Quantum geodesics on quantum Minkowski spacetime

We apply a recent formalism of quantum geodesics to the well-known bicrossproduct model $\lambda$-Minkowski spacetime $[x^i,t]=\imath\lambda_p x^i$ with its flat metric as gravity effects, $\lambda_p$ Planck scale. As examples, geodesic flow plane wave gets an order frequency dependent correction classical velocity. A velocity $v$ Gaussian width $\sqrt{2\beta}$ initially centred at origin changes shape but centre mass moves ${\over}=v(1+{\lambda_p^2\over 2\beta}+O(\lambda^3_p))$, $\lambda_p^2$ correction. This implies, least within perturbation theory, that `point particle' cannot be modelled infinitely sharp due corrections. For contrast, we also look on noncommutative torus 2D curved weak Levi-Civita connection.