NationMaster - Encyclopedia: Metric tensor

In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. In other terms, given a smooth manifold, we make a choice of positive-definite quadratic form on the manifold's tangent spaces which varies smoothly from point to point. The manifold, equipped with the metric tensor (the varying choice of quadratic form), is called a Riemannian manifold and in this context the metric tensor is often called a Riemannian metric. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... âˆ , the angle symbol. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...

Once a local coordinate system xⁱ is chosen, the metric tensor appears as a matrix, conventionally denoted G. The notation g_ij is conventionally used for the components of the metric tensor (i.e. the elements of the matrix). In the following, we use the Einstein notation for implicit sums. For the square matrix section, see square matrix. ... This article or section does not adequately cite its references or sources. ...

The length of a segment of a curve parameterized by t, from a to b, is defined as:

$L = int_a^b sqrt{ g_{ij}{dx^iover dt}{dx^jover dt}}dt$

The angle θ between two tangent vectors, $U=u^i{partialover partial x_i}$ and $V=v^i{partialover partial x_i}$ , is defined as: For other uses, see tangent (disambiguation). ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ...

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

The induced metric tensor for a smooth embedding of a manifold into Euclidean space can be computed by the formula In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

$G = J^T J$

where $J$ denotes the Jacobian of the embedding and $J^T$ its transpose. In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or Aâ€²) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A...

1 Examples
- 1.1 The Euclidean metric
2 The tangent-cotangent isomorphism
3 See also

Examples

The Euclidean metric

Given a two-dimensional Euclidean metric tensor: In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...

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

The length of a curve reduces to the familiar calculus formula: Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ...

$L = int_a^b sqrt{ (dx^1)^2 + (dx^2)^2}$

The Euclidean metric in some other common coordinate systems can be written as follows.

Polar coordinates: $(x^1, x^2)=(r, theta)$ This article describes some of the common coordinate systems that appear in elementary mathematics. ...

$g = begin{bmatrix} 1 & 0 0 & (x^1)^2end{bmatrix}$

Cylindrical coordinates: $(x^1, x^2, x^3)=(r, theta, z)$ This article describes some of the common coordinate systems that appear in elementary mathematics. ...

$g = begin{bmatrix} 1 & 0 & 0 0 & (x^1)^2 & 0 0 & 0 & 1end{bmatrix}$

Spherical coordinates: $(x^1, x^2, x^3)=(r, phi, theta)$ This article describes some of the common coordinate systems that appear in elementary mathematics. ...

$g = begin{bmatrix} 1 & 0 & 0 0 & (x^1)^2 & 0 0 & 0 & (x^1sin x^2)^2end{bmatrix}$

Flat Minkowski space: $(x^0, x^1, x^2, x^3)=(ct, x, y, z)$ In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

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

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: given a manifold M, v ∈ T_pM and a metric tensor g on M, we have that g(v, . ), the mapping that sends another given vector w ∈ T_pM to g(v,w), is an element of the dual space T_p*M. 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. In component terminology, it means that one can identify covariant and contravariant objects i.e. "raise and lower indices." Canonical is an adjective derived from canon. ... It has been suggested that this article or section be merged into Covariant transformation. ...

This has a nice physical interpretation which is often glossed over. The metric tensor obviously has to do with measurement. We 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 "funny units" on R³ which vary from point to point.

Contents

The Euclidean metric

The tangent-cotangent isomorphism

See also