pseudo-Riemannian manifold

A pseudo-Riemannian manifoldMathworldPlanetmath is a manifold M together with a non degenerate (, symmetricMathworldPlanetmathPlanetmath sectionMathworldPlanetmath g of T20(M) (2-covariant tensor bundle over M).

Unlike with a Riemannian manifoldMathworldPlanetmath, g is not positive definitePlanetmathPlanetmath. That is, there exist vectors vTpM such that g(v,v)0.

A well known result from linear algebra ( permits us to make a change of basis such that in the new base g is represented by a diagonal matrixMathworldPlanetmath with -1 or 1 elements in the diagonalMathworldPlanetmath. If there are i, -1 elements in the diagonal and j, 1, the tensor is said to have signaturePlanetmathPlanetmath (i,j)

The signature will be invariant in every connected componentMathworldPlanetmath of M, but usually the restriction that it be a global invariant is added to the definition of a pseudo-Riemannian manifoldMathworldPlanetmath.

Unlike a Riemannian metric, some manifolds do not admit a pseudo-Riemannian metric.

Pseudo-Riemannian manifolds are crucial in Physics and in particular in General Relativity where space-time is modeled as a 4-pseudo Riemannian manifold with signature (1,3)11also referred to as (-+++).

Intuitively pseudo-Riemannian manifolds are generalizations of Minkowski’s space just as a Riemannian manifold is a generalization of a vector spaceMathworldPlanetmath with a positive definite metric.

Title pseudo-Riemannian manifold
Canonical name PseudoRiemannianManifold
Date of creation 2013-03-22 15:44:15
Last modified on 2013-03-22 15:44:15
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 10
Author cvalente (11260)
Entry type Definition
Classification msc 53Z05
Related topic EinsteinFieldEquations
Related topic SylvestersLaw
Related topic MinkowskiSpace
Related topic CategoryOfRiemannianManifolds
Defines pseudo-Riemannian geometry
Defines pseudo-Riemannian manifold