[Paper Review] Equivalence Principle and the Principle of Local Lorentz Invariance

Waldyr A. Rodrigues, M. Sharif|ArXiv.org|Feb 4, 2003

Relativity and Gravitational Theory5 references4 citations

TL;DR

This paper rigorously analyzes the Principle of Local Lorentz Invariance (PLLI) in General Relativity, showing that it does not hold universally by constructing models—specifically Friedmann universes—where Local Lorentz Reference Frames (LLRFγ) with different initial metric velocities exhibit distinct expansion ratios (e.g., 2av²), proving they are physically inequivalent. The authors conclude PLLI is only approximately valid, not a fundamental law, due to limitations in experimental precision.

ABSTRACT

In this paper we scrutinize the so called Principle of Local Lorentz Invariance (\emph{PLLI}) that many authors claim to follow from the Equivalence Principle. Using rigourous mathematics we introduce in the General Theory of Relativity two classes of reference frames (\emph{PIRFs} and \emph{LLRF}$γ$\emph{s}) which natural generalizations of the concept of the inertial reference frames of the Special Relativity Theroy. We show that it is the class of the \emph{LLRF}$γ$\emph{s} that is associated with the \emph{PLLI.} Next we give a defintion of physically equivalent referefrence frames. Then, we prove that there are models of General Relativity Theory (in particular on a Friedmann universe) where the \emph{PLLI}is false. However our find is not in contradiction with the many experimental claims vindicating the \emph{PLLI}, because theses experiments do not have enough accuracy to detect the effect we found. We prove moreover that \emph{PIRFs}are not physically equivalent.

Motivation & Objective

• To rigorously define physically equivalent reference frames in General Relativity using mathematical and physical criteria.
• To classify reference frames in GR that generalize Special Relativity's inertial frames, distinguishing Pseudo Inertial Reference Frames (PIRFs) and Local Lorentz Reference Frames (LLRFγ).
• To test whether the Principle of Local Lorentz Invariance (PLLI) holds universally by examining physical equivalence of LLRFγ frames.
• To demonstrate that PLLI fails in certain GR models, such as Friedmann universes, despite its widespread acceptance.
• To reconcile the theoretical failure of PLLI with the success of experimental tests, explaining their insensitivity to the predicted effect.

Proposed method

• Define reference frames in GR as unit timelike vector fields, with natural generalizations of inertial frames in Minkowski spacetime.
• Introduce two classes of generalized inertial-like frames: Pseudo Inertial Reference Frames (PIRFs) and Local Lorentz Reference Frames (LLRFγ).
• Define physical equivalence via observable, objective quantities such as expansion ratios of reference frame congruences.
• Construct a Friedmann universe model with a specific metric to compute the expansion ratio of LLRFγ frames at a spacetime point.
• Use the Raychaudhuri equation to derive the expansion ratio of a moving LLRFγ′ relative to a stationary one, yielding ΘL′(p) = 2av².
• Compare the physical behavior of PIRFs and LLRFγ frames, showing that even PIRFs are not physically equivalent in curved spacetime.

Experimental results

Research questions

• RQ1 Are Local Lorentz Reference Frames (LLRFγ) physically equivalent in General Relativity, as implied by the Principle of Local Lorentz Invariance (PLLI)?
• RQ2 Can the PLLI be derived from the Equivalence Principle, or does it require additional assumptions?
• RQ3 What is the role of the expansion ratio of a reference frame congruence in determining physical equivalence?
• RQ4 In a Friedmann universe model, do LLRFγ frames with different initial metric velocities relative to a central frame exhibit measurable physical differences?
• RQ5 Why do high-precision experiments not detect the predicted violation of PLLI, despite its theoretical existence?

Key findings

• In a Friedmann universe model, a Local Lorentz Reference Frame (LLRFγ) moving with initial metric velocity v relative to a stationary LLRFγ has an expansion ratio of 2av² at a spacetime point p, where a ≪ 1.
• At point p, one LLRFγ has zero expansion ratio (ΘL(p) = 0), while another (LLRFγ′) has a non-zero expansion ratio ΘL′(p) = 2av², proving physical inequivalence.
• The physical inequivalence of LLRFγ frames implies that the Principle of Local Lorentz Invariance (PLLI) is not a fundamental law of nature but only an approximation.
• The effect is proportional to 2av², which is too small to be detected by current experiments, explaining why PLLI remains empirically viable.
• Even Pseudo Inertial Reference Frames (PIRFs) are not physically equivalent in certain GR models, as shown by differences in their physical behavior.
• Friedman’s formulation of PLLI—that identical experiments in different LLRFγ frames yield identical outcomes—is invalid in this model, as measuring the expansion ratio is an objective physical experiment.