Discretized Finsler Structure: An Approach to Quantizing the First Fundamental Form

Tawfik, Abdel Nasser

doi:10.3390/ecu2023-14066

Cited by 3 publications

(5 citation statements)

References 28 publications

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“…We find that the additional curvature, whose Kretschann invariant scalar [18,19] assures its intrinsicality, essentiality, and reality, is associated with a maximal proper force which apparently emerges quantum-mechanical properties, such as discretizations, noncommutations, and jumps, especially in the relativistic quantum regimes [11,12]. The predicted maximal proper force is introduced as a new physical constant, a new fundamental invariant quantity observed in nature and appears in the fundamental equations of the additional curvatures [16,17]. The maximal proper force is related the maximal proper acceleration which was predicted by Caianellio, about four decades ago [20,21].…”

Section: Of 22mentioning

confidence: 87%

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A Quantum Geometric Approach for the Generalization of Spacetime Curvature

Tawfik,

Dabash

2023

Preprint

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Despite the successful reconciliation of special relativity and quantum mechanics (QM), a good century ago, general relativity (GR) in its current classical formulation still does not compile with QM. While GR gives precise sensical predictions at large scales, QM is obviously dominated at small scales! We think that the everlasting battle for exploring and understanding the Universe and for privileging a consistent perception of reality is therefore awaiting a consolidator rather than a conqueror. This should be capable of either unifying the two different benchmarks or at least bringing one closer to the other! The latter describes the consolidating approach, which is based on extending QM to relativistic energies and gravitational fields and generalizing Riemann to discretized Finsler geometry. Some properties of the additional geometric structures and curvatures which are disclosed by the proposed quantization and apparently overlooked in Einstein's GR, are analytically and numerically characterized. The analytical analyses introduce dynamics with possible nonlinear connections. We conclude that i) additional geometric structures and curvatures are intrinsic essential and ii) the spacetime at the quantum scale is no longer smooth or continuous. A maximal proper force is predicted, which maximally gravitationally accelerates a test particle and importantly manifests the quantum geometric nature of the generalized spacetime curvature.

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Section: Of 22mentioning

confidence: 87%

“…With a postive mass of a test particle, F could be reexpressed in coordinates and momenta instead of coordinates and velocities. This allows for a direct implication for RGUP on F and thereby the derivation of the generalized metric tensor, on which quantum-mechanical ingredients are imposed [14][15][16][17].…”

Section: Of 22mentioning

confidence: 99%

A Quantum Geometric Approach for the Generalization of Spacetime Curvature

Tawfik,

Dabash

2023

Preprint

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“…One substitutes Equation (28) into Equation (34) for the deformed Levi-Civita connection in terms of the classical one,…”

Section: Deformed Riemann Curvature Tensormentioning

confidence: 99%

“…With g ξζ g ξζ = 1, we replace g ξζ,ν ẍξ ẍζ by ẍ2 g ξζ,ν g ξζ in Equations ( 39) and (40). By substituting into Equation (34), the minimal length-induced correction to the Riemann curvature tensor is…”

Section: Deformed Riemann Curvature Tensormentioning

confidence: 99%

“…One of the authors of the present paper (A.T.), in his recent paper [34], introduced an approach to 'quantize' the fundamental tensor (metric tensor) and first fundamental form (line element) by extending the four-dimensional Riemann manifold to the eightdimensional Finsler manifold, in which the quadratic restriction on the length measure is relaxed. A.T. emphasizes that these results represent "quantum-induced corrections but not a full quantization".…”

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confidence: 99%

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On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor

Farouk,

Tawfik,

Tarabia

et al. 2023

Physics

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The minimal length conjecture is merged with a generalized quantum uncertainty formula, where we identify the minimal uncertainty in a particle’s position as the minimal measurable length scale. Thus, we obtain a quantum-induced deformation parameter that directly depends on the chosen minimal length scale. This quantum-induced deformation is conjectured to require the generalization of Riemannian spacetime geometry underlying the classical theory of general relativity to an eight-dimensional spacetime fiber bundle, which dictates the deformation of the line element, metric tensor, Levi-Civita connection, Riemann curvature tensor, etc. We calculate the deformation thus produced in the Levi-Civita connection and find it to explicitly and exclusively depend on the product of the minimum measurable length and the particle’s spacelike four-acceleration vector, L2x¨2. We find that the deformed Levi-Civita connection preserves all properties of its undeformed counterpart, such as torsion freedom and metric compatibility. Accordingly, we have constructed a deformed version of the Riemann curvature tensor whose expression can be factorized in all its terms with different functions of L2x¨2. We also show that the classical four-manifold status of being Riemannian is preserved when the quantum-induced deformation is negligible. We study the dependence of a parallel-transported tangent vector on the spacelike four-acceleration. We illustrate the impact of the minimal-length-induced quantum deformation on the classical geometrical objects of the general theory of relativity using the unit radius two-sphere example. We conclude that the minimal length deformation implies a correction to the spacetime curvature and its contractions, which is manifest in the additional curvature terms of the corrected Riemann tensor. Accordingly, quantum-induced effects endow an additional spacetime curvature and geometrical structure.

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On Possible Minimal Length Deformation of Metric Tensor and Affine Connection

T.¹,

Tawfik²,

Tarabia³

et al. 2023

Preprint

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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.

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Discretized Finsler Structure: An Approach to Quantizing the First Fundamental Form

Abstract: This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Cited by 3 publications

References 28 publications

A Quantum Geometric Approach for the Generalization of Spacetime Curvature

A Quantum Geometric Approach for the Generalization of Spacetime Curvature

On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor

On Possible Minimal Length Deformation of Metric Tensor and Affine Connection

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