Abstract
The classical uncertainty principle inequalities are imposed over the general relativity geodesic equation as a mathematical constraint. In this way, the uncertainty principle is reformulated in terms of proper space–time length element, Planck length and a geodesic-derived scalar, leading to a geometric expression for the uncertainty principle (GeUP). This re-formulation confirms the need for a minimum length of space–time line element in the geodesic, which depends on a Lorentz-covariant geodesic-derived scalar. In agreement with quantum gravity theories, GeUP imposes a perturbation over the background Minkowski metric unrelated to classical gravity. When applied to the Schwarzschild metric, a geodesic exclusion zone is found around the singularity where uncertainty in space-time diverged to infinity.
Highlights
General Relativity (GR) describes gravitation as a dynamical space–time geometry in a pseudo-Riemannian manifold shaped by energy-momentum densities [1]
GR is largely incompatible with quantum mechanics
GR world-lines for particles are defined with infinite precision [1], while this is not allowed in quantum mechanics
Summary
General Relativity (GR) describes gravitation as a dynamical space–time geometry in a pseudo-Riemannian manifold shaped by energy-momentum densities [1]. Considering GUP in the framework of quantum geometry theory, any accelerating particle in the absence of gravity experiences a gravitational field [8] This field corresponds to a perturbation unrelated to classical gravitation over the background Minkowski metric. This approach recovers GUP in the p-quadratic form as a function of the particle mass, m, the proper acceleration, A, and the quadratic form of the space-time length element, δs [8]:. The signs of these metric components change in the interior of the black hole once the event horizon is crossed, with the radial-dependent metric behaving as time rather than space Despite these considerations, the singularity remains at R position 0 where the length of the space–time line element is undefined. When applied to geodesics in the Schwarzschild metric, the presence of an exclusion zone around the singularity was confirmed
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