Metric signature

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The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted. If the matrix is n×n, the possible number of positive signs may take any value p from 0 to n. The signature may be denoted either by a pair of integers such as (p, q), or as an explicit list such as (−,+,+,+).

The signature is said to be indefinite if both p and q are non-zero. A Riemannian metric is a metric with a (positive) definite signature. A Lorentzian metric is one with signature (p, 1) (or sometimes (1, q)).

There is also another definition of signature which uses a single number s defined as the codimension of the biggest (positive or negative) definite subspace. Using the nondegenerate metric tensor from above, the signature is simply the minimum of p and q. For example (+,−,−,−) and (−,+,+,+) have both signature s = 1.

[edit] Signature in physics

In theoretical physics, spacetime is modeled by a pseudo-Riemannian manifold. The signature counts how many dimensions of spacetime have a time-like or space-like character, in the sense defined by special relativity.

The spacetimes with purely space-like directions are said to have Euclidean signature while the spacetimes with signature like (3,1) are said to have Minkowskian signature. The more general signatures are often referred to as Lorentzian signature although this term is often used as a synonyn of the Minkowskian signature.

See also pseudo-Riemannian manifold.