nLab signature of a metric

Idea

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

If $(X,g)$ 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.

Related concepts

• Lorentzian geometry

• Euclidean gravity, Wick rotation

• D=12 supergravity

• D=14 supersymmetry

• relativistic field theory, Euclidean field theory