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

TL;DR

This paper demonstrates that in linear perturbations of FRW universes, the local inertial frame dragging is exactly caused by cosmological energy currents, confirming Mach's principle within a relativistic cosmological context.

Contribution

It provides an exact, linear perturbation-based proof that Mach's principle holds in all FRW universes with different curvatures, linking inertial frame dragging to cosmological energy currents.

Findings

Inertial frame dragging is caused by energy currents in the universe.

The gravitomagnetic field obeys an Ampere-like law with a Yukawa potential in open universes.

Mach's principle is validated for linear perturbations in all FRW cosmologies.

Abstract

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.