Classical aspects of lightlike dimensional reduction

TL;DR

This paper explores classical lightlike dimensional reduction in flat spacetime, revealing how shadows and Galilean symmetries relate to relativistic physics and providing methods to reconstruct trajectories from shadows.

Contribution

It introduces a group theoretical framework for lightlike dimensional reduction, demonstrating the Galilean group as an exact symmetry acting on shadows and solving trajectory reconstruction from shadow data.

Findings

Shadows maintain shape across orthogonal screens.

Simultaneous shadows in one screen are simultaneous in all.

Relativistic collisions project to non-relativistic collisions with energy conservation.

Abstract

Some aspects of lightlike dimensional reduction in flat spacetime are studied with emphasis to classical applications. Among them the Galilean transformation of shadows induced by inertial frame changes is studied in detail by proving that, (i) the shadow of an object has the same shape in every orthogonal-to-light screen, (ii) if two shadows are simultaneous in an orthogonal-to-light screen then they are simultaneous in any such screen. In particular, the Galilean group in 2+1 dimensions is recognized as an exact symmetry of Nature which acts on the shadows of the events instead that on the events themselves. The group theoretical approach to lightlike dimensional reduction is used to solve the reconstruction problem of a trajectory starting from its acceleration history or from its projected (shadow) trajectory. The possibility of obtaining a Galilean projected physics starting from a Poincar\'e invariant physics is stressed through the example of relativistic collisions. In particular, it is shown that the projection of a relativistic collision between massless particles gives a non-relativistic collision in which the kinetic energy is conserved.