Metric Tensor Vs. Metric Extensor

TL;DR

This paper compares metric tensors and metric extensors in finite-dimensional real vector spaces, highlighting the advantages of extensors, such as invertibility, and clarifying their determinant relationship, enabling advanced calculations without matrices.

Contribution

It introduces the concept of metric extensors as an alternative to tensors, emphasizing their invertibility and providing a clear link between their determinants and classical matrix determinants.

Findings

Metric extensors have inverses that are also metric extensors.

Determinant of a metric extensor relates to the classical matrix determinant.

Calculations with metric extensors can be performed without matrix representations.

Abstract

In this paper we give a comparison between the formulation of the concept of metric for a real vector space of finite dimension in terms of \emph{tensors} and \emph{extensors}. A nice property of metric extensors is that they have inverses which are also themselves metric extensors. This property is not shared by metric tensors because tensors do \emph{not} have inverses. We relate the definition of determinant of a metric extensor with the classical determinant of the corresponding matrix associated to the metric tensor in a given vector basis. Previous identifications of these concepts are equivocated. The use of metric extensor permits sophisticated calculations without the introduction of matrix representations.