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

TL;DR

This paper explores whether Planck-scale space quantization can coexist with Lorentz invariance by applying a covariant geometric uncertainty principle to FRW geometries, deriving compatible space-time line elements.

Contribution

It introduces a covariant geometric uncertainty principle applied to FRW geometries, deriving space-time line elements consistent with quantum gravity length scales and experimental constraints.

Findings

Derived a quadratic proper space-time line element proportional to Planck length squared.

Showed compatibility of space quantization with Lorentz invariance within experimental limits.

Provided a covariant framework for quantum gravity effects in cosmological models.

Abstract

Most quantum gravity theories quantize space time on the order of Planck length (lp). Some of these theories, such as loop quantum gravity (LQG), predict that this discreetness could be manifested through Lorentz invariance violations (LIV) over travelling particles at astronomical length distances. However, reports on LIV are controversial, and space discreetness could still be compatible with Lorentz invariance. Here, it is tested whether space quantization on the order of Planck length could still be compatible with Lorentz invariance through the application of a covariant geometric uncertainty principle (GeUP) as a constraint over geodesics in FRW geometries. Space time line elements compatible with the uncertainty principle are calculated for a homogeneous, isotropic expanding Universe represented by the Friedmann Lemaitre Robertson Walker solution to General Relativity (FLRW or FRW metric). A generic expression for the quadratic proper space time line element is derived, proportional to Planck length squared, and dependent on two contributions. The first is associated to the energy time uncertainty, and the second depends on the Hubble function. The results are in agreement with space-time quantization on the expected length orders, according to quantum gravity theories, and within experimental constraints on putative LIV.