On Mach's principle: Inertia as gravitation

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 ${\mathbf{g}}^{\ast }$. 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 ${\mathbf{g}}^{\ast }$ 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 $R_U=c{t}_0$. 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 $\dot{\mathbf{v}}$ in the LHS of Newton's second law can be interpreted as a reactive acceleration, in other words, as minus the acceleration of gravity ${\mathbf{g}}^{\ast }$ in the rest frame of the accelerated particle ({\it i. e.} to know whether or not ${\mathbf{g}}^{\ast }=-\dot{\mathbf{v}}$). The results strongly support Mach's idea since the reactive acceleration for $\hat{I}{\text{c◯}}_{\hat{I}}›=0.7$ turns out to be about ${\mathbf{g}}^{\ast }=-1.1\dot{\mathbf{v}}$, in the first approach, and about ${\mathbf{g}}^{\ast }=-0.7\dot{\mathbf{v}}$, in the second. These results depend little on $\hat{I}{\text{c◯}}_{\hat{I}}›$ if $\hat{I}{\text{c◯}}_{\hat{I}}›<0.9$. Even considering the approximations and idealizations made during the calculations, we deem these results as interesting and encouraging.

17 pages, 2 figures