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Lorentzian geometry is the geometry of Minkowski spacetime, hence essentially of a Euclidean space, but equipped not with the standard Euclidean Riemannian metric of spacetime signature $(+,+,+,\ldots,+)$ (which yields Euclidean geometry) but with the pseudo-Riemannian metric of signature $(-,+,+,\ldots,+)$. This is in particular the context of the theory of physics called “theory of special relativity”, and it is locally the context of the “theory of general relativity”.
Note that while Lorentzian geometry is analogous to Euclidean geometry (as Minkowski space is analogous to a Euclidean space), a Lorentzian manifold is analogous to a Riemannian manifold. Thus, one might use ‘Lorentzian geometry’ analogously to Riemannian geometry (and insist on Minkowski geometry for our topic here), but usually one skips all the way to pseudo-Riemannian geometry (which studies pseudo-Riemannian manifolds, including both Riemannian and Lorentzian manifolds).
Since the isometry group of Minkowski spacetime is the Poincaré group, and since Minkowski spacetime is the quotient of the Poincaré group by the Lorentz group-subgroup, Lorentzian geometry and the study of Lorentzian manifolds is to a large extent the Cartan geometry of the Poincaré group.
Promoting this perspective from global to local symmetry yields the first order formulation of gravity. Promoting this perspective form the Poincaré group to the super Poincaré group yields supergravity. Promoting it further to the Lie n-algebra extensions of the super Poincaré group (from the brane scan/brane bouquet) yields type II supergravity, heterotic supergravity and 11-dimensional supergravity in higher Cartan geometry-formulation (D'Auria-Fré formulation of supergravity).
A synthetic set of axioms for Lorentzian geometry was developed in these articles
Edwin B. Wilson & Gilbert N. Lewis (1912) The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics Proceedings of the American Academy of Arts and Sciences 48:387–507
James Cockle, Synthetic Spacetime, (web)
Introductions and surveys:
Christian Bär, Lorentzian Geometry, lecture notes (2004) [pdf, pdf]
Javayoles, Sánchez, An introduction to Lorentzian Geometry and its application, 2010 (pdf)
Graciela Birman, Katsumi Nomizu, Trigonometry in Lorentzian geometry, The American Mathematical Monthly Vol. 91, No. 9 (Nov., 1984), pp. 543-549 (JSTOR)
See also
Last revised on November 24, 2023 at 20:00:47. See the history of this page for a list of all contributions to it.