Special Relativity: Einstein's Spherical Waves versus Poincare's Ellipsoidal Waves

TL;DR

This paper compares Einstein's and Poincare's interpretations of special relativity, showing that Lorentz transformations of spherical waves become ellipsoidal, reflecting different conventions and geometrical representations of simultaneity and synchronization.

Contribution

It demonstrates that Lorentz transformations of spherical waves are ellipsoidal, highlighting the geometric and conceptual differences between Einstein's and Poincare's approaches to relativity.

Findings

Lorentz transformation of spherical waves results in ellipsoidal waves.

Poincare's ellipsoid geometrically represents relativity of simultaneity.

Einstein's spheres represent synchronization conventions.

Abstract

We show that the image by the Lorentz transformation of a spherical (circular) light wave, emitted by a moving source, is not a spherical (circular) light wave but an ellipsoidal (elliptical) light wave. Poincare's ellipsoid (ellipse) is the direct geometrical representation of Poincare's relativity of simultaneity. Einstein's spheres (circles) are the direct geometrical representation of Einstein's convention of synchronisation. Poincare adopts another convention for the definition of space-time units involving that the Lorentz transformation of an unit of length is directly proportional to Lorentz transformation of an unit of time. Poincare's relativistic kinematics predicts both a dilation of time and an expansion of space as well.