Metric tensor

Definition

In mathematics, a (covariant) metric tensor g is a nonsingular symmetric tensor field of rank 2 that is used to measure distance in a space. In other words, 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. If it is positive, this is just an inner product on each tangent space, which is required to vary smoothly from point to point. There is also a contravariant metric tensor, which is a (2,0) tensor; see below.

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 or zero (again assuming it is applied to two identical non zero vectors), the metric is called pseudo-Riemannian. In special and general relativity, spacetime is assumed to have a pseudo-Riemannian 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 the connection is a metric connection. If this connection also commutes with itself when acting on a scalar function, we say that it is torsion-free, and the manifold is a Riemannian manifold.

More generally, one may speak of a metric in a vector bundle. Thus if E is a vector bundle over a manifold, then a metric is a non-singular bilinear map E ⊕ E → E. Using duality, a metric is often identified with a section of the tensor product bundle E^* ⊗ E^*. In particular, if E is the tangent bundle, then a metric g is a section of the tensor product of the cotangent bundle with itself: i.e., it is a rank 2 covariant tensor on the manifold. (See metric (vector bundle).)

Covariant and contravariant metric tensors

The covariant metric tensor is a structure on the tangent space; the contravariant metric tensor is the corresponding structure on the cotangent space.

A nonsingular form is an isomorphism $V\to V^{*}$; the inverse is a map $V^{*}\to V$, which yields (by the tensor-hom adjunction) a map $V^{*}\otimes V^{*}\to K$, that is, a (2,0) tensor.

In coordinates, if $g_{ij}$ is the matrix for the covariant metric tensor, then the matrix for the contravariant metric tensor is the inverse (sometimes there is a transpose, depending on convention). Symbolically,

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

The covariant and contravariant metric tensors are equivalent data, but they transform differently under change of coordinates; see covariance and contravariance of vectors.

They can be used in raising and lowering indices.

The contravariant metric tensor is also called the conjugate metric tensor.

Matrix properties

The matrix for the metric tensor is diagonal in a system of coordinates if and only if the moving frame is orthogonal with respect to the metric. The square roots of the diagonal entries are called the scale factors.

With respect to an orthonormal basis, the metric tensor is the identity matrix, and you can raise and lower indices indiscriminantly.

Measuring length and angles with a metric

Once a local coordinate system $x^{i}\$ is chosen, the metric tensor appears as a matrix, conventionally denoted $\mathbf {g}$. The notation $g_{ij}\$ is conventionally used for the components of the metric tensor. More precisely, $g_{ij}=\left\langle {\frac {\partial }{\partial x^{i}}},{\frac {\partial }{\partial x^{j}}}\right\rangle$, where the inner product of the riemannian manifold is used and ${\frac {\partial }{\partial x^{i}}}$ is the partial derivative (also called derivation) in direction $x^{i}$.

Note that in the following, we use the Einstein summation notation for implicit sums (i.e., each index below has its counterpart index above).

In a Riemannian manifold, 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

$\mathbf {g} =J^{T}J\$

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

$L=\int _{a}^{b}{\sqrt {\left|g_{ij}{dx^{i} \over dt}{dx^{j} \over dt}\right|}}dt\ .$

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.

The energy, variational principles and geodesics

Given a segment of a curve, another frequently defined quantity is the (kinetic) energy of the curve:

$E={\frac {1}{2}}\int _{a}^{b}g_{ij}{dx^{i} \over dt}{dx^{j} \over dt}dt\$

This usage comes from physics, specifically, classical mechanics, where the integral E can be seen to directly correspond to the kinetic energy of a point particle moving on the surface of a manifold. Thus, for example, in Jacobi's formulation of Maupertuis principle, the metric tensor can be seen to correspond to the mass tensor of a moving particle.

