Minimal length discretization and properties of modified metric tensor and geodesics

Tawfik, Abdel Nasser; T., Farouk, Fady; Tarabia, F. Salah; Maher, Muhammad

doi:10.1142/9789811269776_0339

Cited by 6 publications

(5 citation statements)

References 17 publications

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“…In this respect, we emphasize that the present script extends the approach introduced in Tawfik et al (2021b), by deriving the minimum measurable length from the four‐dimensional relativistic GUP instead of the three‐dimensional non‐relativistic GUP. by introducing Finsler geometry with tangent bundle and structure, where the metric tensor in the Hessian of the structure instead of an arbitrary manifold. by suggesting the minimum measurable length as an adequate quantity suitable to GR instead of the arbitrary three‐dimensional minimum measurable length. …”

Section: Methods and Formalismmentioning

confidence: 67%

“…The present script suggests to impose the deformed noncommutative Heisenberg algebra,

\left[{x}^{\mu },{p}_{\nu}\right]=i\mathrm{\hslash}\left[{\delta}_{\nu}^{\mu }+f{\left[p\right]}_{\nu}^{\mu}\right]

, with

\mu, \nu \in \left\{\mathrm{0,1,2,3}\right\}

, on the GR's curved spacetime manifold, to which a minimum length uncertainty is also applied. We focus on the metric tensor

{g}_{\mu \nu}

and find that an additional term including novel.quantum‐mechanical, such as,

\mathrm{\hslash}

(Planck's constant), gravitational, such as,

G

(Newton's gravitational constant), and geometrical imprints, such as,

{\ddot{x}}^{\lambda }{\ddot{x}}_{\lambda }

(spacelike second derivative of 1‐form coordinates) are added to

{g}_{\mu \nu}

Tawfik et al (2021a), (2021b). …”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On possible quantization of the fundamental tensor in the relativistic regime

Tawfik

2022

Astron Nachr

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Inspired by the quantum gravity prescriptions, for instance, string theory, doubly-special relativity, and quantum loop gravity, we apply the generalized relativistic noncommutative Heisenberg algebra on the eight-dimensional Finsler manifold, the natural generalization of the four-dimensional Riemann manifold. The Finsler manifold is conjectured to relax the quadratic restriction on the length measure, especially in the relativistic regime. The additional curvature coupled with the Finsler manifold is assumed to mimic the quantization on the Riemann manifold. The second ingredient we suggest is the existence of a minimum measurable length, whose value is determined from the relativistic generalized uncertainty principle, and then applying in to the coordinates of the Finsler manifold. The quantum-induced corrections to the fundamental tensor are determined under extreme conditions of a relativistic energy. Whether this proposal is adequate for a full quantization, the various limitations of the proposed theory are elaborated and discussed, especially the limitations of the curved spacetime model assumed for the line element measure. We have conservatively formulated our conclusion that the quantization of the fundamental tensor might be conditioned to an adequate quantization of the line element measure, at least the one on the eight-dimensional Finsler manifold. K E Y W O R D Salternatives to general relativity, general relativity in relativistic quantum regime, generalized noncommutative Heisenberg algebra, reconciling principles of quantum mechanics with general relativity

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Section: Methods and Formalismmentioning

confidence: 67%

“…The present script suggests to impose the deformed noncommutative Heisenberg algebra,

\left[{x}^{\mu },{p}_{\nu}\right]=i\mathrm{\hslash}\left[{\delta}_{\nu}^{\mu }+f{\left[p\right]}_{\nu}^{\mu}\right]

, with

\mu, \nu \in \left\{\mathrm{0,1,2,3}\right\}

, on the GR's curved spacetime manifold, to which a minimum length uncertainty is also applied. We focus on the metric tensor

