The Newtonian limit of orthonormal frames in metric theories of gravity

TL;DR

This paper extends the Newtonian limit analysis from Lorentzian metrics to orthonormal frames, showing convergence to Galilei structures under specific conditions, thus broadening the understanding of the Newtonian limit in metric theories of gravity.

Contribution

It proves that orthonormal frames in Lorentzian metrics converge to Galilei frames in the Newtonian limit, given certain conditions on rotation and boost velocity.

Findings

Orthogonal frames converge pointwise to Galilei frames in the Newtonian limit.

Necessary conditions include bounded rotation and converging boost velocities.

The results generalize the Newtonian limit to orthonormal frames in metric theories.

Abstract

We extend well-known results on the Newtonian limit of Lorentzian metrics to orthonormal frames. Concretely, we prove that, given a one-parameter family of Lorentzian metrics that in the Newtonian limit converges to a Galilei structure, any family of orthonormal frames for these metrics converges pointwise to a Galilei frame, assuming that the two obvious necessary conditions are satisfied: the spatial frame must not rotate indefinitely as the limit is approached, and the frame's boost velocity with respect to some fixed reference observer needs to converge.