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The Newtonian limit of orthonormal frames in metric theories of gravity

Philip K. Schwartz, Arian L. von Blanckenburg

TL;DR

This work extends the Newtonian limit from Lorentzian metrics to orthonormal frames, showing that a one-parameter family of Lorentzian frames converges to a Galilei frame in the Newtonian limit provided the frame’s spatial rotation remains controlled and its boost velocity relative to a fixed observer converges. The authors develop a rigorous, coordinate-free framework for limits of finite-dimensional vector-space data, including order-k convergence and key lemmas on taking square roots of near-identity matrices and tensor products. The main result establishes precise convergence statements for the frame fields, up to a Milne boost and spatial rotation, with explicit limit expressions after λ-scaling. These findings broaden the geometric understanding of Newtonian limits to frame data and have potential applications in teleparallel and related metric theories of gravity, where orthonormal frames are central.

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.

The Newtonian limit of orthonormal frames in metric theories of gravity

TL;DR

This work extends the Newtonian limit from Lorentzian metrics to orthonormal frames, showing that a one-parameter family of Lorentzian frames converges to a Galilei frame in the Newtonian limit provided the frame’s spatial rotation remains controlled and its boost velocity relative to a fixed observer converges. The authors develop a rigorous, coordinate-free framework for limits of finite-dimensional vector-space data, including order-k convergence and key lemmas on taking square roots of near-identity matrices and tensor products. The main result establishes precise convergence statements for the frame fields, up to a Milne boost and spatial rotation, with explicit limit expressions after λ-scaling. These findings broaden the geometric understanding of Newtonian limits to frame data and have potential applications in teleparallel and related metric theories of gravity, where orthonormal frames are central.

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.
Paper Structure (5 sections, 5 theorems, 23 equations)

This paper contains 5 sections, 5 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Technical preliminaries
  4. Limits of orthonormal frames
  5. The boost velocity relating two Lorentzian states of motion

Key Result

Lemma 2.4

Let $\accentset{\lambda}{M}$ be a one-parameter family of real-valued symmetric $n\times n$ matrices satisfying $\accentset{\lambda}{M} \xrightarrow[k\times]{\lambda\to0}\mathbb 1$, where $\mathbb 1$ is the identity matrix. Then for $\lambda$ small enough there is a one-parameter family $\accentset{

Theorems & Definitions (13)

  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 3.1
  • proof : Proof of \ref{['prop:metr_conv_Galilei']}
  • Theorem 3.2
  • Remark 3.3
  • ...and 3 more