Template:Short description 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 Template:Nowrap of the metric tensor with respect to a basis. In relativistic physics, v conventionally represents the number of time or virtual dimensions, and p the number of space or physical dimensions. 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 Template:Nowrap, 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 Template:Nowrap implying r = 0, or as an explicit list of signs of eigenvalues such as Template:Nowrap or Template:Nowrap for the signatures Template:Nowrap and Template:Nowrap, respectively.<ref>Rowland, Todd. "Matrix Signature." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/MatrixSignature.html</ref>
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 Template:Nowrap. A Lorentzian metric is a metric with signature Template:Nowrap, or Template:Nowrap.
There is another notion of signature of a nondegenerate metric tensor given by a single number s defined as Template:Nowrap, 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 Template:Nowrap and its mirroring s' = −s = +2 for Template:Nowrap.
DefinitionEdit
The signature of a metric tensor is defined as the signature of the corresponding quadratic form.<ref>Template:Cite book</ref> It is the number Template:Nowrap 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, Template:Nowrap 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 Template:Nowrap are basis independent.
PropertiesEdit
Signature and dimensionEdit
By the spectral theorem a symmetric Template:Nowrap matrix over the reals is always diagonalizable, and has therefore exactly n real eigenvalues (counted with algebraic multiplicity). Thus Template:Nowrap.
Sylvester's law of inertia: independence of basis choice and existence of orthonormal basisEdit
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 Template:Nowrap there exists a basis such that Template:Nowrap for Template:Nowrap, Template:Nowrap for Template:Nowrap and Template:Nowrap otherwise. It follows that there exists an isometry Template:Nowrap 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 indicesEdit
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 Template:Nowrap of the scalar product. Thus a nondegenerate scalar product has signature Template:Nowrap, with Template:Nowrap. A duality of the special cases Template:Nowrap correspond to two scalar eigenvalues which can be transformed into each other by the mirroring reciprocally.
ExamplesEdit
MatricesEdit
The signature of the Template:Nowrap identity matrix is Template:Nowrap. 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 Template:Nowrap, therefore they are congruent because of Sylvester's law of inertia:
- <math>\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. </math>
Scalar productsEdit
The standard scalar product defined on <math> \mathbb{R}^n </math> has the n-dimensional signatures Template:Nowrap, where Template:Nowrap and rank Template:Nowrap.
In physics, the Minkowski space is a spacetime manifold <math>\R^4</math> with v = 1 and p = 3 bases, and has a scalar product defined by either the <math>\check g</math> matrix:
- <math>\check g=\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} </math>
which has signature <math>(1, 3, 0)^-</math> and known as space-supremacy or space-like; or the mirroring signature <math>(1,3, 0)^+</math>, known as virtual-supremacy or time-like with the <math>\hat g</math> matrix.
- <math>\hat g=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}=-\check g</math>
How to compute the signatureEdit
There are some methods for computing the signature of a matrix.
- For any nondegenerate symmetric Template:Nowrap matrix, 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's 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 physicsEdit
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 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,
- <math> ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2, </math>
the metric signature is <math>(1, 3, 0)^+</math> or (+, −, −, −) if its eigenvalue is defined in the time direction, or <math>(1, 3, 0)^-</math> or (−, +, +, +) if the eigenvalue is defined 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 changeEdit
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.<ref>Template:Cite journal</ref> Such signature changing metrics may possibly have applications in cosmology and quantum gravity.