Relative-locality distant observers and the phenomenology of momentum-space geometry

TL;DR

This paper explores the translational invariance in the relative-locality framework with momentum-space geometry inspired by $oldsymbol{ ext{kappa}- ext{Poincaré}}$, revealing observable effects of torsion on particle detection times.

Contribution

It provides an explicit example ensuring translational invariance for multi-interaction processes and analyzes the phenomenology of momentum-space geometry with torsion and non-metricity.

Findings

Simultaneously emitted massless particles do not arrive simultaneously at distant detectors.

Torsion in the connection causes detectable variations in particle arrival times.

The framework reproduces known relative-locality results in the free-particle limit.

Abstract

We study the translational invariance of the relative-locality framework proposed in arXiv:1101.0931, which had been previously established only for the case of a single interaction. We provide an explicit example of boundary conditions at endpoints of worldlines, which indeed ensures the desired translational invariance for processes involving several interactions, even when some of the interactions are causally connected (particle exchange). We illustrate the properties of the associated relativistic description of distant observers within the example of a $\kappa$-Poincar\'e-inspired momentum-space geometry, with de Sitter metric and parallel transport governed by a non-metric and torsionful connection. We find that in such a theory simultaneously-emitted massless particles do not reach simultaneously a distant detector, as expected in light of the findings of arXiv:1103.5626 on the implications of non-metric connections. We also show that the theory admits a free-particle limit, where the relative-locality results of arXiv:1102.4637 are reproduced. We establish that the torsion of the $\kappa$-Poincar\'e connection introduces a small (but observably-large) dependence of the time of detection, for simultaneously-emitted particles, on some properties of the interactions producing the particles at the source.