The mystical formula and the mystery of Khronos

TL;DR

This paper explores the historical and conceptual development of Minkowski's four-dimensional spacetime formulation, highlighting its significance, contradictions with everyday perception, and implications for modern physics.

Contribution

It provides a comprehensive analysis of Minkowski's spacetime concept, its mathematical formulation, and its influence on contemporary physics and the understanding of space and time.

Findings

Minkowski's formulation unified space and time into a four-dimensional continuum.

The concept has influenced the development of General Relativity.

There are ongoing challenges in reconciling intuitive perceptions with relativistic physics.

Abstract

In 1908, Minkowski put forward the idea that invariance under what we call today the Lorentz group, $GL(1,3, {\bf R})$, would be more meaningful in a four-dimensional space-time continuum. This suggestion implies that space and time are intertwined entities so that, kinematic and dynamical quantities can be expressed as vectors, or more generally by tensors, in the four-dimensional space-time. Minkowski also showed how causality should be structured in the four-dimensional vector space. The mathematical formulation proposed by Minkowski made its generalization to curved spaces quite natural, leaving the doors to the General Theory of Relativity and many other developments ajar. Nevertheless, it is remarkable that this deceptively simple formulation eluded many researchers of space and time, and goes against our every day experience and perception, according to which space and time are distinct entities. In this contribution, we discuss these contradictory views, analyze how they are seen in contemporary physics and comment on the challenges that space-time explorers face.