Projective geometry and special relativity

TL;DR

This paper explores how concepts from projective geometry can be applied to fundamental physics, particularly in understanding Minkowski space, the Lorentz group, and the transition from classical to wave mechanics.

Contribution

It introduces a geometric framework connecting projective geometry with special relativity and electromagnetism, highlighting the transition from infinite to finite light speed and from point to wave mechanics.

Findings

Transition from hyperplane at infinity to light cone

Replacement of rest space with proper time hyperboloid

Geometric perspective on electromagnetism and wave mechanics

Abstract

Some concepts of real and complex projective geometry are applied to the fundamental physical notions that relate to Minkowski space and the Lorentz group. In particular, it is shown that the transition from an infinite speed of propagation for light waves to a finite one entails the replacement of a hyperplane at infinity with a light cone and the replacement of an affine hyperplane - or rest space - with a proper time hyperboloid. The transition from the metric theory of electromagnetism to the pre-metric theory is discussed in the context of complex projective geometry, and ultimately it is proposed that the geometrical issues are more general than electromagnetism, namely, they pertain to the transition from point mechanics to wave mechanics.