nLab
Lorentzian geometry

Contents

Context

Riemannian geometry

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

Lorentzian geometry is the geometry of Minkowski spacetime, hence essentially of a Euclidean space, but equipped not with the standard Euclidean Riemannian metric of 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).

Lorentzian Cartan geometry and first order gravity

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).

geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

References

Introductions and surveys include

  • Javayoles, Sánchez, An introduction to Lorentzian Geometry and its application, 2010 (pdf)

  • Christian Bär, N. Ginaux, Frank Pfäffle, I. Lorentzian geometry (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 September 30, 2013 at 13:14:04. See the history of this page for a list of all contributions to it.