Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle

Version 1 : Received: 28 July 2021 / Approved: 29 July 2021 / Online: 29 July 2021 (10:40:47 CEST)
Version 2 : Received: 31 July 2021 / Approved: 2 August 2021 / Online: 2 August 2021 (13:38:51 CEST)
Version 3 : Received: 24 August 2021 / Approved: 25 August 2021 / Online: 25 August 2021 (09:01:26 CEST)

A peer-reviewed article of this Preprint also exists.

Physics 2021, 3(3), 790-798; Physics 2021, 3(3), 790-798;


General relativity is a theory for gravitation based on Riemannian geometry, difficult to compatibilize with quantum mechanics. This is evident in relativistic problems in which quantum effects cannot be discarded. For example in quantum gravity, gravitation of zero-point energy or events close to a black hole singularity. Here, we set up a mathematical model to select general relativity geodesics according to compatibility with the uncertainty principle. To achieve this, we derived a geometric expression of the uncertainty principle (GeUP). This formulation identified proper space-time length with Planck length by a geodesic-derived scalar. GeUP imposed a minimum allowed value for the interval of proper space-time which depended on the particular space-time geometry. GeUP forced the introduction of a “zero-point” curvature perturbation over flat Minkowski space, caused exclusively by quantum uncertainty but not to gravitation. When applied to the Schwarzschild metric and choosing radial-dependent geodesics, our mathematical model identified a particle exclusion zone close to the singularity, similar to calculations by loop quantum gravity. For a 2 black hole merger, this exclusion zone was shown to have a radius that cannot go below a value proportional to the energy/mass of the incoming black hole multiplied by Planck length.


General relativity; Uncertainty principle; Geodesics; Black hole singularity; zero-point energy


Physical Sciences, Quantum Science and Technology

Comments (1)

Comment 1
Received: 2 August 2021
Commenter: David Escors
Commenter's Conflict of Interests: Author
Comment: Some readers have pointed out a few things to be fixed and induced confusion. First, the term GUP is generally understood as General Uncertainty Principle. We have fixed this by naming our equation GeUP (Geometric Uncertainty Principle). Second, there were a few unintentional mistakes in some equations after bringing up "c" in some examples. For example, in the metric shown in equations 21, where c was shown in some parts, while the -1 would correspond to -c^2 if c is considered. In addition, a factor of 2 from inequation 23 onwards, which was missing. All of these mistakes have been corrected, although they did not change the fundamental equations or the conclusions. Nevertheless, these mistakes needed quick fixing.
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