# Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Derivation of a Relativistic Tensor Expression for the Classical Uncertainty Principle Inequalities

^{μ}. In some specific cases, the temporal coordinate X

^{0}are represented as t for clarity.

^{μ}) which is parametrized by proper time “τ”. Inequalities (5) take the following form in tensor notation:

## 3. Derivation of a Covariant Geometric Form of the Uncertainty Principle

_{geo}):

## 4. Geometric Uncertainty Principle in Minkowski space

_{μ}

_{ν}. This metric has null Christoffel connectors, resulting in the geodesic scalar being 0. Then, inequality (18) represents a contradiction unless Planck length is considered 0 in the non-quantum limit:

_{00}, to avoid confusion with the Planck constant. For a particle at rest, only the X

^{0}coordinate (t) contributes to proper time. The inequality (18) takes the following form in terms of the Planck constant:

^{0}coordinate (Equation (21)):

^{0}, with dε denoting the accuracy on the measurement. High-precision determination of P

^{0}dε implies long intervals of time. Likewise, measurements over increasingly precise intervals of geodesic time correspond to increased fluctuations (dε) in the energy of the particle. These fluctuations of ε alter the background space–time metric (Equation (21)). The relativistic factor omitted in inequality (9) can also be re-introduced with a value of 2 leading then to:

## 5. Application of the Geometric Uncertainty Principle to the Schwarzschild Metric

^{2}) diverges to infinity. Particles close to the singularity are thus highly de-localized. No particle geodesic below the threshold, set by inequality (36), are allowed. This condition defines an exclusion zone around the singularity as a function of uncertainty in the t-coordinate, see Figure 1.

^{−15}to 10

^{−16}cm.

## 6. Discussion

^{−15}cm. A rough estimation on the diameter of a Planck star by loop quantum gravity gives a value of about 10

