# Covariant Space-Time Line Elements in the Friedmann–Lemaitre–Robertson–Walker Geometry

^{*}

## Abstract

**:**

## 1. Introduction

_{geo}:

## 2. Results

#### 2.1. Space-Time Line Element for a Particle at Rest in the FRW Metric

#### 2.2. Generic Solution for the FRW Space-Time Line Element for a Moving Particle in the R Coordinate

#### 2.3. Specific Solutions for the FRW Space-Time Line Element for a Moving Particle When ${E}_{un}$ Is the Dominant Term

#### 2.4. Specific Solutions for the FRW Space-Time Line Element for a Moving Particle When ${H}_{ex}$ Is the Dominant Term

## 3. Discussion

^{−14}on the vacuum birefringence effect was estimated [16]. This constraint on LIV would translate into “spatial volume units” in the order of 10

^{−42}cubic meters or less. Some recent studies are providing LIV violations experimentally at different energy orders, while other studies establish very stringent constraints for LIV, or even fail to detect it [13,15,16,44,45].

^{0}).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Escors, D.; Kochan, G.
Covariant Space-Time Line Elements in the Friedmann–Lemaitre–Robertson–Walker Geometry. *Axioms* **2022**, *11*, 310.
https://doi.org/10.3390/axioms11070310

**AMA Style**

Escors D, Kochan G.
Covariant Space-Time Line Elements in the Friedmann–Lemaitre–Robertson–Walker Geometry. *Axioms*. 2022; 11(7):310.
https://doi.org/10.3390/axioms11070310

**Chicago/Turabian Style**

Escors, David, and Grazyna Kochan.
2022. "Covariant Space-Time Line Elements in the Friedmann–Lemaitre–Robertson–Walker Geometry" *Axioms* 11, no. 7: 310.
https://doi.org/10.3390/axioms11070310