$κ$-General-Relativity I: a Non-Commutative GR Theory with the $κ$-Minkowski Spacetime as its Flat Limit

TL;DR

The paper develops a twist-deformed General Relativity on a local $\kappa$-Minkowski noncommutative spacetime by enlarging the symmetry to $\mathrm{IGL}(3,1)$ and applying an Abelian twist to realize the flat limit $\kappa$-Minkowski. To ensure a meaningful classical limit, an Inönü–Wigner contraction is proposed, yielding a contracted symmetry $\mathrm{IGL}'(3,1)$ that reduces to $\mathrm{ISO}(3,1)$ as the deformation effects vanish; the Planck-frequency scale $\omega_p$ is identified as the fundamental contraction scale, with a system-dependent RL parameter $\epsilon_{RL}=\omega_{\text{local}}/\omega_p$ guiding the quantum-planckian regime. A deformed gravitational theory is constructed via a $\star$-covariant derivative, $\star$-curvature, and a $\star$-Einstein equation, complemented by a deformed Einstein–Hilbert action and discussion of matter coupling, conservation, and Noether charges. The framework preserves twisted general covariance and yields a consistent perturbative expansion in the deformation parameter $\lambda'$, while highlighting subtleties in reality, ordering, and local conservation that motivate further work. Overall, the work lays a foundation for calculable phenomenology in $\kappa$-Minkowski gravity and guides future steps toward explicit predictions and observational tests.

Abstract

We employ a twist deformation of infinitesimal diffeomorphisms to construct a modification of General Relativity on a non-commutative spacetime extending the local kappa-Minkowski geometry. This spacetime arises in Deformed Special Relativity (DSR) models, where a fundamental length scale is incorporated into Special Relativity as an effective description of quantum gravitational effects. To avoid the mathematical and physical inconsistencies associated with twisting the Poincare group, we instead deform the dilatation-enlarged IGL(3,1) group, constructing a covariant and explicitly consistent gravitational theory (distinct from Weyl gravity). The relativistic consistency of the twisted kappa-Minkowski spacetime is demonstrated, including deformed transformations and differential structures. A physically motivated Inonu-Wigner (IW) contraction procedure is suggested to enable a well-defined classical limit, addressing the correspondence issue. This framework provides a consistent foundation for a dynamical sector of DSR and allows, in future treatment, explicit computations that could advance phenomenological predictions.