Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

On Possible Minimal Length Deformation of Metric Tensor and Affine Connection

Version 1 : Received: 6 June 2023 / Approved: 6 June 2023 / Online: 6 June 2023 (09:33:57 CEST)

A peer-reviewed article of this Preprint also exists.

Farouk, F.T.; Tawfik, A.N.; Tarabia, F.S.; Maher, M. On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor. Physics 2023, 5, 983-1002. Farouk, F.T.; Tawfik, A.N.; Tarabia, F.S.; Maher, M. On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor. Physics 2023, 5, 983-1002.

Abstract

When the minimal length approach emerging from noncommutative Heisenberg algebra, generalized uncertainty principle (GUP), and thereby integrating gravitational fields to this fundamental theory of quantum mechanics (QM) is thoughtfully extended to Einstein field equations, the possible deformation of the metric tensor could be suggested. This is a complementary term combining the effects of QM and general relativity (GR) and comprising noncommutative algebra together with maximal spacelike four–acceleration. This deformation compiles with GR as curvature in relativistic eight–dimensional spacetime tangent bundle, generalization of Riemannian spacetime, is the recipe applied to derive the deformed metric tensor. This dictates how the affine connection on Riemannian manifold is straightforwardly deformed. We have discussed the symmetric property of deformed metric tensor and affine connection. Also, we have evaluated the dependence of a parallel transported tangent vector on the spacelike four–acceleration given in units of L, where L = rℏcG3 is a universal constant, c is speed of light, and ℏ is Planck constant, and G is Newton’s gravitational constant.

Keywords

deformed theories of gravity; Noncommutative geometry; Curved spacetime; Relativity and gravitation

Subject

Physical Sciences, Theoretical Physics

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