Metric tensor
From Freepedia
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.
Once a local coordinate system <math> x^i \ </math> is chosen, the metric tensor appears as a matrix, conventionally denoted G. The notation <math>g_{ij} \ </math> 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.
The length of a segment of a curve parameterized by t, from a to b, is defined as:
- <math>L = \int_a^b \sqrt{ g_{ij}{dx^i\over dt}{dx^j\over dt}}dt \ </math>
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>G = J^T J \ </math>
where <math>J \ </math> denotes the Jacobian of the embedding and <math>J^T \ </math> its transpose.
Examples
The Euclidean metric
Given a two-dimensional Euclidean metric tensor:
- <math>g = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} \ </math>
The length of a curve reduces to the familiar calculus formula:
- <math>L = \int_a^b \sqrt{ (dx^1)^2 + (dx^2)^2} \ </math>
The Euclidean metric in some other common coordinate systems can be written as follows.
Polar coordinates: <math>(x^1, x^2)=(r, \theta) \ </math>
- <math>g = \begin{bmatrix} 1 & 0 \\ 0 & (x^1)^2\end{bmatrix} \ </math>
Cylindrical coordinates: <math>(x^1, x^2, x^3)=(r, \theta, z) \ </math>
- <math>g = \begin{bmatrix} 1 & 0 & 0\\ 0 & (x^1)^2 & 0 \\ 0 & 0 & 1\end{bmatrix} \ </math>
Spherical coordinates: <math>(x^1, x^2, x^3)=(r, \phi, \theta) \ </math>
- <math>g = \begin{bmatrix} 1 & 0 & 0\\ 0 & (x^1)^2 & 0 \\ 0 & 0 & (x^1\sin x^2)^2\end{bmatrix} \ </math>
Flat Minkowski space: <math>(x^0, x^1, x^2, x^3)=(t, 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>
