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Concept# Metric tensor (general relativity)

Summary

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. Gutfreund and Renn say "that in general relativity the gravitational potential is represented by the metric tensor."
This article works with a metric signature that is mostly positive (− + + +); see sign convention. The gravitation constant will be kept explicit. This article employs the Einstein summation convention, where repeated indices are automatically summed over.
Mathematically, spacetime is represented by a four-dimensional differentiable manifold and the metric tensor is given as a covariant, second-degree, symmetric tensor on , conventionally denoted by . Moreover, the metric is required to be nondegenerate with signature (− + + +). A manifold 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 that varies in a smooth (or differentiable) manner from point to point. Given two tangent vectors and at a point in , the metric can be evaluated on and 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.
Physicists usually work in local coordinates (i.e. coordinates defined on some local patch of ). In local coordinates (where is an index that runs from 0 to 3) the metric can be written in the form
The factors are one-form gradients of the scalar coordinate fields .

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