[Paper Review] A First-Principles Implementation of Scale Invariance Using Best Matching

Hans Westman|ArXiv.org|Oct 8, 2009

Mathematics and Applications18 references3 citations

TL;DR

This paper presents a first-principles gauge theory of spatial scale invariance using the best matching method, introducing a dynamical 3-vector potential $A_k$ as a fundamental field alongside the 3-metric. Unlike Weyl’s original theory, the equations of motion are second-order in time derivatives, avoiding unphysical fourth-order behavior, and the vector potential is argued to be incompatible with electromagnetism but potentially linked to dark matter or cosmological scale effects.

ABSTRACT

We present a first-principles implementation of spatial scale invariance as a local gauge symmetry in geometry dynamics using the method of best matching . In addition to the 3-metric, the proposed scale invariant theory also contains a 3-vector potential $A_k$ as a dynamical variable. Although some of the mathematics is similar to Weyl's ingenious but physically questionable theory, the equations of motion of this new theory are second order in time-derivatives. Thereby we avoid the problems associated with fourth order time derivatives that plague Weyl's original theory. It is tempting to try to interpret the vector potential $A_k$ as the electromagnetic field. We exhibit four independent reasons for not giving into this temptation. A more likely possibility is that it can play the role of "dark matter". Indeed, as noted in scale invariance seems to play a role in the MOND phenomenology. Spatial boundary conditions are derived from the free-endpoint variation method and a preliminary analysis of the constraints and their propagation in the Hamiltonian formulation is presented.

Motivation & Objective

• To develop a first-principles implementation of spatial scale invariance as a local gauge symmetry in geometry dynamics.
• To avoid the unphysical fourth-order time derivatives present in Weyl’s original scale-invariant theory.
• To explore whether the introduced vector potential $A_k$ can play a role analogous to dark matter or explain cosmological scale effects.
• To derive spatial boundary conditions via the free-endpoint variation method and analyze constraints in the Hamiltonian formulation.
• To investigate whether scale invariance can be implemented without relying on conformalizing the Baierlein-Sharp-Wheeler action.

Proposed method

• The theory is constructed using the best matching method to enforce scale invariance as a local gauge symmetry.
• A 3-vector potential $A_k$ is introduced as a new dynamical variable alongside the 3-metric, distinct from Weyl’s original formulation.
• The equations of motion are derived to be second-order in time derivatives, avoiding the problematic fourth-order behavior of Weyl’s theory.
• The method of free-endpoint variation is used to derive spatial boundary conditions for the theory.
• The Hamiltonian formulation is analyzed for constraints and their propagation, with an emphasis on foliation invariance and potential emergence of proper time.
• The theory is compared to Weyl’s original framework, highlighting key mathematical differences such as the Lie group manifold being $\mathbb{R}$ rather than $S^1$, indicating a non-Abelian-like structure distinct from electromagnetism.

Experimental results

Research questions

• RQ1 Can spatial scale invariance be implemented as a local gauge symmetry from first principles without relying on conformalized actions?
• RQ2 Why is the introduced vector potential $A_k$ incompatible with identification as the electromagnetic field?
• RQ3 How can a scale-invariant theory account for the observed expansion of the universe without violating scale invariance?
• RQ4 What are the implications of non-zero scale curvature $F_{ij} = \partial_i A_j - \partial_j A_i \neq 0$ for global geometric effects?
• RQ5 Can the vector potential $A_k$ provide a viable alternative explanation for dark matter or galaxy rotation curves?

Key findings

• The theory introduces a 3-vector potential $A_k$ as a dynamical field, distinct from Weyl’s original formulation, with second-order equations of motion that avoid unphysical fourth-order derivatives.
• The vector potential cannot be identified with the electromagnetic field due to its universal coupling, absence of charge dependence, and the fact that its gauge group is $\mathbb{R}$ rather than $U(1)$.
• The theory allows for a relational, scale-invariant description of the universe’s expansion, where apparent expansion can be interpreted as a universal shrinking of objects relative to the Hubble radius.
• Non-trivial scale curvature $F_{ij} \neq 0$ could lead to observable global effects, such as rulers changing length after parallel transport around closed loops, and shifts in spectral lines.
• The spectrum of the cosmic microwave background being scale-invariant at large wavelengths is naturally explained in this framework, especially in regions of approximately homogeneous mass density.
• The theory may support the emergence of proper time via lightclocks if the Hamiltonian constraint propagates and the Dirac-Teitelboim algebra closes, ensuring foliation invariance and a universal lightcone structure.