Abstract
We derive the non-relativistic c→∞ and ultra-relativistic c→0 limits of the κ-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the κ-(A)dS quantum algebra, and quantize the resulting contracted Poisson–Hopf algebras, thus giving rise to the κ-deformation of the Newtonian (Newton–Hooke and Galilei) and Carrollian (Para-Poincaré, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corresponding κ-Newtonian and κ-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the κ-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter κ, the curvature parameter η and the speed of light parameter c.
Highlights
Quantum gravity research usually focusses on the “relativistic regime”, namely it considers scenarios that at low energies/large distances reduce either to general or special relativity, with finite speed of light
The non-relativistic contracted cocommutator coincides with the κ-AdSΛ one given by (21). Note that this new fundamental Lie bialgebra contraction (LBC) in which the deformation parameter does not change leads to a divergence for the c → ∞ limit of the r-matrix (22), which is consistent with the fact that the Newtonian Lie bialgebra defined by the cocommutator (21) together with the commutation rules (6) is not a coboundary Lie bialgebra
While it has been understood for a while that Planck scale effects might have a non-trivial interplay with curvature effects, their relation to relativistic effects was not investigated before
Summary
Quantum gravity research usually focusses on the “relativistic regime”, namely it considers scenarios that at low energies/large distances reduce either to general or special relativity, with finite speed of light. We show that studying the non-relativistic and ultra-relativistic limits of these models allows us to investigate the possible interplay between the noncommutativity parameter, the cosmological constant and the speed of light all at once. We compute for the first time the Poisson version of the κ-deformed Newtonian and Carrollian quantum algebras, including their extension with non-vanishing curvature These can be obtained as the c → ∞ and c → 0 contractions of the κ-(A)dS algebra, respectively..
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