  We show that the dragging of the axis directions of local inertial frames by
a weighted average of the energy currents in the universe is exact for all
linear perturbations of any Friedmann-Robertson-Walker (FRW) universe with K =
(+1, -1, 0) and of Einstein's static closed universe. This includes FRW
universes which are arbitrarily close to the Milne Universe, which is empty,
and to the de Sitter universe. Hence the postulate formulated by E. Mach about
the physical cause for the time-evolution of the axis directions of inertial
frames is shown to hold in cosmological General Relativity for linear
perturbations. The time-evolution of axis directions of local inertial frames
(relative to given local fiducial axes) is given experimentally by the
precession angular velocity of gyroscopes, which in turn is given by the
operational definition of the gravitomagnetic field. The gravitomagnetic field
is caused by cosmological energy currents via the momentum constraint. This
equation for cosmological gravitomagnetism is analogous to Ampere's law, but it
holds also for time-dependent situtations. In the solution for an open universe
the 1/r^2-force of Ampere is replaced by a Yukawa force which is of identical
form for FRW backgrounds with $K = (-1, 0).$ The scale of the exponential
cutoff is the H-dot radius, where H is the Hubble rate, and dot is the
derivative with respect to cosmic time. Analogous results hold for energy
currents in a closed FRW universe, K = +1, and in Einstein's closed static
universe.
