Anatomy of Null Contractions

TL;DR

This paper explores null contractions of Poincare and conformal algebras, revealing how these contractions produce lower-dimensional Carroll and Galilean algebras, and discusses their implications in conformal cases.

Contribution

It introduces null contractions of relativistic algebras, demonstrating how they produce lower-dimensional Carroll and Galilean sub-algebras and analyzing their structure in conformal contexts.

Findings

Null contractions produce lower-dimensional Carroll and Galilean algebras.

Longitudinal contraction disentangles Carroll algebras at null boundaries.

Transverse contraction separates overlapping Galilean sub-algebras.

Abstract

We introduce null contractions of the Poincare and relativistic conformal algebras. The longitudinal null contraction involves writing the algebra in lightcone coordinates and contracting one of the null directions. For the Poincare algebra, this yields two non-overlapping co-dimension one Carroll algebras. The transverse contraction is a limit on the spatial dimensions and yields two non-overlapping co-dimension one Galilean algebras. We find, similar to Susskind's original observation of the non-relativistic case, that the Poincare algebra, written in the lightcone coordinates, naturally contains Carrollian sub-algebras in one lower dimension. The effect of the longitudinal contraction, which essentially focusses on the null direction, is to disentangle the two Carroll algebras that now correspond to the symmetries of the two null boundaries. The transverse contraction similarly separates the overlapping Galilean sub-algebras of the original Poincare algebra. We discuss aspects of the conformal case, where we get lower dimensional Carroll Conformal algebras and Schrodinger algebras.