Metric signature

In mathematics, the signature of a metric tensor is the number of positive, negative and zero eigenvalues of the real symmetric matrix 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, 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 implying, or as an explicit list of signs of eigenvalues such as or for the signatures and, respectively. The choice of the variable names and reflects the convention in relativistic physics that represents the number of time or virtual dimensions, and the number of space or physical dimensions.
The signature is said to be indefinite or mixed if both and are nonzero, and degenerate if is nonzero. A Riemannian metric is a metric with a positive definite signature. A Lorentzian metric is a metric with signature, or.
There is another notion of signature of a nondegenerate metric tensor given by a single number defined as, where and are as above, which is equivalent to the above definition when the dimension is given or implicit. For example, for and its mirroring for.

Definition

The signature of a metric tensor is defined as the signature of the corresponding quadratic form. It is the number of positive, negative and zero eigenvalues of any matrix representing the form, counted with their algebraic multiplicities. Usually, 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 are basis independent.

Properties

Signature and dimension

By the spectral theorem a symmetric matrix over the reals is always diagonalizable, and has therefore exactly real eigenvalues. Thus.

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, does not depend on the choice of basis. Moreover, for every metric of signature there exists a basis such that
for, for and otherwise. It follows that there exists an isometry if and only if the signatures of and 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 on the space of symmetric rank 2 contravariant tensors and classifies each orbit.

Geometrical interpretation of the indices

The number is the maximal dimension of a vector subspace on which the scalar product is positive-definite, and is the dimension of the radical of the scalar product or the null subspace of symmetric matrix of the scalar product. Thus a nondegenerate scalar product has signature, with. A duality of the special cases correspond to two scalar eigenvalues which can be transformed into each other by the mirroring reciprocally.

Examples

Matrices

The signature of the identity matrix is. 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, therefore they are congruent because of Sylvester's law of inertia:

Scalar products

The standard scalar product defined on has the -dimensional signatures, where and rank.
In physics, the Minkowski space is a spacetime manifold with and bases, and has a scalar product defined by either the matrix:
which has signature and known as space-supremacy or space-like; or the mirroring signature, known as virtual-supremacy or time-like with the matrix.

How to compute the signature

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

For any nondegenerate symmetric matrix, diagonalize 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's algorithm gives a way to compute an orthogonal basis, and thus compute a diagonal matrix congruent 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.
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 has an eigenvalue on the time-like subspace, and its mirroring eigenvalue on the space-like subspace.
In the specific case of the Minkowski metric,
the metric signature is or if its eigenvalue is defined in the time direction, or or if the eigenvalue is defined in the three spatial directions, and.

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.