Relativistic Generalized Uncertainty Principle and Maximum Acceleration near a Black Hole Horizon

We investigate the maximum acceleration allowed by the Relativistic Generalized Uncertainty Principle (RGUP) and its implications for black hole (BH) physics. Previous analyses based on the nonrelativistic Generalized Uncertainty Principle (GUP) yield the universal scale \( a_{max}\approx c{{}^2}/l_{P} \), where c is the speed of light and l_P is the Planck length. We extend this analysis to relativistic quantum mechanics and Schwarzschild spacetime. For an observer in flat spacetime or in free fall in a gravitational field, the RGUP reduces to the GUP and the same finite maximum acceleration is recovered. In contrast, for a fiducial observer held at fixed Schwarzschild radius r_g, the metric dependence of the RGUP qualitatively changes the result: the quantum-corrected maximum proper acceleration remains finite away from the horizon but diverges as the Schwarzschild radial coordinate approaches the gravitational radius. Thus, the RGUP does not remove the classical divergence for a static observer at the horizon. This difference between freely falling and fiducial observers provides a quantum-mechanical manifestation of the observer dependence associated with the BH horizon and suggests a form of BH complementarity at the level of maximum acceleration.