nLab Lorentzian geometry

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 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).

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 context	gauge group	stabilizer subgroup	local model space	local geometry	global geometry	differential cohomology	first order formulation of gravity
differential geometry	Lie group/algebraic group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	Klein geometry	Cartan geometry	Cartan connection
examples	Euclidean group $Iso(d)$	rotation group $O(d)$	Cartesian space $\mathbb{R}^d$	Euclidean geometry	Riemannian geometry	affine connection	Euclidean gravity
	Poincaré group $Iso(d-1,1)$	Lorentz group $O(d-1,1)$	Minkowski spacetime $\mathbb{R}^{d-1,1}$	Lorentzian geometry	pseudo-Riemannian geometry	spin connection	Einstein gravity
	anti de Sitter group $O(d-1,2)$	$O(d-1,1)$	anti de Sitter spacetime $AdS^d$				AdS gravity
	de Sitter group $O(d,1)$	$O(d-1,1)$	de Sitter spacetime $dS^d$				deSitter gravity
	linear algebraic group	parabolic subgroup/Borel subgroup	flag variety	parabolic geometry
	conformal group $O(d,t+1)$	conformal parabolic subgroup	Möbius space $S^{d,t}$		conformal geometry	conformal connection	conformal gravity
supergeometry	super Lie group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	super Klein geometry	super Cartan geometry	Cartan superconnection
examples	super Poincaré group	spin group	super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$	Lorentzian supergeometry	supergeometry	superconnection	supergravity
	super anti de Sitter group		super anti de Sitter spacetime
higher differential geometry	smooth 2-group $G$	2-monomorphism $H \to G$	homotopy quotient $G//H$	Klein 2-geometry	Cartan 2-geometry
	cohesive ∞-group	∞-monomorphism (i.e. any homomorphism) $H \to G$	homotopy quotient $G//H$ of ∞-action	higher Klein geometry	higher Cartan geometry	higher Cartan connection
examples			extended super Minkowski spacetime		extended supergeometry		higher supergravity: type II, heterotic, 11d

Related concepts

• timelike curve, spacelike curve, lightlike curve

• causet

• Lorentzian manifold

• hyperbolic geometry

References

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

• Geometry Group Berlin, Lorentzian geometry