^{−10}cm [16,20]. Both calculations provide sizes several orders of magnitude larger than Planck length.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Einstein, A. Grundlage der allgemeinen Relativitätstheorie. Ann. Der Phys.
**1916**, 49, 769–822. [Google Scholar] [CrossRef][Green Version] - Werner, R.F.; Farrely, T. Uncertainty from Heisenberg to today. Found. Phys.
**2019**, 49, 460–491. [Google Scholar] [CrossRef][Green Version] - Ozawa, M. Heisenberg’s original derivation of the uncertainty principle and its universally valid reformulations. Curr. Sci.
**2015**, 109, 2006–2016. [Google Scholar] [CrossRef][Green Version] - Das, S.; Vanegas, E.C. Phenomenological implications of the generalized uncertainty principle. Can. J. Phys.
**2009**, 87, 233–240. [Google Scholar] [CrossRef] - Gine, J. Hawking effect and Unruh effect from the uncertainty principle. EPL
**2018**, 121, 10001. [Google Scholar] [CrossRef] - Hamber, H.W. Quantum Gravitation; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar] [CrossRef]
- Quach, J.Q. Fisher information and the weak equivalence principle of a quantum particle in a graviational wave. Eur. Phys. J. C
**2020**, 80, 987. [Google Scholar] [CrossRef] - Capozziello, S.; Lambiase, G.; Scarpetta, G. Generalized uncertainty principle from quantum geometry. Int. J. Theor. Phys.
**2000**, 39, 15–22. [Google Scholar] [CrossRef] - Ashtekar, A.; Lewandoski, J. Background Independent Quantum Gravity: A Status Report. Class. Quantum Grav.
**2004**, 21, R53. [Google Scholar] [CrossRef] - Magueijo, J.; Smolin, L. String theories with deformed energy momentum relations, and a possible non-tachyonic bosonic string. Phys. Rev. D
**2005**, 71, 026010. [Google Scholar] [CrossRef][Green Version] - Kempf, A.; Mangano, G.; Mann, R.B. Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D
**1995**, 52, 1108–1118. [Google Scholar] [CrossRef][Green Version] - Todorinov, V.; Bosso, P.; Das, S. Relativistic generalized uncertainty principle. Ann. Phys.
**2019**, 405, 92–100. [Google Scholar] [CrossRef][Green Version] - Schwarzschild, K. On the gravitational field of a mass point according to Einstein’s theory. Sitzungsber. Preuss. Akad.Wiss. Berl. (Math. Phys.)
**1916**, 1916, 189–196. (In German). English translation: Gen. Rel. Grav.**2003**, 35, 951–959 [Google Scholar] [CrossRef] - Teukolsky, S.A. The Kerr metric. Class. Quantum Grav.
**2015**, 32, 124006. [Google Scholar] [CrossRef] - Townsend, P.K. Black holes. arXiv
**1997**, arXiv:gr-qc/9707012. [Google Scholar] - Rovelli, C.; Vidotto, F. Planck stars. Int. J. Mod. Phys. D.
**2014**, 23, 1442026. [Google Scholar] [CrossRef] - Mathur, S.D. The Fuzzball proposal for black holes: An Elementary review. Fortsch. Phys.
**2005**, 53, 793–827. [Google Scholar] [CrossRef][Green Version] - Dai, X. The Black Hole Paradoxes and Possible Solutions. J. Phys. Conf. Ser.
**2020**, 1634, 012088. [Google Scholar] [CrossRef] - Susskind, L.; Uglum, J. String physics and black holes. Ucl Phys. Proc. Suppl.
**1996**, 45BC, 115–134. [Google Scholar] [CrossRef][Green Version] - Rovelli, C. Loop Quantum Gravity. Living Rev. Relativ.
**1998**, 1. [Google Scholar] [CrossRef] [PubMed][Green Version] - Casares, P.A.M. A review on Loop Quantum Gravity. arXiv
**2018**, arXiv:1808.01252. [Google Scholar] - Gross, D.J.; Mende, P.F. String theory beyond the Planck scale. Nucl. Phys. B
**1988**, 303, 407–454. [Google Scholar] [CrossRef] - Aharony, O.; Gubser, S.S.; Maldacena, J.; Ooguri, H.; Oz, Y. Large N field theories, string theory and gravity. Phys. Rep.
**2000**, 323, 183–386. [Google Scholar] [CrossRef][Green Version] - Amelino-Camelia, G. Doubly-special relativity: Facts, myths and some key open issues. Symmetry
**2010**, 2, 230. [Google Scholar] [CrossRef][Green Version] - Hossenfelder, S. Minimal length scale scenarios for quantum gravity. Living Rev. Relativ.
**2013**, 16. [Google Scholar] [CrossRef][Green Version] - Haghani, Z.; Harko, T. Effects of quantum metric fluctuations on the cosmological evolution in Friedmann-Lemaitre-Robertson-Walker geometries. Physics
**2021**, 3, 689–714. [Google Scholar] [CrossRef] - Aghanim, N.; Akrami, Y.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; Basak, S.; et al. Planck 2018 results: VI. Cosmological parameters. Astron. Astrophys.
**2020**, 641, A6. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Black hole singularity exclusion zone. Relative uncertainty, dR

^{2}, in the radiial coordinate (R) as a function of R (Equation (36)). The uncertainty diverges to infinity towards the singularity (R = 0). Geodesics with dR

^{2}values below the curve are not allowed, and define a particle exclusion zone in the interior of the black hole.

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Escors, D.; Kochan, G. Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle. *Physics* **2021**, *3*, 790-798.
https://doi.org/10.3390/physics3030049

**AMA Style**

Escors D, Kochan G. Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle. *Physics*. 2021; 3(3):790-798.
https://doi.org/10.3390/physics3030049

**Chicago/Turabian Style**

Escors, David, and Grazyna Kochan. 2021. "Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle" *Physics* 3, no. 3: 790-798.
https://doi.org/10.3390/physics3030049