Lorentz signature
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the signature of a
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
''g'' (or equivalently, a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...
thought of as a real
symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear ...
on a
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
) is the number (counted with multiplicity) of positive, negative and zero
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of the real
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...
of the metric tensor with respect to a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
. In
relativistic physics In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanics, quantum mechanical description of a system of particles, or of a fluid, in cases where ...
, the ''v'' represents the time or virtual dimension, and the ''p'' for the space and physical dimension. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By
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 number, real quadratic form that remain invariant (mathematics), invariant under a change of basis. Namely, if ''A'' is the symme ...
these numbers do not depend on the choice of basis. The signature thus classifies the metric up to a choice of basis. The signature is often denoted by a pair of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s implying ''r''= 0, or as an explicit list of signs of eigenvalues such as or for the signatures and , respectively. 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 In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
is a metric with a
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
signature . A
Lorentzian metric In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...
is a metric with signature , or . There is another notion of signature of a nondegenerate metric tensor given by a single number ''s'' defined as , 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 and its mirroring ''s' '' = −''s'' = +2 for .


Definition

The signature of a metric tensor is defined as the signature of the corresponding
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...
. It is the number of positive, negative and zero
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of any matrix (i.e. in any basis for the underlying vector space) 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 In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
a symmetric matrix over the reals is always
diagonalizable In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
, and has therefore exactly ''n'' real eigenvalues (counted with
algebraic multiplicity In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
). Thus .


Sylvester's law of inertia: independence of basis choice and existence of orthonormal basis

According to
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 number, real quadratic form that remain invariant (mathematics), invariant under a change of basis. Namely, if ''A'' is the symme ...
, 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 there exists a basis such that for , for and otherwise. It follows that there exists an
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
if and only if the signatures of ''g''1 and ''g''2 are equal. Likewise the signature is equal for two
congruent matrices In mathematics, two Square matrix, square matrices ''A'' and ''B'' over a Field (mathematics), field are called congruent if there exists an invertible matrix ''P'' over the same field such that :''P''T''AP'' = ''B'' where "T" denotes the matri ...
and classifies a matrix up to congruency. Equivalently, the signature is constant on the orbits of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
GL(''V'') on the space of symmetric rank 2 contravariant tensors ''S''2''V'' and classifies each orbit.


Geometrical interpretation of the indices

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 Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
of the scalar product ''g'' or the null subspace of
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...
of the
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
. 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 In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
is . The signature of a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
is the number of positive, negative and zero numbers on its
main diagonal In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matrix. ...
. The following matrices have both the same signature , therefore they are congruent because of
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 number, real quadratic form that remain invariant (mathematics), invariant under a change of basis. Namely, if ''A'' is the symme ...
: :\begin 1 & 0 \\ 0 & -1 \end, \quad \begin 0 & 1 \\ 1 & 0 \end.


Scalar products

The standard
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
defined on \mathbb^n has the ''n''-dimensional signatures , where and rank . In physics, the
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
is a spacetime manifold \R^4 with ''v'' = 1 and ''p'' = 3 bases, and has a scalar product defined by either the \check g matrix: :\check g=\begin -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end 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=\begin 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end=-\check g


How to compute the signature

There are some methods for computing the signature of a matrix. * For any
nondegenerate In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. T ...
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
matrix, diagonalize it (or find all of
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of it) and count the number of positive and negative signs. * For a symmetric matrix, the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
will have all real roots whose signs may in some cases be completely determined by
Descartes' rule of signs In mathematics, Descartes' rule of signs, first described by René Descartes in his work ''La Géométrie'', is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots i ...
. * Lagrange's algorithm gives a way to compute an
orthogonal basis In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V is a basis for V whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal 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
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
s of its main minors are positive.


Signature in physics

In mathematics, the usual convention for any
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
is to use a positive-definite
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
(meaning that after diagonalization, elements on the diagonal are all positive). In
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
,
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
is modeled by a
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
. The signature counts how many time-like or space-like characters are in the spacetime, in the sense defined by
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
: as used in
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, 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 In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
, : ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^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 ''x'', ''y'' and ''z''. (Sometimes the opposite
sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
convention is used, but with the one given here ''s'' directly measures
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
.)


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 () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount (lexicographer), Thomas Blount's ''Glossographia'', and in 1731 taken up in ...
and
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
.


See also

*
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
*
Sign convention In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly describ ...


Notes

{{Riemannian geometry Differential geometry *2