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.

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).

Calculating the metric tensor

To calculate the metric tensor from a set of equations relating the space to cartesian space, first compute the Jacobian of the set of equations:

J={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\end{bmatrix}}.

then multiply the transpose of the Jacobian by the Jacobian:

G=J^{T}J\

the result is the metric tensor G.

Using the metric tensor to calculate distance

In the following, we use 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\

where (x₁(t), ..., x_n(t)) is the equation describing this curve in the local coordinate system.

Using the metric tensor to calculate angle

In the following, we use Einstein notation for implicit sums.

The angle θ 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|}}}\

Examples

The Euclidean metric

In cartesian coordinates, the Euclidean metric tensor is simply the covariant Kronecker delta:

g_{ij}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}\ =\delta _{ij}

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_{ij}={\begin{bmatrix}1&0\\0&(x^{1})^{2}\end{bmatrix}}\

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

g_{ij}={\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_{ij}={\begin{bmatrix}1&0&0\\0&(x^{1})^{2}&0\\0&0&(x^{1}\sin x^{2})^{2}\end{bmatrix}}\

Arbitrary Coordinates: The covariant metric tensor can always be found for an arbitrary coordinate system in Euclidean space by applying the covariant tensor transformation rule:

{\bar {g}}_{kl}={\frac {\partial x_{i}}{\partial {\bar {x}}_{k}}}{\frac {\partial x_{j}}{\partial {\bar {x}}_{l}}}g_{ij}

An easy starting point for this transformation is often the Cartesian coordinates, where the $x_{i}$ and $x_{j}$ are the familiar coordinates, and $g_{ij}=\delta _{ij}$

The Pseudo-Euclidean metric

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

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

Covariant and Contravariant Metric Tensors

There are two version of the metric tensor, a covariant tensor $g_{ij}$ and a contravariant tensor $g^{ij}$ . These two tensors must satisfy the identity:

g^{ik}g_{kj}=\delta _{j}^{i}

These two tensors are used to transform between the covariant and contravariant forms of tensors by raising and lowering indices, as follows:

A^{j}=A_{i}g^{ij}

A_{j}=A^{i}g_{ij}

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.