In many cases, whenever a calculation calls for the length to be used, a similar calculation using the energy may be done as well. This often leads to simpler formulas by avoiding the need for the square-root. Thus, for example, the geodesic equations may be obtained by applying variational principles to either the length or the energy. In the later case, the geodesic equations are seen to arise from the principle of least action: they describe the motion of a "free particle" (a particle feeling no forces) that is confined to move on the manifold, but otherwise moves freely, with constant momentum, within the manifold.

Examples

The Euclidean metric

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

$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)^{2}+(dy)^{2}}}\$

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

Polar coordinates: $(r,\theta )\$

$x=r\cos \theta$

$y=r\sin \theta$

$J={\begin{bmatrix}\cos \theta &-r\sin \theta \\\sin \theta &r\cos \theta \end{bmatrix}}$

So

$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}}\$

by trigonometric identities.

In general, if the $x^{i}$ are Cartesian (i.e. orthogonal) coordinates on a Euclidean space, the metric tensor with respect to arbitrary (possibly curvilinear) coordinates $q^{i}$ is given by:

$g_{ij}={\partial x^{k} \over \partial q^{i}}{\partial x^{k} \over \partial q^{j}}$

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 $(\theta ,\phi )$ the metric takes the form

$g=\left[{\begin{array}{cc}1&0\\0&\sin ^{2}\theta \end{array}}\right]$

This is usually written in the form

$ds^{2}=d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}.$

Lorentzian metrics from relativity

In flat Minkowski space (special relativity), with coordinates $r^{\mu }\rightarrow (x^{0},x^{1},x^{2},x^{3})=(ct,x,y,z)\ ,$ the metric is

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

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.

In this case, the spacetime interval is written as

$ds^{2}=-c^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}=dr^{\mu }dr_{\mu }=g_{\mu \nu }dr^{\mu }dr^{\nu }\ .$

The Schwarzschild metric describes the spacetime around a spherically symmetric body, such as a planet, or a black hole. With coordinates $(x^{0},x^{1},x^{2},x^{3})=(ct,r,\theta ,\phi )$, we can write the metric as

$G={\begin{bmatrix}-(1-{\frac {2GM}{rc^{2}}})&0&0&0\\0&(1-{\frac {2GM}{rc^{2}}})^{-1}&0&0\\0&0&r^{2}&0\\0&0&0&r^{2}\sin ^{2}\theta \end{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. For each contravariant vector $A^{\mu }$ there is a covariant vector $A_{\nu }$ which is related by a metric tensor g:

$A_{\nu }=g_{\nu \mu }A^{\mu }.\,$

This also works the other way:

$A^{\mu }=g^{\mu \gamma }A_{\gamma }.\,$

From these definitions it also becomes obvious that

$g_{\nu \mu }g^{\mu \gamma }=\delta _{\nu }^{\gamma }\,$

where $\delta \,$ is the four dimensional Kronecker delta defined by

$\delta _{\nu }^{\gamma }={\begin{cases}1&\mathrm {if} \ \nu =\gamma \\0&\mathrm {otherwise} \end{cases}}\,$

In more mathematical terms: 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. This is the linear algebra version of Riesz representation theorem. In component terminology, it means that one can identify covariant and contravariant objects i.e. "raise and lower indices."

This has a physical interpretation. The metric tensor defines units, and unit vectors, of physical measurement. For instance, one 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 different units on R³ which vary from point to point.

Conjugate Metric Tensor

The conjugate metric tensor is defined as the transposed cofactor of the metric tensor divided by the determinant of the metric tensor.

$q^{ij}={\frac {\operatorname {Cof} (q_{ij})^{T}}{\operatorname {Det} (q_{ij})}}$

a short calculation shows that

$q_{jk}q^{pk}=\delta _{j}^{p}$

where $\delta$ is the Kronecker delta.

See also

• Basic introduction to the mathematics of curved spacetime
• Pseudo-Riemannian metric
• Metric tensor (general relativity)
• List of coordinate charts
• Musical isomorphism
• Finsler manifold

External links

• Caltech Tutorial on Relativity — A simple introduction to the basics of metrics in the context of relativity.