Relativistic generalized uncertainty principle for a test particle in four-dimensional spacetime

The generalized uncertainty principle provides a promising phenomenological approach to reconciling the fundamentals of general relativity with quantum mechanics. As a result, noncommutativity, measurement uncertainty, and the fundamental theory of quantum mechanics are subject to finite gravitational fields. The generalized uncertainty principle (RGUP) on curved spacetime allows for the imposition of quantum-induced elements on general relativity (GR) in four dimensions. In the relativistic regimes, the determination of a test particle’s spacetime coordinates, Formula: see text, becomes uncertain. There exists a specific range of coordinates where the accessibility to Formula: see text is notably limited. Consequently, the spacetime coordinates lack both smoothness and continuity. The quantum-mechanical calculations of the spacetime coordinates are directly linked to their measurement through the expectation value Formula: see text. This expectation value is dependent on Formula: see text, which is itself limited by a minimum measurable length Formula: see text. A crucial finding presented in this script is the existence of a lower bound for the Hamiltonian, which implies the stability of the quantum nature of spacetime. The direct correlation between Formula: see text and Formula: see text is a significant discovery, suggesting a proportional relationship. The conjecture is made that the primary metric g carries all essential information regarding spacetime curvature and serves a role akin to the Jacobian determinant in general relativity. Moreover, the linear relationship between Formula: see text and the Planck length Formula: see text is established with a proportionality factor of Formula: see text, where Formula: see text denotes the RGUP parameter. The discretization of spacetime coordinates results in the discontinuity of the test particle’s wavefunction Formula: see text, leading to an unrealistic Formula: see text. It is also noted that the lower limit of Formula: see text is directly proportional to the fundamental tensor.