Metric tensor

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In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. In other terms, given a smooth manifold, we make a choice of (0,2) tensor on the manifold's tangent spaces. 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. This concept is just like a dot product or inner product. This function from vectors into the real numbers is required to vary smoothly from point to point.

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. More generally, when the metric may give a negative value, the metric is called pseudo-Riemannian. In Special and General Relativity, spacetime is assumed to have a pseudo-Riemmanian metric (more specifically, a Lorentzian metric).

The manifold may also be given an affine connection, which is roughly an idea of change from one point to another. If the metric doesn't "vary from point to point" under this connection, we say that the metric and connection are compatible, and we have a Riemann-Cartan manifold. If this connection is also self-commuting when acting on a scalar function, we say that it is torsion-free, and the manifold is a Riemannian manifold.

1 Measuring length and angles with a metric
2 Examples
3 The tangent-cotangent isomorphism
4 See also
5 External links

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Measuring length and angles with a metric

Once a local coordinate system <math> x^i \ </math> is chosen, the metric tensor appears as a matrix, conventionally denoted <math>\mathbf{g}</math>. The notation <math>g_{ij} \ </math> is conventionally used for the components of the metric tensor (i.e., the elements of the matrix). Note that in the following, we use the Einstein summation notation for implicit sums.

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

The angle <math> \theta \ </math> between two tangent vectors, <math>U=u^i{\partial\over \partial x_i} \ </math> and <math>V=v^i{\partial\over \partial x_i} \ </math>, is defined as:

<math>

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

\ </math>

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

<math>\mathbf{g} = J^T J \ </math>

where <math>J \ </math> denotes the Jacobian of the embedding and <math>J^T \ </math> its transpose.

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

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

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.

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Examples

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The Euclidean metric

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

<math>g = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} \ </math>

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

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

Polar coordinates: <math>(r, \theta) \ </math>

<math>x = r \cos\theta</math>

<math>y = r \sin\theta</math>

<math>J = \begin{bmatrix}\cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta\end{bmatrix}</math>

<math>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} \ </math>

by trig identities.

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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 <math>(\theta,\phi)</math> the metric takes the form

<math>g = \begin{bmatrix} 1 & 0 \\ 0 & \sin^2 \theta\end{bmatrix}</math>

This is usually written in the form

<math>g = d\theta^2 + \sin^2\theta\,d\phi^2.</math>

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Lorentzian metrics from relativity

Flat Minkowski space (special relativity) : <math>(x^0, x^1, x^2, x^3)=(ct, x, y, z) \ </math>

<math>g = \begin{bmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix} \ </math>

For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. For a timelike curve, the length formula gives the proper time along the curve.

The Schwarzschild metric describes the spacetime around a spherically symmetric body. With coordinates <math>(x^0, x^1, x^2, x^3)=(ct, r, \theta, \phi) </math>, we can write the metric as

<math>G = \begin{bmatrix} -1+\frac{2GM}{rc^2} & 0 & 0 & 0\\ 0 & \frac{1}{1-\frac{2GM}{r c^2}} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2 \theta \end{bmatrix} \ </math>

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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."

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.

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