Cosmological gravitomagnetism and Mach's principle

TL;DR

This paper investigates how cosmological vorticity perturbations influence gyroscope axes in FRW universes, demonstrating that local inertial frames are precisely determined by the average motion of cosmic matter, supporting Mach's principle.

Contribution

It provides a detailed analysis of cosmological gravitomagnetism showing that local inertial frames are governed by the average cosmic matter motion, extending Mach's principle within linear perturbation theory.

Findings

Gyroscope axis dragging by distant rotational perturbations is exponentially suppressed beyond the H-dot radius.

Homogeneous rotation inside a radius causes rapid exact dragging as the radius exceeds the H-dot radius.

Gyroscope axes follow a weighted average of cosmic matter energy currents, analogous to Ampere's law with a Yukawa potential.

Abstract

The spin axes of gyroscopes experimentally define local non-rotating frames. But what physical cause governs the time-evolution of gyroscope axes? We consider linear perturbations of Friedmann-Robertson-Walker cosmologies with k=0. We ask: Will cosmological vorticity perturbations exactly drag the spin axes of gyroscopes relative to the directions of geodesics to quasars in the asymptotic unperturbed FRW space? Using Cartan's formalism with local orthonormal bases we cast the laws of linear cosmological gravitomagnetism into a form showing the close correspondence with the laws of ordinary magnetism. Our results, valid for any equation of state for cosmological matter, are: 1) The dragging of a gyroscope axis by rotational perturbations of matter beyond the Hubble-dot radius from the gyroscope is exponentially suppressed, where dot is the derivative with respect to cosmic time. 2) If the perturbation of matter is a homogeneous rotation inside some radius around a gyroscope, then exact dragging of the gyroscope axis by the rotational perturbation is reached exponentially fast as the rotation radius grows beyond the H-dot radius. 3) For the most general linear cosmological perturbations the time-evolution of all gyroscope spin axes exactly follow a weighted average of the energy currents of cosmological matter. The weight function is the same as in Ampere's law except that the inverse square law is replaced by the Yukawa force with the Hubble-dot cutoff. Our results demonstrate (in first order perturbation theory for FRW cosmologies with k = 0) the validity of Mach's hypothesis that axes of local non-rotating frames precisely follow an average of the motion of cosmic matter.