Curvature of \k{appa}-Poincare and Doubly Special Relativity

TL;DR

This paper investigates the geometric properties of momentum space in ppa-Poincare and Magueijo-Smolin DSR theories within the relative locality framework, revealing non-zero connection and torsion, and their implications for non-commutative spacetime structures.

Contribution

It provides a comprehensive calculation of connection, torsion, and curvature for these theories at all orders of the Planck length, highlighting differences and implications for spacetime geometry.

Findings

Connection can be non-zero in ppa-Poincare and MS DSR.

Torsion is non-zero in ppa-Poincare but zero in MS DSR.

Curvature is zero in both theories.

Abstract

We study the \k{appa}-Poincare and the Magueijo-Smolin (MS) DSR in the context of the relative locality theory. This theory assigns connection, torsion and curvature to momentum space of every modified theory beyond special relativity. We obtain these quantities for the \k{appa}-Poincare and the MS DSR in all order of the Planck length, at the every point of the momentum space. The connection for the \k{appa}-Poincare theory and the MS DSR can be non-zero. The torsion for the \k{appa}-Poincare theory can also be non-zero, but it is zero for the MS DSR. The curvature for the \k{appa}-Poincare theory and the MS DSR are zero. We will find that the non-zero torsion and curvature of the momentum space implies a non-commutative spactime which is tangent to this momentum space. Also, we show that the torsion for every Abelian DSR theory is zero at the origin of the momentum space. At the end, we will discus dual spacetime transformations for the \k{appa}-Poincare theory and MS-DSR.