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 sometimes (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 (−, +, +, +).

Definition

Let A be a symmetric matrix of reals. More generally, the metric signature (p, q, r) of A is a group of three natural numbers defined as the number of positive, negative and zero eigenvalues of the matrix counted with regard to their algebraic multiplicity. When r is nonzero the matrix A is called degenerate.

If φ is a scalar product on a finite-dimensional vector space V, the signature of V is the signature of the matrix that represents φ with respect to a chosen basis. According to Sylvester's law of inertia, the signature does not depend on the choice of basis.

Properties

Spectral theorem

Due to the spectral theorem a symmetric matrix of reals is always diagonalizable. Moreover, it has exactly n eigenvalues (counted according by their algebraic multiplicity). Thus p + q + r = n.

Sylvester's law of inertia

According to Sylvester's law of inertia two scalar products are isometric if and only if they have the same signature. This means that the signature is an invariant for scalar products on isometric transformations. In the same way two symmetric matrices are congruent if and only if they have the same signature.

Geometrical interpretation of the indices

The numbers p and q are the dimensions of the two vector subspaces on which the scalar product is positive-definite and negative-definite respectively, and r is the dimension of the radical of the scalar product φ or the null subspace of symmetric matrix A of the bilinear form. Thus a non degenerate scalar product has signature (p, q, 0), with p + q = n. So the values p, q and r are also called the dimensions of the positive-definite, negative-definite and null vector subspaces of the whole vector space V that correspond to the matrix A. The special cases (n, 0, 0) and (0, n, 0) correspond to the two equivalent vector spaces on which the scalar product is positive-definite and negative-definite respectively, and can be transformed into each other by negating their scalar product.

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:

$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.$

Scalar products

The standard scalar product defined on $\mathbb{R}^n$ has the signature (n, 0, 0). A scalar product has this signature if and only if it is a positive definite scalar product.

A negative definite scalar product has the signature (0, n, 0). A positive semi-definite scalar product has a signature (n, 0, m).

The Minkowski space is $\R^4$ and has a scalar product defined by the matrix

$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}$

and has signature (1, 3, 0). Sometimes it is used with the opposite signs, thus obtaining the signature (3, 1, 0).

How to compute the signature

There are some methods for computing the signature of a matrix.

• For any nondegenerate symmetric matrix of n × n, diagonalize it (or find all of eigenvalues of it) and count the number of positive and negative signs, and get p and q = n − p, they may take a pair of values from 0 to n, then the signature will be s = p − q.
• The sign of the roots of the characteristic polynomial may be determined by Cartesius' sign rule as long as all roots are reals.
• Lagrange algorithm avails a way to compute an orthogonal basis, and thus compute a diagonal matrix congruent (thus, with the same signature) to the other one: the signature of a diagonal matrix is the number of positive, negative and zero elements on its diagonal.
• According to Jacobi's criterion, a symmetric matrix is positive-definite if and only if all the determinants of its main minors are positive.

Signature in physics

In theoretical physics, spacetime is modeled by a pseudo-Riemannian manifold. The signature counts how many time-like or space-like characters are in the spacetime, in the sense defined by special relativity: the Riemannian metric is positive definite on the space-like subspace, and negative definite on the time-like subspace.

In the specific case of the Minkowski metric, whose metric has coordinates

ds² = dx² + dy² + dz² − c²dt² ,

the metric signature is (3, 1, 0), since it is positive definite in the x, y and z directions, and negative definite in the time direction.

The spacetimes with purely space-like directions (i.e., all positive definite) are said to have Euclidean signature, while the spacetimes with signature (3, 1) (i.e., (3, 1, 0)) are said to have Minkowskian signature in analogy to the Minkowski metric discussed above. The more general signatures are often referred to as Lorentzian signature although this term is often used as a synonym of the Minkowskian signature.

Signature change

If a metric is regular everywhere then the signature of the metric is constant. However if one allows for metrics that are degenerate or discontinuous on some hypersurfaces, then signature of the metric may change at these surfaces. Such signature changing metrics may possibly have applications in cosmology and quantum gravity.

See also

• pseudo-Riemannian manifold
• Sign convention