# Relativistic compatibility of the interacting $\kappa$-Poincar\'e model and implications for the relative locality framework

## TL;DR

This paper demonstrates that the $ppa$-Poincare9 model with multiple causally connected interactions remains relativistic under Lorentz boosts when a total boost incorporating all causally connected particles is used, within the relative locality framework.

## Contribution

It extends previous single-vertex analyses by showing the relativistic invariance of the model with multiple vertices using a total boost concept in the relative locality framework.

## Key findings

- Relativistic invariance holds with a total boost in multi-vertex scenarios.
- The model remains Lorentz invariant when using an appropriate total boost.
- The approach allows defining distant observers consistently in the relative locality framework.

## Abstract

We investigate the relativistic properties under boost transformations of the $\kappa$-Poincar\'e model with multiple causally connected interactions, both at the level of its formulation in momentum space only and when it is endowed with a full phase space construction, provided by the relative locality framework. Previous studies focussing on the momentum space picture showed that in presence of just one interaction vertex the model is relativistic, provided that the boost parameter acting on each given particle receives a "backreaction" from the momenta of the other particles that participate in the interaction. Here we show that in presence of multiple causally-connected vertices the model is relativistic if the boost parameter acting on each given particle receives a backreaction from the total momentum of all the particles that are causally connected, even those that do not directly enter the vertex. The relative locality framework constructs spacetime by defining a set of dual coordinates to the momentum of each particle and interaction coordinates as Lagrange multipliers that enforce momentum conservation at interaction events. We show that the picture is Lorentz invariant if one uses an appropriate "total boost" to act on the particles' worldlines and on the interaction coordinates. The picture we develop also allows for a reinterpretation of the backreaction as the manifestation of the "total boost" action. Our findings provide the basis to consistently define distant relatively boosted observers in the relative locality framework.