Metric tensor - Citizendia

1 Definition
2 Covariant and contravariant metric tensors
- 2.1 Matrix properties
3 Measuring length and angles with a metric
4 The energy, variational principles and geodesics
5 Examples
6 The tangent-cotangent isomorphism
7 Conjugate Metric Tensor
8 See also
9 External links

Definition

In mathematics, a (covariant) metric tensor g is a nonsingular symmetric tensor field of rank 2 that is used to measure distance in a space. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and For other uses of "covariant" or "contravariant" see Covariance and contravariance. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually Distance is a numerical description of how far apart objects are Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another In other words, given a smooth manifold, we make a choice of (0,2) tensor on the manifold's tangent spaces. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since At a given point in the manifold, this tensor takes a pair of vectors in the tangent space to that point, and gives a real number. If it is positive, this is just an inner product on each tangent space, which is required to vary smoothly from point to point. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. There is also a contravariant metric tensor, which is a (2,0) tensor; see below.

Suppose we feed two copies of the same non zero vector into the metric. If the metric will only ever give us back positive numbers, we say that the metric is positive definite. In this case, the metric is called a Riemannian metric. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M More generally, when the metric may give a negative value or zero (again assuming it is applied to two identical non zero vectors), the metric is called pseudo-Riemannian. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. In special and general relativity, spacetime is assumed to have a pseudo-Riemannian metric (more specifically, a Lorentzian metric). SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold.

The manifold may also be given an affine connection, which is roughly an idea of change from one point to another. In the mathematical field of Differential geometry, an affine connection is a geometrical object on a Smooth manifold which connects nearby Tangent If the metric doesn't "vary from point to point" under this connection, we say that the metric and connection are compatible, and the connection is a metric connection. In Mathematics, a metric connection is a connection in a Vector bundle E equipped with a metric If this connection also commutes with itself when acting on a scalar function, we say that it is torsion-free, and the manifold is a Riemannian manifold. In Differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a Moving frame around a curve

More generally, one may speak of a metric in a vector bundle. In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space Thus if E is a vector bundle over a manifold, then a metric is a non-singular bilinear map E ⊕ E → E. Using duality, a metric is often identified with a section of the tensor product bundle E^* ⊗ E^*. In the Mathematical field of Topology, a section (or cross section) of a Fiber bundle, &pi: E &rarr B In particular, if E is the tangent bundle, then a metric g is a section of the tensor product of the cotangent bundle with itself: i. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the In Mathematics, especially Differential geometry, the cotangent bundle of a Smooth manifold is the Vector bundle of all the Cotangent spaces e. , it is a rank 2 covariant tensor on the manifold. (See metric (vector bundle). In Differential geometry, the notion of a Metric tensor can be extended to an arbitrary Vector bundle. )

Covariant and contravariant metric tensors

The covariant metric tensor is a structure on the tangent space; the contravariant metric tensor is the corresponding structure on the cotangent space. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Differential geometry, one can attach to every point x of a smooth (or differentiable Manifold a Vector space called the cotangent space

A nonsingular form is an isomorphism $V \to V^*$ ; the inverse is a map $V^* \to V$ , which yields (by the tensor-hom adjunction) a map $V^* \otimes V^* \to K$ , that is, a (2,0) tensor. The Tensor - Hom adjunction in Computer science the analogous concept is called Currying.

In coordinates, if $g i j$ is the matrix for the covariant metric tensor, then the matrix for the contravariant metric tensor is the inverse (sometimes there is a transpose, depending on convention). Symbolically,

$g^{ik}g_{kj}=\delta^i_j$

The covariant and contravariant metric tensors are equivalent data, but they transform differently under change of coordinates; see covariance and contravariance of vectors. For other uses of "covariant" or "contravariant" see Covariance and contravariance.

They can be used in raising and lowering indices. In mathematics and mathematical physics given a Tensor on a manifold M, in the presence of a nonsingular form on M (such as a Riemannian metric or

The contravariant metric tensor is also called the conjugate metric tensor.

Matrix properties

The matrix for the metric tensor is diagonal in a system of coordinates if and only if the moving frame is orthogonal with respect to the metric. In Mathematics, a moving frame is a flexible generalization of the notion of an Ordered basis of a Vector space often used to study the extrinsic differential The square roots of the diagonal entries are called the scale factors.

With respect to an orthonormal basis, the metric tensor is the identity matrix, and you can raise and lower indices indiscriminantly.

Measuring length and angles with a metric

Once a local coordinate system $x^i \$ is chosen, the metric tensor appears as a matrix, conventionally denoted $\mathbf{g}$ . In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally The notation $g_{ij} \$ is conventionally used for the components of the metric tensor. More precisely, $g_{ij}=\left\langle\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right\rangle$ , where the inner product of the riemannian manifold is used and $\frac{\partial}{\partial x^i}$ is the partial derivative (also called derivation) in direction $x i$ . In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant

Note that in the following, we use the Einstein summation notation for implicit sums (i. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational e. , each index below has its counterpart index above).

