Fate of $κ$-Minkowski space-time in non-relativistic (Galilean) and ultra-relativistic (Carrollian) regimes

Abstract

Here, we present an algebraic and kinematical analysis of non-commutative $κ$-Minkowski spaces within Galilean (non-relativistic) and Carrollian (ultra-relativistic) regimes. Utilizing the theory of Wigner-Inönu contractions, we begin with a brief review of how one can apply these contractions to the well-known Poincaré algebra, yielding the corresponding Galilean (both massive and mass-less) and Carrollian algebras as $c \to \infty$ and $c\to 0$, respectively. Subsequently, we methodically apply these contractions to non-commutative $κ$-deformed spaces, revealing compelling insights into the interplay among the non-commutative parameters $a^μ$ (with $|a^ν|$ being of the order of Planck length scale) and the speed of light $c$ as it approaches both infinity and zero. Our exploration predicts a sort of "branching" of the non-commutative parameters $a^μ$, leading to the emergence of a novel length scale and time scale in either limit. Furthermore, our investigation extends to the examination of curved momentum spaces and their geodesic distances in appropriate subspaces of the $κ$-deformed Newtonian and Carrollian space-times. We finally delve into the study of their deformed dispersion relations, arising from these deformed geodesic distances, providing a comprehensive understanding of the nature of these space-times.