${\displaystyle {\begin{pmatrix}g_{00}&g_{01}&g_{02}&g_{03}\\g_{10}&g_{11}&g_{12}&g_{13}\\g_{20}&g_{21}&g_{22}&g_{23}\\g_{30}&g_{31}&g_{32}&g_{33}\\\end{pmatrix}}}$ Metric tensor of spacetime in general relativity written as a matrix

In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.

In general relativity, the metric tensor plays the role of the gravitational potential in the classical theory of gravitation, although the physical content of the associated equations is entirely different.^[1] Gutfreund and Renn say "that in general relativity the gravitational potential is represented by the metric tensor."^[2]

Notation and conventions

This article works with a metric signature that is mostly positive (− + + +); see sign convention. The gravitation constant ${\displaystyle G}$ will be kept explicit. This article employs the Einstein summation convention, where repeated indices are automatically summed over.

Definition

Mathematically, spacetime is represented by a four-dimensional differentiable manifold ${\displaystyle M}$ and the metric tensor is given as a covariant, second-degree, symmetric tensor on ${\displaystyle M}$, conventionally denoted by ${\displaystyle g}$. Moreover, the metric is required to be nondegenerate with signature (− + + +). A manifold ${\displaystyle M}$ equipped with such a metric is a type of Lorentzian manifold.

Explicitly, the metric tensor is a symmetric bilinear form on each tangent space of ${\displaystyle M}$ that varies in a smooth (or differentiable) manner from point to point. Given two tangent vectors ${\displaystyle u}$ and ${\displaystyle v}$ at a point ${\displaystyle x}$ in ${\displaystyle M}$, the metric can be evaluated on ${\displaystyle u}$ and ${\displaystyle v}$ to give a real number:

This is a generalization of the dot product of ordinary Euclidean space. Unlike Euclidean space – where the dot product is positive definite – the metric is indefinite and gives each tangent space the structure of Minkowski space.

Local coordinates and matrix representations

Physicists usually work in local coordinates (i.e. coordinates defined on some local patch of ${\displaystyle M}$). In local coordinates ${\displaystyle x^{\mu }}$ (where ${\displaystyle \mu }$ is an index that runs from 0 to 3) the metric can be written in the form

The factors ${\displaystyle dx^{\mu }}$ are one-form gradients of the scalar coordinate fields ${\displaystyle x^{\mu }}$. The metric is thus a linear combination of tensor products of one-form gradients of coordinates. The coefficients ${\displaystyle g_{\mu \nu }}$ are a set of 16 real-valued functions (since the tensor ${\displaystyle g}$ is a tensor field, which is defined at all points of a spacetime manifold). In order for the metric to be symmetric giving 10 independent coefficients.

If the local coordinates are specified, or understood from context, the metric can be written as a 4 × 4 symmetric matrix with entries ${\displaystyle g_{\mu \nu }}$. The nondegeneracy of ${\displaystyle g_{\mu \nu }}$ means that this matrix is non-singular (i.e. has non-vanishing determinant), while the Lorentzian signature of ${\displaystyle g}$ implies that the matrix has one negative and three positive eigenvalues. Physicists often refer to this matrix or the coordinates ${\displaystyle g_{\mu \nu }}$ themselves as the metric (see, however, abstract index notation).

With the quantities ${\displaystyle dx^{\mu }}$ being regarded as the components of an infinitesimal coordinate displacement four-vector (not to be confused with the one-forms of the same notation above), the metric determines the invariant square of an infinitesimal line element, often referred to as an interval. The interval is often denoted

The interval ${\displaystyle ds^{2}}$ imparts information about the causal structure of spacetime. When ${\displaystyle ds^{2}<0}$, the interval is timelike and the square root of the absolute value of ${\displaystyle ds^{2}}$ is an incremental proper time. Only timelike intervals can be physically traversed by a massive object. When ${\displaystyle ds^{2}=0}$, the interval is lightlike, and can only be traversed by (massless) things that move at the speed of light. When ${\displaystyle ds^{2}>0}$, the interval is spacelike and the square root of ${\displaystyle ds^{2}}$ acts as an incremental proper length. Spacelike intervals cannot be traversed, since they connect events that are outside each other's light cones. Events can be causally related only if they are within each other's light cones.

The components of the metric depend on the choice of local coordinate system. Under a change of coordinates ${\displaystyle x^{\mu }\to x^{\bar {\mu }}}$, the metric components transform as