On a Possible Modification of the Gravitational Radius: ## An Analysis Based on Planck Units and a Proposal for a Decisive Atomic Clock Experiment

Abstract Within the framework of General Relativity (GR), the Schwarzschild radius( R_s = 2GM/c^2 )is traditionally regarded as a fundamental scale characterizing the event horizon. However, an analysis of the system of Planck units leads to the relation( l_P = GM_P / c^2 ),which formally corresponds to a gravitational radius without the numerical factor 2. This circumstance suggests that the coefficient appearing in the expression for the gravitational radius may not be fixed directly by Planck-scale considerations and may depend on the physical interpretation of the horizon criterion. In this work, an alternative expression( R_g = GM/c^2 )is considered, possessing both Planck-scale and kinematic motivation. It is emphasized that the question of the numerical coefficient is empirical in nature. The central result of the paper is the proposal of a direct terrestrial experiment measuring the difference in clock rates between the Earth’s pole and equator using optical atomic clocks. It is shown that three logically self-consistent physical scenarios lead to clearly distinguishable predictions:(+98.1) ns/day (standard GR),(-3.0) ns/day (alternative radius with independent superposition of effects), and(+101.0) ns/day (a sequential interpretation of the alternative radius).The separation between these predictions, of order (10^2) ns/day, exceeds the precision of modern chronometric measurements by three orders of magnitude, rendering the proposed experiment technically feasible and potentially decisive for an empirical test of the coefficient in the gravitational radius formula.