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.

Once a local coordinate system $x^{i}\$ 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.

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^{i}v^{j}}{\sqrt {\left|g_{ij}u^{i}u^{j}\right|\left|g_{ij}v^{i}v^{j}\right|}}}\$

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

$G=J^{T}J\$

where $J\$ denotes the Jacobian of the embedding and $J^{T}\$ its transpose.

Examples

The Euclidean metric

Given a two-dimensional Euclidean metric tensor:

$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^{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 )\$

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

Cylindrical coordinates: $(x^{1},x^{2},x^{3})=(r,\theta ,z)\$

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

Spherical coordinates: $(x^{1},x^{2},x^{3})=(r,\phi ,\theta )\$

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

Flat Minkowski space: $(x^{0},x^{1},x^{2},x^{3})=(t,x,y,z)\$

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

See also

• pseudo-Riemannian metric
• metric tensor (general relativity)