Metric Signature

Metric Signature

The signature of a metric tensor (or more generally a symmetric bilinear form, thought of as a quadratic form) is the number of positive, negative and zero eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted. If the matrix of the metric tensor is n × n, then the number of positive, negative and zero eigenvalues p, q and r may take values from 0 to n with p + q + r = n. The signature may be denoted by a pair of integers (p, q) implying r = 0, a triple (p, q, r), or as an explicit list such as (+, −, −, −) or (−, +, +, +), in this case (1, 3) resp. (3, 1).

The signature is said to be indefinite or mixed if both p and q are nonzero, and degenerate if r is nonzero. A Riemannian metric is a metric with a (positive) definite signature. A Lorentzian metric is one with signature (p, 1), or (1, q).

There is another definition of signature of a nondegenerate metric tensor given by a single number s defined as p − q, where p and q are as above. For example, s = 1 − 3 = −2 for (+, −, −, −) and s = 3 − 1 = +2 for (−, +, +, +).