# The distance formula in algebraic spacetime theories

## TL;DR

This paper derives an observer-independent algebraic distance formula in algebraic spacetime theories, connecting it with 2-spinor geometry and confirming a conjecture by Parfionov and Zapatrin.

## Contribution

It provides a new derivation of the Lorentzian distance formula using 2-spinor geometry, demonstrating its observer independence.

## Key findings

- The algebraic distance formula is observer-independent.
- The derivation connects algebraic and geometric approaches.
- The Lorentzian distance formula confirms the conjecture by Parfionov and Zapatrin.

## Abstract

The Lorentzian distance formula, conjectured several years ago by Parfionov and Zapatrin, has been recently proved by the second author. In this work we focus on the derivation of an equivalent expression in terms of the geometry of 2-spinors by using a partly original approach due to the first author. Our calculations clearly show the independence of the algebraic distance formula of the observer.