The Cosmological Origin of Inertia: Mach's Principle
(September 2004)Abstract
The axes of gyroscopes experimentally define local non-rotating frames, i.e.
the time-evolution of axes of inertial frames. But what physical cause governs
the time-evolution of gyroscope axes? Starting from an unperturbed FRW
cosmology with k=0 we consider linear cosmological vorticity perturbations and
ask: Will cosmological vorticity perturbations exactly drag the axes of
gyroscopes relative to the directions of geodesics to galaxies in the
asymptotic FRW space? Using Cartan's formalism with local orthonormal bases we
cast the laws of gravitomagnetism into a form showing the close correspondence
with the laws of ordinary magnetism. Our results, valid for any equation of
state, are: 1) The dragging of a gyroscope axis by rotational perturbations
beyond the H-dot radius (H=Hubble constant) is exponentially suppressed. 2) If
the perturbation is a homogeneous rotation inside a perturbation radius, then
exact dragging of the gyroscope axis by the rotational perturbation is reached
exponentially fast, as the perturbation radius gets larger than the H-dot
radius. 3) The time-evolution of a gyroscope axis exactly follows a specific
average of the matter inside the H-dot radius. In this sense Mach's Principle
is a consequence of cosmology with Einstein gravity.
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