nLab
signature of a metric

Contents

Contents

Idea

Given a pseudo-Riemannian manifold (X,g)(X, g) with metric tensor gg, its signature is the signature of gg as a quadratic form on any tangent space of XX, in the sense of signature of a quadratic form.

If (X,g)(X,g) is regarded as a model for spacetime, then this is also called spacetime signature.

Here the case of signature (,+,,+)(-,+, \cdots, +) corresponds to Lorentzian geometry, being the basis of Einstein-gravity in the sense of “general relativity”.

The case of signature (+,+,,+)(+,+, \cdots, +) corresponds to Riemannian geometry (as opposed to pseudo-Riemannian). In a context of spacetime models this pertains for instance to the usual fibers in KK-compactifications, or to all of spacetime after “Wick rotation” (Euclidean gravity).

Spacetime signatures of the form (,,,+,,+)(-,\cdots,-, +,\cdots, +) have been considered in the context of D=12 supergravity and D=14 supersymmetry.

Created on November 28, 2020 at 07:53:22. See the history of this page for a list of all contributions to it.