On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor
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 lengt...
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MDPI AG
2023-10-01
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| Online Access: | https://www.mdpi.com/2624-8174/5/4/64 |
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| author | Fady Tarek Farouk Abdel Nasser Tawfik Fawzy Salah Tarabia Muhammad Maher |
| author_facet | Fady Tarek Farouk Abdel Nasser Tawfik Fawzy Salah Tarabia Muhammad Maher |
| author_sort | Fady Tarek Farouk |
| collection | DOAJ |
| description | 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, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><msup><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mn>2</mn></msup></mrow></semantics></math></inline-formula>. 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><msup><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mn>2</mn></msup></mrow></semantics></math></inline-formula>. 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. |
| first_indexed | 2024-03-08T20:25:57Z |
| format | Article |
| id | doaj.art-5655929b405f4417aca636b0553fa70a |
| institution | Directory Open Access Journal |
| issn | 2624-8174 |
| language | English |
| last_indexed | 2024-03-08T20:25:57Z |
| publishDate | 2023-10-01 |
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| spelling | doaj.art-5655929b405f4417aca636b0553fa70a2023-12-22T14:33:10ZengMDPI AGPhysics2624-81742023-10-0154983100210.3390/physics5040064On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature TensorFady Tarek Farouk0Abdel Nasser Tawfik1Fawzy Salah Tarabia2Muhammad Maher3Physics Department, Faculty of Science, Helwan University, Ain Helwan, Cairo 11792, EgyptResearch Center, Faculty of Engineering, Future University in Egypt (FUE), New Cairo 11835, EgyptPhysics Department, Faculty of Science, Helwan University, Ain Helwan, Cairo 11792, EgyptPhysics Department, Faculty of Science, Helwan University, Ain Helwan, Cairo 11792, EgyptThe 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, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><msup><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mn>2</mn></msup></mrow></semantics></math></inline-formula>. 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><msup><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mn>2</mn></msup></mrow></semantics></math></inline-formula>. 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.https://www.mdpi.com/2624-8174/5/4/64modified gravityminimal length scalegeneralized uncertainty principlegeneral relativitydeformed phase space |
| spellingShingle | Fady Tarek Farouk Abdel Nasser Tawfik Fawzy Salah Tarabia Muhammad Maher On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor Physics modified gravity minimal length scale generalized uncertainty principle general relativity deformed phase space |
| title | On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor |
| title_full | On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor |
| title_fullStr | On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor |
| title_full_unstemmed | On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor |
| title_short | On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor |
| title_sort | on possible minimal length deformation of metric tensor levi civita connection and the riemann curvature tensor |
| topic | modified gravity minimal length scale generalized uncertainty principle general relativity deformed phase space |
| url | https://www.mdpi.com/2624-8174/5/4/64 |
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