[Paper Review] Metric and Gauge Extensors

A. M. Moya, Waldyr A. Rodrigues+1 more|ArXiv.org|Jan 31, 2005

Algebraic and Geometric Analysis2 references3 citations

TL;DR

This paper introduces metric and gauge extensors to reformulate arbitrary Clifford algebras $σ\ell(V,G)$ as deformations of the Euclidean Clifford algebra $σ\ell(V,G_E)$, enabling calculations in general metric geometries using the simpler Euclidean framework. The central contribution is the 'golden formula,' which relates Clifford products under different metrics via an invertible extensor transformation.

ABSTRACT

In this paper, the second in a series of eight we continue our development of the basic tools of the multivector and extensor calculus which are used in our formulation of the differential geometry of smooth manifolds of arbitrary topology . We introduce metric and gauge extensors, pseudo-orthogonal metric extensors, gauge bases, tetrad bases and prove the remarkable golden formula, which permit us to view any Clifford algebra Cl(V,G) as a deformation of the euclidean Clifford algebra Cl(V,G_{E}) discussed in the first paper of the series and to easily perform calculations in Cl(V,G) using Cl(V,G_{E}).

Motivation & Objective

• To develop a formalism for handling arbitrary metric signatures in geometric algebra using a reference Euclidean structure.
• To define metric extensors as linear mappings that relate scalar products under different metrics, enabling metric-dependent algebraic operations.
• To introduce gauge extensors and tetrad bases as tools for formulating geometric theories of gravity.
• To establish a systematic deformation framework for Clifford algebras via the golden formula, reducing complex metric calculations to Euclidean ones.
• To provide a foundation for deformed geometry theories in subsequent papers of the series.

Proposed method

• Define the metric extensor $g$ as the unique linear map satisfying $v \cdot_G w = g(v) \cdot_{G_E} w$ for all $v,w \in V$, linking the metric $G$ to the reference Euclidean metric $G_E$.
• Use the extended operator $\underline{g}$ to lift $g$ to the full exterior algebra $\bigwedge V$, enabling transformation of multivector operations.
• Derive the metric adjoint formula $t^{\dagger(g)} = \underline{g}^{-1} \circ t^{\dagger} \circ \underline{g}$, relating adjoints under different metrics.
• Introduce orthogonal and Lorentz extensors as special cases of metric extensors with applications in relativistic geometry.
• Construct gauge bases and tetrad bases using invertible extensors to relate orthonormal frames to general bases.
• Prove the golden formula: $X \underset{g}{} Y = \underline{h}^{-1}[\underline{h}(X) \underset{\eta}{} \underline{h}(Y)]$, where $h$ is a gauge extensor mapping $G$ to the Minkowski metric $\eta$.

Experimental results

Research questions

• RQ1 How can Clifford algebra operations under an arbitrary metric $G$ be related to those under a reference Euclidean metric $G_E$?
• RQ2 What is the precise algebraic relationship between the adjoint operators of a linear map under different metrics?
• RQ3 How can gauge extensors be used to define orthonormal bases (tetrad bases) in a general metric space?
• RQ4 What is the mathematical structure that allows the deformation of a Euclidean Clifford algebra into one with arbitrary signature?
• RQ5 Can the full Clifford product under a general metric be computed using only the Euclidean product and a transformation via an extensor?

Key findings

• The metric extensor $g$ provides a unique isomorphism between the metric structures $(V,G)$ and $(V,G_E)$, enabling the transfer of geometric operations.
• The golden formula $X \underset{g}{} Y = \underline{h}^{-1}[\underline{h}(X) \underset{\eta}{} \underline{h}(Y)]$ allows all Clifford algebra operations under a general metric $G$ to be computed using the Minkowski metric $\eta$ and the gauge extensor $h$.
• The metric adjoint satisfies $t^{\dagger(g)} = \underline{g}^{-1} \circ t^{\dagger} \circ \underline{g}$, showing how adjoints transform under metric deformation.
• Gauge extensors enable the construction of tetrad bases and provide a bridge between orthonormal and general coordinate frames in geometric gravity theories.
• Pseudo-orthogonal metric extensors, including Lorentz extensors, are shown to be special cases of the general metric extensor formalism.
• The formalism establishes that any Clifford algebra $\mathcal{C}\ell(V,G)$ is a precise gauge deformation of the Euclidean algebra $\mathcal{C}\ell(V,G_E)$, with the deformation encoded in the extensor $\underline{g}$.