Metric signature

Template:Short description In mathematics, the signature of a metric tensor Template:Mvar (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:Math 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 Template:Math, 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:Math implying Template:Math, or as an explicit list of signs of eigenvalues such as Template:Math or Template:Math for the signatures Template:Math and Template:Math, respectively. The choice of the variable names Template:Mvar and Template:Mvar reflects the convention in relativistic physics that Template:Mvar represents the number of time or virtual dimensions, and Template:Mvar the number of space or physical dimensions.

The signature is said to be indefinite or mixed if both Template:Mvar and Template:Mvar are nonzero, and degenerate if Template:Mvar is nonzero. A Riemannian metric is a metric with a positive definite signature Template:Math. A Lorentzian metric is a metric with signature Template:Math, or Template:Math.

There is another notion of signature of a nondegenerate metric tensor given by a single number Template:Mvar defined as Template:Math, where Template:Mvar and Template:Mvar are as above, which is equivalent to the above definition when the dimension Template:Math is given or implicit. For example, Template:Math for Template:Math and its mirroring Template:Math for Template:Math.

Definition

The signature of a metric tensor is defined as the signature of the corresponding quadratic form. It is the number Template:Math 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:Math 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:Math are basis independent.

Properties

Signature and dimension

By the spectral theorem a symmetric Template:Math matrix over the reals is always diagonalizable, and has therefore exactly Template:Mvar real eigenvalues (counted with algebraic multiplicity). Thus Template:Math.

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

Geometrical interpretation of the indices

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

Examples

Matrices

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

$\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right).$

Scalar products

The standard scalar product defined on ${ℝ}^n$ has the Template:Mvar-dimensional signatures Template:Math, where Template:Math and rank Template:Math.

In physics, the Minkowski space is a spacetime manifold ${ℝ}^4$ with Template:Math and Template:Math bases, and has a scalar product defined by either the $\check{g}$ matrix:

$\check{g}=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$

which has signature $(1,3,0{)}^{-}$ and known as space-supremacy or space-like; or the mirroring signature $(1,3,0{)}^{+}$, known as virtual-supremacy or time-like with the $\hat{g}$ matrix.

$\hat{g}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)=-\check{g}$

How to compute the signature

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

• For any nondegenerate symmetric Template:Math 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 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 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,

$d{s}^2=c^{2}d{t}^2-d{x}^2-d{y}^2-d{z}^2,$

the metric signature is $(1,3,0{)}^{+}$ or (+, −, −, −) if its eigenvalue is defined in the time direction, or $(1,3,0{)}^{-}$ or (−, +, +, +) if the eigenvalue is defined in the three spatial directions Template:Mvar, Template:Mvar and Template:Mvar. (Sometimes the opposite sign convention is used, but with the one given here Template:Mvar 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. Such signature changing metrics may possibly have applications in cosmology and quantum gravity.

See also

• pseudo-Riemannian manifold
• Sign convention