In a Riemannian manifold, the length of a segment of a curve parameterized by t, from a to b, is defined as:

$L = \int_a^b \sqrt{ g_{ij}{dx^i\over dt}{dx^j\over dt}}dt \$

The angle $\theta \$ between two tangent vectors, $U=u^i{\partial\over \partial x^i} \$ and $V=v^i{\partial\over \partial x^i} \$ , is defined as:

$\cos \theta = \frac{g_{ij}u^iv^j} {\sqrt{ \left| g_{ij}u^iu^j \right| \left| g_{ij}v^iv^j \right|}} \$

The induced metric tensor for a smooth embedding of a manifold into Euclidean space can be computed by the formula

$\mathbf{g} = J^T J \$

where $J \$ denotes the Jacobian of the embedding and $J^T \$ its transpose. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a

For a pseudo-Riemannian metric, the length formula above is not always defined, because the term under the square root may become negative. We generally only define the length of a curve when the quantity under the square root is always of one sign or the other. In this case, we define

$L = \int_a^b \sqrt{ \left|g_{ij}{dx^i\over dt}{dx^j\over dt}\right|}dt \ .$

Note that, while these formulas use coordinate expressions, they are in fact independent of the coordinates chosen; they depend only on the metric, and the curve along which the formula is integrated.

The energy, variational principles and geodesics

Given a segment of a curve, another frequently defined quantity is the (kinetic) energy of the curve:

$E = \frac{1}{2} \int_a^b g_{ij}{dx^i\over dt}{dx^j\over dt}dt \$

This usage comes from physics, specifically, classical mechanics, where the integral E can be seen to directly correspond to the kinetic energy of a point particle moving on the surface of a manifold. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects The kinetic energy of an object is the extra Energy which it possesses due to its motion Thus, for example, in Jacobi's formulation of Maupertuis principle, the metric tensor can be seen to correspond to the mass tensor of a moving particle. In Classical mechanics, Maupertuis' principle (named after Pierre Louis Maupertuis) is an integral equation that determines the path followed by a physical

In many cases, whenever a calculation calls for the length to be used, a similar calculation using the energy may be done as well. This often leads to simpler formulas by avoiding the need for the square-root. Thus, for example, the geodesic equations may be obtained by applying variational principles to either the length or the energy. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces A variational principle is a principle in Physics which is expressed in terms of the Calculus of variations. In the later case, the geodesic equations are seen to arise from the principle of least action: they describe the motion of a "free particle" (a particle feeling no forces) that is confined to move on the manifold, but otherwise moves freely, with constant momentum, within the manifold. This article discusses the history of the principle of least action

Examples

The Euclidean metric

The most familiar example is that of basic high-school geometry: the two-dimensional Euclidean metric tensor. In the usual $x$ - $y$ coordinates, we can write

$g = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} \$

The length of a curve reduces to the familiar calculus formula:

$L = \int_a^b \sqrt{ (dx)^2 + (dy)^2} \$

The Euclidean metric in some other common coordinate systems can be written as follows. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

Polar coordinates: $(r, \theta) \$

x = r cosθ

y = r sinθ

$J = \begin{bmatrix}\cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta\end{bmatrix}$

$g = J^T J = \begin{bmatrix}\cos^2\theta+\sin^2\theta & -r\sin\theta \cos\theta + r\sin\theta\cos\theta \\ -r\cos\theta\sin\theta + r\cos\theta\sin\theta & r^2 \sin^2\theta + r^2\cos^2\theta\end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & r^2\end{bmatrix} \$

by trigonometric identities. In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables

In general, if the $x i$ are Cartesian (i. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane e. orthogonal) coordinates on a Euclidean space, the metric tensor with respect to arbitrary (possibly curvilinear) coordinates $q i$ is given by:

$g_{ij} = {\partial x^k \over \partial q^i} {\partial x^k \over \partial q^j}$

The round metric on a sphere

The unit sphere in R³ comes equipped with a natural metric induced from the ambient Euclidean metric. In standard spherical coordinates $(θ,φ)$ the metric takes the form

$g = \left[\begin{array}{cc} 1 & 0 \\ 0 & \sin^2 \theta\end{array}\right]$

This is usually written in the form

$ds^2 = d\theta^2 + \sin^2\theta\,d\phi^2.$

Lorentzian metrics from relativity

In flat Minkowski space (special relativity), with coordinates $r^\mu \rightarrow (x^0, x^1, x^2, x^3)=(ct, x, y, z) \ ,$ the metric is

$g = \begin{bmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \$

For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial For a timelike curve, the length formula gives the proper time along the curve. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In relativity, proper time is Time measured by a single Clock between events that occur at the same place as the clock

In this case, the spacetime interval is written as

$ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 = dr^\mu dr_\mu = g_{\mu \nu} dr^\mu dr^\nu\ .$

