Change of basis and Gram-Schmidt orthonormalization in special relativity

TL;DR

This paper explores the use of basis change and Gram-Schmidt orthonormalization in special relativity, emphasizing the conceptual advantages of explicit basis inclusion and deriving Lorentz transformations through these methods.

Contribution

It introduces the reciprocal basis and applies Gram-Schmidt orthonormalization to derive Lorentz transformations in special relativity.

Findings

Lorentz transformations can be viewed as change of basis operations.

The Lorentz boost in one dimension is derived using Gram-Schmidt orthonormalization.

Other Lorentz transformations are obtained via Gram-Schmidt procedure.

Abstract

While an explicit basis is common in the study of Euclidean spaces, it is usually implied in the study of inertial relativistic systems. There are some conceptual advantages to including the basis in the study of special relativistic systems. A Minkowski metric implies a non-orthonormal basis, and to deal with this complexity the concepts of reciprocal basis and the vector dual are introduced. It is shown how the reciprocal basis is related to upper and lower index coordinate extraction, the metric tensor, change of basis, projections in non-orthonormal bases, and finally the Gram-Schmidt procedure. It will be shown that Lorentz transformations can be viewed as change of basis operations. The Lorentz boost in one spatial dimension will be derived using the Gram-Schmidt orthonormalization algorithm, and it will be shown how other Lorentz transformations can be derived using the Gram-Schmidt procedure.