Metric tensor (general relativity)

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

In general relativity, the metric tensor (or simply, the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separating the future and the past.

Throughout this article we work with a metric signature that is mostly positive (− + + +); see sign convention. The gravitation constant G will be kept explicit. The summation convention, where repeated indices are automatically summed over, is employed.

Mathematically, spacetime is represented by a 4-dimensional differentiable manifold M and the metric is given as a covariant, second-order, symmetric tensor on M, conventionally denoted by g. Moreover, the metric is required to be nondegenerate with signature (-+++). A manifold M equipped with such a metric is a type of Lorentzian manifold.

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