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unification_of_spacetime_symmetries

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unification_of_spacetime_symmetries [2016/12/30 09:03] jakobadmin [Methods] |
unification_of_spacetime_symmetries [2017/12/06 09:33] (current) |
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[{{ :cartangeometrydiagram.jpg?nolink |Source: https://arxiv.org/pdf/1103.0731v1.pdf}}] | [{{ :cartangeometrydiagram.jpg?nolink |Source: https://arxiv.org/pdf/1103.0731v1.pdf}}] | ||

- | Euclidean geometry can be understood in the sense of [[http://notes.jakobschwichtenberg.com/doku.php?id=unification_of_spacetime_symmetries:methods:klein_geometry|Klein]] as the study of invariants of subgroups of the Euclidean symmetry group. Thus, if we want to study generalized geometries, we can simply allow different symmetry groups instead of the Euclidean symmetry group. This is the basic idea of [[http://notes.jakobschwichtenberg.com/doku.php?id=unification_of_spacetime_symmetries:methods:klein_geometry|Klein geometry]]. However, one can generalize Euclidean geometry in a different direction by allowing instead of the flat Euclidean space curved spaces, which are commonly called Riemannian manifolds. The defining feature of a manifold is that it looks in the neighborhood of each point like the usual Euclidean space. A possible generalization is then, of course, that we also allow more general tangent spaces instead of the Euclidean space. This is idea behind [[[[unification_of_spacetime_symmetries:methods:cartan_geometry]]]. An important example is that the tangent space is the curved deSitter space, which is the [[https://arxiv.org/abs/gr-qc/0611154|idea behind MacDowell-Mansouri gravity]]. | + | Euclidean geometry can be understood in the sense of [[http://notes.jakobschwichtenberg.com/doku.php?id=unification_of_spacetime_symmetries:methods:klein_geometry|Klein]] as the study of invariants of subgroups of the Euclidean symmetry group. Thus, if we want to study generalized geometries, we can simply allow different symmetry groups instead of the Euclidean symmetry group. This is the basic idea of [[http://notes.jakobschwichtenberg.com/doku.php?id=unification_of_spacetime_symmetries:methods:klein_geometry|Klein geometry]]. However, one can generalize Euclidean geometry in a different direction by allowing instead of the flat Euclidean space curved spaces, which are commonly called Riemannian manifolds. The defining feature of a manifold is that it looks in the neighborhood of each point like the usual Euclidean space. A possible generalization is then that we also allow more general tangent spaces instead of the Euclidean space. This is idea behind [[[[unification_of_spacetime_symmetries:methods:cartan_geometry]]]. An important example is that the tangent space is the curved deSitter space, which is the [[https://arxiv.org/abs/gr-qc/0611154|idea behind MacDowell-Mansouri gravity]]. |

unification_of_spacetime_symmetries.txt · Last modified: 2017/12/06 09:33 (external edit)