# 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

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**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