abstract

In order to test the validity of Mach's principle, we calculate the action of the entire universe on a test mass in its rest frame, which is an acceleration ${\bf g}^*$. We show the dependence of the inertia principle on the lapse and the shift. Using the formalism of linearized gravitation, we obtain the non-relativistic limit of ${\bf g}^*$ in terms of two integrals. We follow then two approaches. In the first one, these integrals are calculated in the actual time section $t=t_0$ up to the distance $R_U=ct_0$. In the more exact and satisfactory second approach, they are calculated over the past light cone using the formalism of the retarded potentials. The aim is to find whether the acceleration $\dot{\bf v}$ in the LHS of Newton's second law can be interpreted as a reactive acceleration, in other words, as minus the acceleration of gravity ${\bf g}^*$ in the rest frame of the accelerated particle ({\it i. e.} to know whether or not ${\bf g}^*=-\dot{\bf v}$). The results strongly support Mach's idea since the reactive acceleration for $\Omega_\Lambda =0.7$ turns out to be about ${\bf g}^*=-1.1 \dot{\bf v}$, in the first approach, and about ${\bf g}^*= -0.7 \dot{\bf v}$, in the second. These results depend little on $\Omega_\Lambda$ if $\Omega_\Lambda<0.9$. Even considering the approximations and idealizations made during the calculations, we deem these results as interesting and encouraging.