A new derivation of the Minkowski metric

TL;DR

This paper derives Minkowski spacetime from Clifford geometric algebra, embedding it in an eight-dimensional structure, leading to natural emergence of relativity laws and new insights into the nature of time.

Contribution

It introduces a geometric algebra-based derivation of Minkowski spacetime, extending it within an eight-dimensional framework and deriving key physical laws intrinsically.

Findings

Minkowski spacetime is a natural property of 3D space modeled with Clifford algebra.

The invariant interval and Lorentz transformations are generalized within this framework.

The approach yields a two-dimensional concept of time and offers philosophical insights.

Abstract

The four dimensional spacetime continuum, as first conceived by Minkowski, has become the dominant framework within which to describe physical laws. In this paper, we show how this four-dimensional structure is a natural property of physical three-dimensional space, if modeled with Clifford geometric algebra $ C\ell(\Re^3) $. We find that Minkowski spacetime can be embedded within a larger eight dimensional structure. This then allows a generalisation of the invariant interval and the Lorentz transformations. Also, with this geometric oriented approach the fixed speed of light, the laws of special relativity and a generalised form of Maxwell's equations, arise naturally from the intrinsic properties of the algebra without recourse to physical arguments. We also find new insights into the nature of time, which can be described as two-dimensional. Some philosophical implications of this approach as it relates to the foundations of physical theories are also discussed.