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

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.

The Uncertainty Principle and the Minimal Space-Time Length Element

“…Nevertheless, space-time quantization can be compatible with a minimal, Lorentzcovariant length element, as shown by GUP and other covariant formulations including ours [56,57]. Hence, it is yet unclear whether LIVs could be definitely detected within our current energy scales.…”

“…However, it turned out that such formulation did not recover the two classical inequalities. Hence, we decided to re-express the classical inequalities in a covariant form, allowing its application as a mathematical constraint over GR geodesics [56,57]. This formulation extended the uncertainty inequality to a differential length of relativistic proper space-time line element (𝑑𝑑𝜏𝜏 2 ) as a function of Planck length, ℓ 𝑝𝑝 , and a geodesic-related scalar (𝐺𝐺 geo ) as follows:…”

“…This covariant reformulation of the classical uncertainty principle sets a length limit for the quadratic proper space-time line element. Its application as a constraint to Minkowski space, required the introduction of a time-dependent differential perturbation (𝜀𝜀) to the g00 component of the metric [56,57]:…”

“…) arising from the uncertainty principle as follows: When applied to the metric of an expanding universe, as represented by the FRW metric [58][59][60], the quadratic space-time line element was calculated in terms of two functions [57]. The first one derived from energy fluctuations from the uncertainty principle (𝐸𝐸 𝑢𝑢𝑚𝑚 ) and the second from the expansion rate of the universe (𝐻𝐻 𝑒𝑒𝑒𝑒 ):…”