Born reciprocity and relativistic generalized uncertainty principle in Finsler structure: Fundamental tensor in discretized curved spacetime

Abstract

The main findings of the generalized uncertainty principle (GUP), the phenomenological approach, for instance, the emergence of a minimal measurable length uncertainty, are obtained in various versions from theories of quantum gravity, such as string theory, loop quantum gravity, doubly special relativity and black hole physics. GUP counts for impacts of relativistic energies and finite gravitational fields on the fundamental theories of quantum mechanics (QM), the noncommutation and measurement uncertainty. Utilizing GUP in reconciling principles of general relativity (GR) and QM, thereby enables to draw convincing conclusions about quantum gravity. To resolve the shortcuts reported with the nonrelativistic three-dimensional GUP, namely, violation of Lorentz covariance, dependence on frame of reference, and violation of the linear additional law of momenta, we introduce relativistic four-dimensional generalized uncertainty principle (RGUP) to curved spacetime. To unify GR and QM, we apply the Born reciprocity principle (BRP), distance-momentum duality symmetry and RGUP to estimate the fundamental tensor in discretized curved spacetime. To this end, we generalize Riemann geometry. The Finsler geometry, which is characterized by manifold and Finsler structure, allows to directly apply RGUP to the Finsler structure of a free particle so that [Formula: see text] can be expressed as [Formula: see text], from which the metric tensor in discretized Riemann spacetime could be deduced. We conclude that [Formula: see text] is homogeneous with degree [Formula: see text] in [Formula: see text], while [Formula: see text] is [Formula: see text]-homogeneous resulting in [Formula: see text]. Despite, the astonishing similarity with the conformal transformation, know as Weyl tensor, this study suggests that principles of QMs could be unambiguously imposed on the resulting fundamental tensor. Also, we conclude that the features of Finsler geometry assumed in this study are likely the ones of the duel Hamilton geometry.

A Quantum Geometric Approach for the Generalization of Spacetime Curvature

“…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 combination of both attempts results in discretized quantum geometry, additional curvature, and importantly generalized fundamental tensor. The resulting generalized fundamental tensor, which becomes quantum [11,12] and dynamical [13], represents Disclaimer/Publisher's Note: The statements, opinions, and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions, or products referred to in the content.…”

On pseudo-Riemannian quartics in Finsler geometry

“…Here, χ ijkl is a constitutive pseudo-tensor that describes the physical parameters of media, see [11]. The field equations (5) together with the constitutive relation (7) form a well-posed system of partial differential equations. This system is completely independent of the geometric structure of the manifold.…”

“…Possible pseudo-Finsler modifications of gravity were discussed in [5]. For recent progress in the Finsler construction in quantum gravity; see [6,7]. Finsler geometry is a widely used construction in quantum field theories as well.…”

Finslerian wormhole solution in the framework of modified gravity

“…Tawfik et al [29] have explored approaches to quantizing curvature tensors in the context of GR on three-sphere models. Furthermore, Tawfik et al [30] have investigated the implications of Born reciprocity and the relativistic generalized uncertainty principle within Finsler structures, analyzing their influence on emerged curvatures and their role in regulating space-time and addressing initial singularities. Additionally, Tawfik et al [31] have examined the behavior of timelike geodesic congruence within the simplest solutions derived from the Einstein field equations.…”