{g}_{\mu \nu}

and find that an additional term including novel.quantum‐mechanical, such as,

\mathrm{\hslash}

(Planck's constant), gravitational, such as,

G

(Newton's gravitational constant), and geometrical imprints, such as,

{\ddot{x}}^{\lambda }{\ddot{x}}_{\lambda }

(spacelike second derivative of 1‐form coordinates) are added to

{g}_{\mu \nu}

Tawfik et al (2021a), (2021b). …”

Section: Introductionmentioning

confidence: 99%

On possible quantization of the fundamental tensor in the relativistic regime

Tawfik

2022

Astron Nachr

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“…from the spacetime discretization on the relativistic eight‐dimensional manifold, the quantum‐induced revisiting fundamental tensor is suggested as Tawfik et al (2021a), (2021b)

{\tilde{g}}_{\mu \nu}=\left(1+\mathcal{T}\kern0.1em {\left|\ddot{x}\right|}^2\right)\kern0.1em {g}_{\mu \nu}.

…”

Section: Methods and Formalismmentioning

confidence: 99%

On quantum‐induced revisiting Einstein tensor in the relativistic regime

Tawfik

2022

Astron Nachr

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For a rigorous approach to reconcile principles of general relativity (GR) and quantum mechanics (QM), we suggest applying the generalized relativistic noncommutative Heisenberg algebra on spacetime coordinates and momenta of a test particle on curved Riemann geometry manifold, which in turn is extended to an eight-dimensional manifold. The natural generalization of the four-dimensional Riemann geometry is the Finsler geometry, in which the quadratic restriction on the length measure is relaxed. With the minimum measurable length derived from the relativistic generalized four-dimensional uncertainty principle, the quantum-induced corrections to the fundamental tensor could be determined in the relativistic regime. Accordingly, the affine connections could be revisited. With the quantum-induced revisiting Riemann curvature tensor and its contractions, the Ricci curvature tensor, and then the Ricci scalar, we have been able to construct a quantum-induced revision of the Einstein tensor, in which besides quantum-induced corrections, additional geometric structures emerge. On the surface of the 2-sphere, we compared the quantum-induced revisiting and non-revisiting Einstein tensor and concluded that the difference between both versions strongly depends on the minimum measurable length and the local geodesics of the test particle through the additional curvature. K E Y W O R D Salternatives to general relativity, general relativity in relativistic quantum regime, generalized noncommutative Heisenberg algebra, reconciling principles of quantum mechanics with general relativity

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“…The latter are essential components of the underlying quantum theory. In other words, GUP helps explaining the origin of the gravitational field and how a particle behaves in it [23,24].…”

Section: Of 15mentioning

confidence: 99%

Minimal Length Deformation of Spacetime Curvature

Maher¹,

Tawfik²,

Tarabia³

et al. 2023

Preprint

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When generalized noncommutative Heisenberg algebra accommodating impacts of finite gravitational fields as specified by loop quantum gravity, doubly–special relativity, and string theory, for instance, is thoughtfully applied to the eight-dimensional manifold (M8), the generalization of the Riemannian manifold becomes imminent. By constructing the deformed affine connections on a four-dimensional Riemannian manifold, we have determined the minimal length deformation of Riemann curvature tensor and its contractions, the Ricci curvature tensor, and Ricci scalar. Consequently, we have been able to construct the deformed Einstein tensor. As in Einstein’s classical theory of general relativity, we have proved that the covariant derivative of the deformed Einstein tensor vanishes, as well. We conclude that the minimal length correction suggests a correction to the spacetime curvature which is manifest in the additional curvature terms of the corrected Riemann tensor and its contractions. Accordingly, the spacetime curvature endows additional curvature and geometrical structure.

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Minimal length discretization and properties of modified metric tensor and geodesics

Cited by 6 publications

References 17 publications

On possible quantization of the fundamental tensor in the relativistic regime

On possible quantization of the fundamental tensor in the relativistic regime

On quantum‐induced revisiting Einstein tensor in the relativistic regime

Minimal Length Deformation of Spacetime Curvature

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