2+1 gravity and Doubly Special Relativity

TL;DR

The paper shows that gravity in $2+1$ dimensions coupled to point particles furnishes a concrete $DSR$ framework, by constructing an explicit map between the $2+1$ gravity phase space and the $\kappa$-Poincar\'e/DSR algebra. It demonstrates that the momentum space is curved (AdS) and the coordinates are noncommutative, with an energy bound set by $G$, thereby providing a nontrivial, solvable example of a $DSR$ theory. The authors further argue that, by dimensional reduction, similar $DSR$-type structures may describe ultra-high-energy kinematics in $3+1$ dimensions, with the deformation scale $\kappa$ set by $G$ and the system's longitudinal momentum. This work clarifies the physical content and basis dependence of $DSR$ theories, addresses criticisms of nontriviality, and suggests observable implications for Planck-scale physics. Overall, it positions $2+1$ gravity as a transparent, explicit realization of $DSR$ and points to a potential route for $3+1$ dimensional high-energy kinematics to be governed by a deformed relativistic symmetry.

Abstract

It is shown that gravity in 2+1 dimensions coupled to point particles provides a nontrivial example of Doubly Special Relativity (DSR). This result is obtained by interpretation of previous results in the field and by exhibiting an explicit transformation between the phase space algebra for one particle in 2+1 gravity found by Matschull and Welling and the corresponding DSR algebra. The identification of 2+1 gravity as a $DSR$ system answers a number of questions concerning the latter, and resolves the ambiguity of the basis of the algebra of observables. Based on this observation a heuristic argument is made that the algebra of symmetries of ultra high energy particle kinematics in 3+1 dimensions is described by some DSR theory.