The Schwarzschild metric describes the spacetime around a spherically symmetric body, such as a planet, or a black hole. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e With coordinates $(x 0, x 1, x 2, x 3) = (c t, r,θ,φ)$ , we can write the metric as

$G = \begin{bmatrix} -(1-\frac{2GM}{rc^2}) & 0 & 0 & 0\\ 0 & (1-\frac{2GM}{r c^2})^{-1} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2 \theta \end{bmatrix} \$

The tangent-cotangent isomorphism

In tensor analysis, the metric tensor is often used to provide a canonical isomorphism from the tangent space to the cotangent space. Canonical is an Adjective derived from canon. Canon comes from the Greek word kanon, "rule" (perhaps originally from In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Differential geometry, one can attach to every point x of a smooth (or differentiable Manifold a Vector space called the cotangent space For each contravariant vector $A μ$ there is a covariant vector $A ν$ which is related by a metric tensor g:

$A_\nu = g_{\nu \mu} A^\mu. \,$

This also works the other way:

$A^\mu = g^{\mu \gamma} A_\gamma. \,$

From these definitions it also becomes obvious that

$g_{\nu \mu} g^{\mu \gamma} = \delta_\nu^\gamma \,$

where $\delta \,$ is the four dimensional Kronecker delta defined by

$\delta_\nu^\gamma = \begin{cases} 1 & \mathrm{if} \ \nu=\gamma \\ 0 & \mathrm{otherwise} \end{cases} \,$

In more mathematical terms: given a manifold M, v ∈ T_pM and a metric tensor g on M, we have that g(v, . For other uses of "covariant" or "contravariant" see Covariance and contravariance. For other uses of "covariant" or "contravariant" see Covariance and contravariance. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two ), the mapping that sends another given vector w ∈ T_pM to g(v,w), is an element of the dual space T_p*M. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals The nondegeneracy of the metric tensor makes it a one-to-one correspondence, and the fact that g itself is a tensor means that this identification is independent of coordinates. This is the linear algebra version of Riesz representation theorem. There are several well-known theorems in Functional analysis known as the Riesz representation theorem. In component terminology, it means that one can identify covariant and contravariant objects i. e. "raise and lower indices. "

This has a physical interpretation. The metric tensor defines units, and unit vectors, of physical measurement. For instance, one may ask, what is the scale for these measurements? A choice of basis defines the system of units on our manifold. The notions of contravariance and covariance correspond to quantities whose components transform "inversely" or "with" the coordinate system, hence the names. For example, consider R³ with the standard coordinate chart. If we transform the coordinate system by scaling the unit distance (say meters) down by a factor of 1000, the displacement vector (1,2,3) becomes (1000,2000,3000). On the other hand, if (1,2,3) represents a dual vector (for example, electric field strength), an object which takes a displacement vector and yields a scalar (in the example: potential difference in, say, volts), then the transformed coordinates become (0. 001,0. 002,0. 003). What does the Euclidean metric on R³ do? (1,2,3) becoming (1000,2000,3000) makes sense because scaling down by 1000 takes meters to millimeters. For the field strength vector, (1,2,3) becoming (0. 001, 0. 002, 0. 003) is a reflection of field strength going from volts per meter to volts per millimeter.

But what is the contravariant version of the field strength? How can we make a field strength vector's coordinates go from (1,2,3) to (1000,2000,3000)? The solution is to view the scale-down-by-1000 transformation as affecting the volts on the units V/m instead of the meters so that our new strength is measured in millivolts per meter. The metric tensor tells us precisely that we are still dealing with the same object; that is, it identifies the scaling of the basis vectors for the units "in the denominator" with a corresponding inverse change "in the numerator. " Although somewhat trivial for R³, for general manifolds M it is very important since one can only define things locally. One can also imagine, for example, defining different units on R³ which vary from point to point.

Conjugate Metric Tensor

The conjugate metric tensor is defined as the transposed cofactor of the metric tensor divided by the determinant of the metric tensor. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

$q^{ij} = \frac{ \operatorname{Cof}(q_{ij})^T }{ \operatorname{Det}(q_{ij}) }$

a short calculation shows that

$q_{jk} q^{pk} = \delta^{p}_{j}$

where $δ$ is the Kronecker delta. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two

External links

Caltech Tutorial on Relativity — A simple introduction to the basics of metrics in the context of relativity. An understanding of Calculus and Differential equations is necessary for the understanding of Nonrelativistic physics. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. This article discusses metrics in General relativity, for a discussion of metrics in general see Metric tensor. This article attempts to conveniently list articles on some of the most useful coordinate charts in some of the most useful examples of Riemannian manifolds The notion of a Coordinate In Mathematics, the musical isomorphism (or canonical isomorphism is an Isomorphism between the Tangent bundle TM and the Cotangent In Mathematics, particularly Differential geometry, a Finsler manifold is a Differentiable manifold M with a Banach norm defined over

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