  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.
