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

We employ a twist deformation of infinitesimal diffeomorphisms to construct a modification of General Relativity on a non-commutative spacetime extending the local κ-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 Poincaré 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 κ-Minkowski spacetime is demonstrated, including deformed transformations and differential structures. A physically motivated Inönü–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.