Metric Tensor · aiwiki.page

A metric tensor is a mathematical structure that specifies local geometric measurements on a smooth manifold. It assigns a symmetric, nondegenerate bilinear form to each tangent space, varying smoothly from point to point. A positive-definite metric defines lengths and angles in Riemannian geometry; an indefinite metric underlies pseudo-Riemannian geometry, including the geometry of spacetime. It is a central object in differential geometry and general relativity. (damtp.cam.ac.uk)

Definition and algebraic structure

At a point $p$ of an $n$ -dimensional manifold $M$ , the metric is a map

g_p:T_pM\times T_pM\longrightarrow\mathbb R.

It is linear in each argument and symmetric:

g_p(u,v)=g_p(v,u).

Nondegeneracy means that if $g_p(u,v)=0$ for every $v$ , then $u=0$ . The tangent space is a vector space, so these conditions are statements about bilinear algebra at each point. Globally, $g$ is a covariant tensor field of type $(0,2)$ , rather than merely an array of numbers. (damtp.cam.ac.uk)

A Riemannian metric additionally satisfies $g_p(v,v)>0$ for every nonzero $v$ . Thus it supplies an inner product on every tangent space. A pseudo-Riemannian metric permits positive and negative squared norms while retaining nondegeneracy. Its signature records the numbers of positive and negative directions in a diagonalized basis. A Lorentzian metric has one direction of one sign and all remaining directions of the opposite sign. Nonzero vectors with zero squared norm are possible in this indefinite setting. (damtp.cam.ac.uk)

Coordinate representation

In local coordinates $x^1,\ldots,x^n$ , the coordinate vectors $\partial_i=\partial/\partial x^i$ form a basis, and

g_{ij}=g(\partial_i,\partial_j),\qquad g=g_{ij}\,dx^i\otimes dx^j.

Here $\otimes$ denotes the tensor product, and repeated indices are summed according to the Einstein summation convention. The components constitute a symmetric matrix with $n(n+1)/2$ independent entries. For tangent vectors $u$ and $v$ ,

g(u,v)=g_{ij}u^iv^j.

The associated line element is conventionally written $ds^2=g_{ij}dx^idx^j$ . (damtp.cam.ac.uk)

Under a coordinate change $x\mapsto x'$ ,

g'_{ab} =\frac{\partial x^i}{\partial x'^a} \frac{\partial x^j}{\partial x'^b}\,g_{ij}.

Although the entries change, the bilinear pairing and line element do not. The transformation uses the Jacobian matrix of the inverse coordinate change. Consequently, a metric tensor is a coordinate-independent geometric object; its component matrix is only one representation of it. (damtp.cam.ac.uk)

Length, angle, and distance

For a Riemannian metric, a tangent vector has norm

\|v\|_g=\sqrt{g(v,v)},

and nonzero vectors have an angle determined by

\cos\theta=\frac{g(u,v)}{\|u\|_g\|v\|_g}.

The length of a piecewise smooth curve $\gamma:[a,b]\to M$ is the integral

L_g(\gamma)=\int_a^b \sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt.

This length is unchanged by regular reparametrization. On a connected manifold, taking the infimum of curve lengths between two points defines a distance function. (damtp.cam.ac.uk)

The distinction from a metric-space metric is important: $g_p$ pairs tangent vectors at one point, whereas a distance function pairs points of the manifold. Riemannian geometry constructs the latter from the former. An indefinite spacetime metric does not directly define an ordinary metric-space distance through the same positive-length construction. (people.maths.ox.ac.uk)

Examples

In Cartesian coordinates on Euclidean space, $g_{ij}=\delta_{ij}$ , the identity matrix, producing the usual Euclidean distance. In polar coordinates on the plane,

ds^2=dr^2+r^2d\phi^2.

These nonconstant components describe the same flat geometry, demonstrating that coordinate-dependent entries do not by themselves establish curvature. (people.maths.ox.ac.uk)

For a sphere of radius $R$ , the metric inherited from surrounding Euclidean space is

ds^2=R^2d\theta^2+R^2\sin^2\theta\,d\phi^2,

where $\theta$ is colatitude. The spherical-coordinate description breaks down at the poles, although the metric itself remains smooth there. More generally, restricting an ambient Riemannian metric to tangent vectors of an embedded submanifold produces an induced metric. (people.maths.ox.ac.uk)

Inverse metric and geometric differentiation

Nondegeneracy guarantees an inverse matrix $g^{ij}$ , characterized by

g^{ik}g_{kj}=\delta^i{}_j.

The metric identifies a tangent space with its dual space, converting vectors into covectors and conversely:

v_i=g_{ij}v^j,\qquad v^i=g^{ij}v_j.

These operations are called lowering and raising indices. They also enable tensor contractions involving two indices of the same variance. (preposterousuniverse.com)

A metric uniquely determines the Levi-Civita connection, which is torsion-free and metric-compatible: $\nabla g=0$ . Its coefficients are

\Gamma^k{}_{ij} =\frac12g^{k\ell} (\partial_i g_{j\ell}+\partial_j g_{i\ell}-\partial_\ell g_{ij}).

This connection defines the covariant derivative. Its curvature is the Riemann curvature tensor, which depends on the metric, its inverse, and its first and second coordinate derivatives. (people.maths.ox.ac.uk)

Affinely parametrized geodesics satisfy

\ddot x^k+\Gamma^k{}_{ij}\dot x^i\dot x^j=0.

Riemannian geodesics minimize length locally over sufficiently short segments, but need not minimize it globally. The metric also determines the invariant volume density

d\mu_g=\sqrt{|\det(g_{ij})|}\,|dx^1\cdots dx^n|,

where the determinant compensates for coordinate changes. (people.maths.ox.ac.uk)

Role in general relativity

General relativity describes spacetime using a four-dimensional Lorentzian metric. With signature $(-+++)$ , flat spacetime has

ds^2=-c^2dt^2+dx^2+dy^2+dz^2,

where $c$ is the speed of light. Negative, positive, and zero squared intervals correspond locally to timelike, spacelike, and null directions. Along a timelike worldline, proper time satisfies $d\tau=\sqrt{-ds^2}/c$ . (preposterousuniverse.com)

The metric represents the gravitational field, rather than serving only as a fixed background. The Einstein field equations relate curvature derived from it to matter and energy. Freely falling test particles follow timelike geodesics, while light propagation follows null geodesics in the geometric-optics approximation. (damtp.cam.ac.uk)