Editing Line element (section)

==General formulation==

{{for|notation used|Ricci calculus|Einstein notation}}

===Definition of the line element and arc length===

The [[coordinate]]-independent definition of the square of the line element ''ds'' in an ''n''-[[dimension]]al [[Riemannian manifold|Riemannian]] or [[Pseudo Riemannian manifold]] (in physics usually a [[spacetime manifold|Lorentzian manifold]]) is the  "square of the length"  of an infinitesimal displacement <math>d\mathbf{q}</math><ref name="Kay">Tensor Calculus, D.C. Kay, Schaum’s Outlines, McGraw Hill (USA), 1988, {{isbn|0-07-033484-6}}</ref> (in pseudo Riemannian manifolds possibly negative) whose square root should be used for computing curve length: 
<math display="block"> ds^2 = d\mathbf{q}\cdot d\mathbf{q} = g(d\mathbf{q},d\mathbf{q})</math> 
where ''g'' is the [[metric tensor]], '''·''' denotes [[inner product]], and ''d'''''q''' an [[infinitesimal]] [[Displacement (vector)|displacement]] on  the (pseudo) Riemannian manifold. By parametrizing a curve <math>\mathbf{q}(\lambda)</math>, we can define the [[arc length]] of the curve length of the curve between <math>\mathbf{q}_1=\mathbf{q}(\lambda_1)</math>, and <math>\mathbf{q}_2=\mathbf{q}(\lambda_2)</math> as the [[integral]]:<ref name="SpiegelLipschutzSpellman">Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipcshutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, {{isbn|978-0-07-161545-7}}</ref>
<math display="block"> s = \int_{\mathbf{q}_1}^{\mathbf{q}_2}\sqrt{ \left|ds^2\right|} = \int_{\lambda_1}^{\lambda_2} d\lambda \sqrt{ \left|g\left(\frac{d\mathbf{q}}{d\lambda},\frac{d\mathbf{q}}{d\lambda}\right)\right|} = \int_{\lambda_1}^{\lambda_2} d\lambda \sqrt{ \left|g_{ij}\frac{dq^i}{d\lambda}\frac{dq^j}{d\lambda}\right|}.</math>

To compute a sensible length of curves in pseudo Riemannian manifolds, it is best to assume that the infinitesimal displacements have the same sign everywhere. E.g. in physics the square of a line element along a timeline curve would (in the <math>-+++</math> signature convention) be negative and the negative square root of the square of the line element along the curve would measure the proper time passing for an observer moving along the curve.
From this point of view, the metric also defines in addition to line element the [[surface (topology)|surface]] and [[volume element]]s etc.

===Identification of the square of the line element with the metric tensor===
Since <math>d\mathbf{q}</math> is an arbitrary "square of the arc length", <math>ds^2</math> completely defines the metric, and it is therefore usually best to consider the expression for <math>ds^2</math> as a definition of the metric tensor itself, written in a suggestive but non tensorial notation:
<math display="block">ds^2 = g</math> 
This identification of the square of arc length <math>ds^2</math> with the metric is even more easy to see in ''n''-dimensional general [[curvilinear coordinates]] {{nowrap|1='''q''' = (''q''<sup>1</sup>, ''q''<sup>2</sup>, ''q''<sup>3</sup>, ..., ''q<sup>n</sup>'')}}, where it is  written as a symmetric rank 2 tensor<ref name="SpiegelLipschutzSpellman"/><ref>An introduction to Tensor Analysis: For Engineers and Applied Scientists, J.R. Tyldesley, Longman, 1975, {{isbn|0-582-44355-5}}</ref> coinciding with the metric tensor:
<math display="block"> ds^2= g_{ij} dq^i dq^j = g .</math>

Here the [[Ricci calculus|indices]] ''i'' and ''j'' take values 1, 2, 3, ..., ''n'' and [[Einstein summation convention]] is used. Common examples of (pseudo) Riemannian spaces include [[three-dimensional]] [[space]] (no inclusion of [[time]] coordinates), and indeed [[four-dimensional]] [[spacetime]].