Indefinite quadratic forms and the invariance of the interval in Special Relativity

TL;DR

This paper proves a theorem on indefinite quadratic forms and uses it to clarify the invariance of the interval in Special Relativity, addressing common student misconceptions and generalizing the proof with advanced algebraic tools.

Contribution

It introduces a new theorem on indefinite quadratic forms and applies Hilbert's Nullstellensatz to generalize the invariance proof in Special Relativity.

Findings

Clarified the proof of interval invariance in Special Relativity.

Generalized the proof using Hilbert's Nullstellensatz.

Provided conditions for diagonalizability of semi-definite quadratic functions.

Abstract

A simple theorem on proportionality of indefinite real quadratic forms is proved, and is used to clarify the proof of the invariance of the interval in Special Relativity from Einstein's postulate on the universality of the speed of light; students are often rightfully confused by the incomplete or incorrect proofs given in many texts. The result is illuminated and generalized using Hilbert's Nullstellensatz, allowing one form to be a homogeneous polynomial which is not necessarily quadratic. Also a condition for simultaneous diagonalizabilityof semi-definite real quadratic functions is given.