Metric signature

The signature (p, q, r) of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix g_ab of the metric tensor with respect to a basis. Alternatively, it can be defined as the dimensions of a maximal positive, negative and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis. The signature thus classifies the metric up to a choice of basis. The signature is often denoted by a pair of integers (p, q) implying r = 0 or as an explicit list of signs of eigenvalues such as (+, −, −, −) or (−, +, +, +) for the signature (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 notion of signature of a nondegenerate metric tensor given by a single number s defined as p − q, where p and q are as above, which is equivalent to the above definition when the dimension n = p + q is given or implicit. For example, s = 1 − 3 = −2 for (+, −, −, −) and s = 3 − 1 = +2 for (−, +, +, +).

The signature of a metric tensor is defined as the signature of the corresponding quadratic form. It is the number (p, q, r) of positive, negative and zero eigenvalues of any matrix (i.e. in any basis for the underlying vector space) representing the form, counted with their algebraic multiplicity. Usually, r = 0 is required, which is the same as saying a metric tensor must be nondegenerate, i.e. no nonzero vector is orthogonal to all vectors.

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