Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle

Abstract

The classical uncertainty principle inequalities are imposed over the general relativity geodesic equation as a mathematical constraint. In this way, the uncertainty principle is reformulated in terms of proper space–time length element, Planck length and a geodesic-derived scalar, leading to a geometric expression for the uncertainty principle (GeUP). This re-formulation confirms the need for a minimum length of space–time line element in the geodesic, which depends on a Lorentz-covariant geodesic-derived scalar. In agreement with quantum gravity theories, GeUP imposes a perturbation over the background Minkowski metric unrelated to classical gravity. When applied to the Schwarzschild metric, a geodesic exclusion zone is found around the singularity where uncertainty in space-time diverged to infinity.

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

“…The constraints to the space-time line elements calculated in this paper are strictly obtained using a relativistic covariant formulation of the uncertainty principle in momentum and position 4-vectors. To obtain such constraints, GeUP [29] was applied to the FRW metric as an approximation to current cosmological models [31]. More specifically, a flat geometry condition was applied to the solution as it agrees with current observations [32].…”

“…The classical uncertainty principle can also be reformulated as a relativistic covariant form in terms of the proper space-time line element (𝜏 ) and Planck length, ℓ [29]. This reformulation allows its application as a mathematical constraint over GR geodesics without an explicit quantization of space-time.…”

“…This covariant form of the classical uncertainty principle can be applied to any GR solution. For example, a welldefined minimum space-time line element in a Schwarzschild black hole singularity at radial position R=0 was calculated in [29]:…”

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

“…Therefore, the classical FRW metric is incompatible with the uncertainty principle unless a t-dependent differential perturbation function, 𝜀, is introduced in the 𝑔 component of the metric, following a similar approach by semi-classical quantum gravity and developed for Minkowski space in [29]:…”

“…The term (1 + 𝛾) in inequality (7) takes a value of 2 for a particle at rest. After the calculation of the geodesic scalar and Christoffel connector as described in [29], one obtains inequality (7) as:…”