Born reciprocity and discretized Finsler structure: An approach to quantize GR curvature tensors on three-sphere

Abstract

At relativistic energies and finite magnetic fields, the noncommutative relation of distance and momentum, the Heisenberg uncertainty principle, the fundamental theory of quantum mechanics, is conjectured to get modifications. Results from various rigorous approaches to quantum gravity, such as string theory, loop quantum gravity and doubly special relativity support the generalization of the noncommutative relation of the distance and momentum operators and the emergence of a minimal measurable length. With the relativistic four-dimensional generalized uncertainty principle (RGUP) in curved spacetime and Born reciprocity principle, the distance–momentum duality symmetry, we suggest to generalize Riemann to Finsler geometry. The Finsler structure allows the direct implementation of RGUP with its quantum-mechanical nature on a free particle with mass [Formula: see text], so that the Finsler structure [Formula: see text] can be expressed as [Formula: see text], from which the quantized fundamental tensor can be deduced. We present a systematic analytic and numerical evaluation of the additional geometric structures and connections which exclusively emerged from the proposed quantization approach on three-sphere. When limiting the discussion on the Einstein tensor, we find that the emerged curvatures, i.e. additional sources of gravitation, are dominant almost everywhere on the three-sphere. The nature of those curvatures is radically distinct from the ones of the classical Einstein tensor. For instance, the additional curvatures are no longer smooth or continuous.

Finslerian wormhole solution in the framework of modified gravity

“…By substituting expressions equation ( 35)- (38) and equation (29) in equation (8), we can determine the scalar curvature for the Finslerian wormhole space-time as…”

“…Abdel Nasser Tawfikʼs contributions lie in the conceptualization of Born reciprocity within discretized Finsler structures. 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.…”

Singularity attenuation with quantum‐mechanically revisited metric tensor

“…As for the quantum-mechanically revisited theory of general relativity (GR), we recall that the generalized geometric structures were introduced by one of the authors (AT) (Farouk et al 2023;Tawfik 2023aTawfik , 2023bTawfik & Dabash 2023a, 2023b. Also, the same author (AT) constructed a theory for the kinematics of quantum particles in the generalized Finsler-Hamilton space, in which additional geometric structures and curvatures emerge.…”

“…Also, the same author (AT) constructed a theory for the kinematics of quantum particles in the generalized Finsler-Hamilton space, in which additional geometric structures and curvatures emerge. As for the gravitationally imposed revision of quantum mechanics (QM), we recall the relativistic generalized uncertainty principle (RGUP) (Farouk et al 2023;Tawfik 2023aTawfik , 2023b, the fundamental theory of QM. RGUP is introduced to count for finite gravitational fields and relativistic energies.…”

On pseudo-Riemannian quartics in Finsler geometry

“…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.…”