In mathematics, the signature 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 gab of the metric tensor with respect to a basis. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis and thus can be used to classify the metric. It is denoted by three integers (v, p, r), where v is the number of positive eigenvalues, p is the number of negative ones and r is the number of zero eigenvalues of the metric tensor. It can also be denoted (v, p) implying r = 0, or as an explicit list of signs of eigenvalues such as (+, −, −, −) or (−, +, +, +) for the signatures (1, 3, 0) and (3, 1, 0), respectively. The choice of the variable names v and p reflects the convention in relativistic physics that v represents the number of time or virtual dimensions, and p the number of space or physical dimensions. The signature is said to be indefinite or mixed if both v and p are nonzero, and degenerate if r is nonzero. A Riemannian metric is a metric with a positive definite signature (v, 0). A Lorentzian metric is a metric with signature (p, 1), or (1, p). There is another notion of signature of a nondegenerate metric tensor given by a single number s defined as (v − p), where v and p are as above, which is equivalent to the above definition when the dimension n = v + p is given or implicit. For example, s = 1 − 3 = −2 for (+, −, −, −) and its mirroring s′ = −s = +2 for (−, +, +, +).
Definition The signature of a metric tensor is defined as the signature of the corresponding quadratic form. It is the number (v, p, 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 multiplicities. 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. By Sylvester's law of inertia, the numbers (v, p, r) are basis independent.
Properties
Signature and dimension By the spectral theorem a symmetric n × n matrix over the reals is always diagonalizable, and has therefore exactly n real eigenvalues (counted with algebraic multiplicity). Thus v + p = n = dim(V).
Sylvester's law of inertia: independence of basis choice and existence of orthonormal basis According to Sylvester's law of inertia, the signature of the scalar product (a.k.a. real symmetric bilinear form), g does not depend on the choice of basis. Moreover, for every metric g of signature (v, p, r) there exists a basis such that gab = +1 for a = b = 1, ..., v, gab = −1 for a = b = v + 1, ..., v + p and gab = 0 otherwise. It follows that there exists an isometry (V1, g1) → (V2, g2) if and only if the signatures of g1 and g2 are equal. Likewise the signature is equal for two congruent matrices and classifies a matrix up to congruency. Equivalently, the signature is constant on the orbits of the general linear group GL(V) on the space of symmetric rank 2 contravariant tensors S2V∗ and classifies each orbit.
Geometrical interpretation of the indices The number v (resp. p) is the maximal dimension of a vector subspace on which the scalar product g is positive-definite (resp. negative-definite), and r is the dimension of the radical of the scalar product g or the null subspace of symmetric matrix gab of the scalar product. Thus a nondegenerate scalar product has signature (v, p, 0), with v + p = n. A duality of the special cases (v, p, 0) correspond to two scalar eigenvalues which can be transformed into each other by the mirroring reciprocally.
Examples
Matrices The signature of the n × n identity matrix is (n, 0, 0). The signature of a diagonal matrix is the number of positive, negative and zero numbers on its main diagonal. The following matrices have both the same signature (1, 1, 0), therefore they are congruent because of Sylvester's law of inertia:
( 1 0 0 − 1 ) , ( 0 1 1 0 ) . {\displaystyle {\begin{pmatrix}1&0\\0&-1\end{pmatrix}},\quad {\begin{pmatrix}0&1\\1&0\end{pmatrix}}.}
Scalar products The standard scalar product defined on R n {\displaystyle \mathbb {R} ^{n}} has the n-dimensional signatures (v, p, r), where v + p = n and rank r = 0. In physics, the Minkowski space is a spacetime manifold R 4 {\displaystyle \mathbb {R} ^{4}} with v = 1 and p = 3 bases, and has a scalar product defined by either the g ˇ {\displaystyle {\check {g}}} matrix:
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