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).^[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 (p, 0). A Lorentzian metric is a metric 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 (−, +, +, +).

Definition

The signature of a metric tensor is defined as the signature of the corresponding quadratic form.^[2] 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.

By Sylvester's law of inertia, the numbers $(p, q, r)$ 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 p + q + r = 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 (p, q, r) there exists a basis such that g_ab = +1 for a = b = 1, ..., p, g_ab = −1 for a = b = p + 1, ..., p + q and g_ab = 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 p (resp. q) 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 g_ab of the scalar product. Thus a nondegenerate scalar product has signature (p, q, 0), with p + q = n. The special cases (n, 0, 0) and (0, n, 0) correspond to positive-definite and negative-definite scalar products which can be transformed into each other by negation.

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 (p, 0, r), where p + r = n.

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 (3, 1, 0), which is known as space-positive.

Sometimes it is used with the opposite signs, thus obtaining the signature (1, 3, 0), which is known as time-positive.

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

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.
For a symmetric matrix, the characteristic polynomial will have all real roots whose signs may in some cases be completely determined by Descartes' rule of signs.
Lagrange algorithm gives 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 mathematics, the usual convention for any Riemannian manifold is to use a positive-definite metric tensor (meaning that after diagonalization, elements on the diagonal are all positive).

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: as used in particle physics, the metric is positive definite on the time-like subspace, and negative definite on the space-like subspace. In the specific case of the Minkowski metric,

ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2

the metric signature is (1, 3, 0) or (+, −, −, −), since it is positive definite in the time direction, and negative definite in the three spatial directions x, y and z. (Sometimes the opposite sign convention is used, but with the one given here s directly measures proper time.)

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.^[3] Such signature changing metrics may possibly have applications in cosmology and quantum gravity.

Notes

↑ Rowland, Todd. "Matrix Signature." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/MatrixSignature.html
↑ Landau, L.D.; Lifshitz, E.M. (2002) [1939]. The Classical Theory of Fields. Course of Theoretical Physics. 2 (4th ed.). Butterworth–Heinemann. pp. 245–246. ISBN 0 7506 2768 9.
↑ Dray, Tevian; Ellis, George; Hellaby, Charles; Manogue, Corinne A. (1997). "Gravity and signature change". General Relativity and Gravitation. 29: 591–597. Bibcode:1997GReGr..29..591D. arXiv:gr-qc/9610063 . doi:10.1023/A:1018895302693.

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