A Quantum-Gravity-Motivated GUP Effective Metric
Recent critiques have addressed certain aspects of the generalized uncertainty principle (GUP) effective metric (Ong 2023). This study presents a scale-dependent quadratic GUP effective metric, constructed through analyses of the gravity-induced phase shift (COW experiment) and the Einstein-Bohr photon box Gedanken experiment. In contrast to the procedure outlined in (Xiang et al. 2018), the momentum-dependent metric is improved by introducing an interpolating function $\Delta p (r)$, which employs the effective distance concept to accurately capture the distinct behavior of $\Delta p (r)$ in both short and long distance regimes. The resulting effective metric exhibits the same structure as that derived from the Renormalization Group (RG) theory. However, the RG parameter $\hat{\gamma}$ can now be related to the dimensionless GUP parameter $\beta_0$, thereby distinguishing this metric from the RG-based approach. The corresponding effective metric prediction demonstrates internal consistency of the model, and phenomenologically the predictions are in agreement with some quantum black hole models in some limits. The effective metric improved black hole thermodynamics and shadow predictions compared to the heuristic approach and other proposed GUP effective metrics. Furthermore, the relationship between GUP and $f(R)$ gravity (D'Agostino et al. 2026) may provide a possible future route toward a more fundamental description of GUP.