Minimizing Curvature in Euclidean and Lorentz Geometry
In this paper, an interesting symmetry in Euclidean geometry, which is broken in Lorentz geometry, is studied. As it turns out, attempting to minimize the integral of the square of the scalar curvature leads to completely different results in these two cases. The main concern in this paper is about...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-08-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/8/1433 |
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| author | Martin Tamm |
| author_facet | Martin Tamm |
| author_sort | Martin Tamm |
| collection | DOAJ |
| description | In this paper, an interesting symmetry in Euclidean geometry, which is broken in Lorentz geometry, is studied. As it turns out, attempting to minimize the integral of the square of the scalar curvature leads to completely different results in these two cases. The main concern in this paper is about metrics in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula>, which are close to being invariant under rotation. If we add a time-axis and let the metric start to rotate with time, it turns out that, in the case of (locally) Euclidean geometry, the (four-dimensional) scalar curvature will increase with the speed of rotation as expected. However, in the case of Lorentz geometry, the curvature will instead initially decrease. In other words, rotating metrics can, in this case, be said to be less curved than non-rotating ones. This phenomenon seems to be very general, but because of the enormous amount of computations required, it will only be proved for a class of metrics which are close to the flat one, and the main (symbolic) computations have been carried out on a computer. Although the results here are purely mathematical, there is also a connection to physics. In general, a deeper understanding of Lorentz geometry is of fundamental importance for many applied problems. |
| first_indexed | 2024-03-10T08:20:19Z |
| format | Article |
| id | doaj.art-3dea23e925014d34ba6bc18066dbeace |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T08:20:19Z |
| publishDate | 2021-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-3dea23e925014d34ba6bc18066dbeace2023-11-22T10:01:26ZengMDPI AGSymmetry2073-89942021-08-01138143310.3390/sym13081433Minimizing Curvature in Euclidean and Lorentz GeometryMartin Tamm0Department of Mathematics, University of Stockholm, 106 91 Stockholm, SwedenIn this paper, an interesting symmetry in Euclidean geometry, which is broken in Lorentz geometry, is studied. As it turns out, attempting to minimize the integral of the square of the scalar curvature leads to completely different results in these two cases. The main concern in this paper is about metrics in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula>, which are close to being invariant under rotation. If we add a time-axis and let the metric start to rotate with time, it turns out that, in the case of (locally) Euclidean geometry, the (four-dimensional) scalar curvature will increase with the speed of rotation as expected. However, in the case of Lorentz geometry, the curvature will instead initially decrease. In other words, rotating metrics can, in this case, be said to be less curved than non-rotating ones. This phenomenon seems to be very general, but because of the enormous amount of computations required, it will only be proved for a class of metrics which are close to the flat one, and the main (symbolic) computations have been carried out on a computer. Although the results here are purely mathematical, there is also a connection to physics. In general, a deeper understanding of Lorentz geometry is of fundamental importance for many applied problems.https://www.mdpi.com/2073-8994/13/8/1433rotationcurvatureLorentz geometry |
| spellingShingle | Martin Tamm Minimizing Curvature in Euclidean and Lorentz Geometry Symmetry rotation curvature Lorentz geometry |
| title | Minimizing Curvature in Euclidean and Lorentz Geometry |
| title_full | Minimizing Curvature in Euclidean and Lorentz Geometry |
| title_fullStr | Minimizing Curvature in Euclidean and Lorentz Geometry |
| title_full_unstemmed | Minimizing Curvature in Euclidean and Lorentz Geometry |
| title_short | Minimizing Curvature in Euclidean and Lorentz Geometry |
| title_sort | minimizing curvature in euclidean and lorentz geometry |
| topic | rotation curvature Lorentz geometry |
| url | https://www.mdpi.com/2073-8994/13/8/1433 |
| work_keys_str_mv | AT martintamm minimizingcurvatureineuclideanandlorentzgeometry |