Mach’s principle: Exact frame-dragging via gravitomagnetism in perturbed Friedmann-Robertson-Walker universes with $K=(\pm 1,0)$

We show that there is exact dragging of the axis directions of local inertial frames by a weighted average of the cosmological energy currents via gravitomagnetism for all linear perturbations of all Friedmann-Robertson-Walker (FRW) universes and of Einstein’s static closed universe, and for all energy-momentum-stress tensors and in the presence of a cosmological constant. This includes FRW universes arbitrarily close to the Milne Universe and the de Sitter universe. Hence the postulate formulated by Ernst Mach about the physical cause for the time-evolution of inertial axes is shown to hold in general relativity for linear perturbations of FRW universes.—The time-evolution of local inertial axes (relative to given local fiducial axes) is given experimentally by the precession angular velocity ${\overrightarrow{\Omega }}_{\mathrm{gyro}}$ of local gyroscopes, which in turn gives the operational definition of the gravitomagnetic field: ${\overrightarrow{B}}_{\mathrm{g}}\equiv -2{\overrightarrow{\Omega }}_{\mathrm{gyro}}$. The gravitomagnetic field is caused by energy currents ${\overrightarrow{J}}_{\epsilon }$ via the momentum constraint, Einstein’s $G^{\widehat{0}}{}_{\widehat{i}}$ equation, $(-\Delta +{\mu }^2){\overrightarrow{A}}_{\mathrm{g}}=-16\pi G_{\mathrm{N}}{\overrightarrow{J}}_{\epsilon }$ with ${\overrightarrow{B}}_{\mathrm{g}}=\mathrm{curl}{\overrightarrow{A}}_{\mathrm{g}}$. This equation is analogous to Ampère’s law, but it holds for all time-dependent situations. $\Delta$ is the de Rham-Hodge Laplacian, and $\Delta =-\mathrm{curl}\mathrm{curl}$ for the vorticity sector in Riemannian 3-space.—In the solution for an open universe the $1/r^2$-force of Ampère is replaced by a Yukawa force $Y_{\mu }(r)=(-d/dr)[(1/R)\exp (-\mu r)]$, form-identical for FRW backgrounds with $K=(-1,0)$. Here $r$ is the measured geodesic distance from the gyroscope to the cosmological source, and $2\pi R$ is the measured circumference of the sphere centered at the gyroscope and going through the source point. The scale of the exponential cutoff is the $H$-dot radius, where $H$ is the Hubble rate, dot is the derivative with respect to cosmic time, and ${\mu }^2=-4(dH/dt)$. Analogous results hold in closed FRW universes and in Einstein’s closed static universe.—We list six fundamental tests for the principle formulated by Mach: all of them are explicitly fulfilled by our solutions.—We show that only energy currents in the toroidal vorticity sector with $\ell =1$ can affect the precession of gyroscopes. We show that the harmonic decomposition of toroidal vorticity fields in terms of vector spherical harmonics ${\overrightarrow{X}}_{\ell m}^{-}$ has radial functions which are form-identical for the 3-sphere, the hyperbolic 3-space, and Euclidean 3-space, and are form-identical with the spherical Bessel-, Neumann-, and Hankel functions.—The Appendix gives the de Rham-Hodge Laplacian on vorticity fields in Riemannian 3-spaces by equations connecting the calculus of differential forms with the curl notation. We also give the derivation the Weitzenböck formula for the difference between the de Rham-Hodge Laplacian $\Delta$ and the “rough” Laplacian ${\nabla }^2$ on vector fields.