Metric signature

The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative 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 and negative eigenvalues p and q = n − p may take a pair of values from 0 to n. The signature may be denoted either by a pair of integers such as (p, q), or as an explicit list such as (+, −, −, −) or (−, +, +, +), in this case (1,3) resp. (3,1).^[1]

The signature is said to be indefinite or mixed if both p and q are non-zero. 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 also another definition of signature which uses a single number s defined as the number p − q, where the p and q are the number of positive and negative eigenvalues of the metric tensor. Using the nondegenerate metric tensor from above, the signature is simply the sum of p and −q. For example, s = (1 − 3) = −2 for (+, −, −, −) and s = (3 − 1) = +2 for (−, +, +, +).

1 Definition
2 Properties
3 Examples
- 3.1 Matrices
- 3.2 Scalar products
4 How to compute the signature
5 Signature in physics
6 Signature change
7 See also
8 Notes

Definition

Let A be a symmetric matrix of reals. More generally, the metric signature (i₊,i₋,i₀) of A is a group of three natural numbers can be defined as the number of positive, negative and zero-valued eigenvalues of the matrix counted with regard to their algebraic multiplicity. In the case $i_0$ is non-zero, the matrix A called degenerate.

If $\phi$ is a scalar product on a finite-dimensional vector space V, the signature of V is the signature of the matrix which represents $\phi$ with respect to a chosen basis. According to Sylvester's law of inertia, the signature does not depend on the 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 $i_%2B %2B i_- %2B i_0 = n$

Sylvester's law of inertia

According to Sylvester's law of inertia two scalar products are isometrical if and only if they have the same signature. This means that the signature is a complete 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 indices $i_%2B$ and $i_-$ are the dimensions of the two vector subspaces on which the scalar product is positive-definite and negative-definite respectively. And the $i_0$ is the dimension of the radical of the scalar product $\phi$ or the null subspace of symmetric matrix A of the bilinear form. Thus a non degenerate scalar product has signature $(i_%2B, i_-, 0)$ , with $i_- = n - i_%2B$ . So the values $i_%2B, i_-$ and $i_0$ are also called the dimensions of the positive-definite, negative-definite and null vector subspaces of the whole vector space V which 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 transform each other by multiplying -1 to their scalar product.

Examples

Matrices

The signature of the identity matrix $n\times n$ is $(n, 0, 0)$ . More generally, 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 $(n, 0, 0)$ signature. A scalar product has this signature if and only if it is a positive definite scalar product.

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

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 $(3, 1, 0)$ signature.

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

$ds^2 = dx^2 %2B dy^2 %2B dz^2 - c^2 dt^2$ ,

the metric signature is evidently $(3,1,0)$ , since it is positive definite in the xyz-directions (in fact this restriction makes it equal to the standard Euclidean metric) 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.^[2] 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
^ Dray, Tevian; Ellis, George; Hellaby, Charles; Manogue, Corinne A. (1997). "Gravity and signature change". General Relativity and Gravity 29: 591–597. arXiv:gr-qc/9610063. doi:10.1023/A:1018895302693.