Metric signature

Metric signature

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The signature of a metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

 (or more generally a nondegenerate symmetric bilinear form
Symmetric bilinear form
A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....

, thought of as quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

) 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
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 of the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

 is n×n, then the number of positive and negative eigenvalues p and may take a pair of values from 0 to n. The signature may be denoted either by a pair of integers such as , or as an explicit list such as or , in this case (1,3) resp. (3,1).

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 , or sometimes .

There is also another definition of signature which uses a single number s defined as the number pq, where the p and q are the number of positive and negative eigenvalues of the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

. Using the nondegenerate metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

 from above, the signature is simply the sum of p and −q. For example, for and for .

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 is non-zero, the matrix A called degenerate
Degenerate form
In mathematics, specifically linear algebra, a degenerate bilinear form ƒ on a vector space V is one such that the map from V to V^* given by v \mapsto is not an isomorphism...

.

If is a scalar product on a finite-dimensional
Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...

 vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 V, the signature of V is the signature of the matrix which represents with respect to a chosen basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

. According to Sylvester's law of inertia
Sylvester's law of inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of coordinates...

, the signature does not depend on the basis.

Spectral theorem


Due to the spectral theorem
Spectral theorem
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...

 a symmetric matrix of reals is always diagonalizable. Moreover, it has exactly n eigenvalues (counted according by their algebraic multiplicity). Thus

Sylvester's law of inertia


According to Sylvester's law of inertia
Sylvester's law of inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of coordinates...

 two scalar products are isometrical
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

 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
Congruence relation
In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...

 if and only if they have the same signature.

Geometrical interpretation of the indices


The indices and are the dimensions of the two vector subspaces on which the scalar product is positive-definite and negative-definite respectively. And the is the dimension of the radical of the scalar product or the null subspace
Null space
In linear algebra, the kernel or null space of a matrix A is the set of all vectors x for which Ax = 0. The kernel of a matrix with n columns is a linear subspace of n-dimensional Euclidean space...

 of symmetric matrix A of the bilinear form. Thus a non degenerate scalar product has signature , with .
So the values and are also called the dimensions of the positive-definite, negative-definite and null vector subspaces of the whole vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 V which correspond to the matrix A.
The special cases and correspond to the two equivalent vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s on which the scalar product is positive-definite and negative-definite respectively, and can transform each other by multiplying -1 to their scalar product.

Matrices


The signature of the identity matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

  is . More generally, the signature of a diagonal matrix
Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...

 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 signature. A scalar product has this signature if and only if it is a positive definite scalar product.

A negative definite scalar product has signature. A semi-definite positive scalar product has signature.

The Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

 is and has a scalar product defined by the matrix
and has signature .
Sometimes it is used with the opposite signs, thus obtaining 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 , they may take a pair of values from 0 to n, then the signature will be .
  • The sign of the roots of the characteristic polynomial may be determined by Cartesius' sign rule
    Descartes' rule of signs
    In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for determining the number of positive or negative real roots of a polynomial....

     as long as all roots are reals.
  • Lagrange algorithm avails a way to compute an orthogonal basis
    Orthogonal basis
    In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal...

    , and thus compute a diagonal matrix
    Diagonal matrix
    In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...

     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
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

, spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

 is modeled by a pseudo-Riemannian manifold
Pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...

. The signature counts how many time-like or space-like characters are in the spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

, in the sense defined by special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

: 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
,

the metric signature is evidently , 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 (i.e., ) 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. Such signature changing metrics may possibly have applications in cosmology
Cosmology
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...

 and quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

.