Editing Line element

{{Short description|Line segment of infinitesimally small length}}
{{about|lines in mathematics|Long Interspersed Nuclear Elements in DNA|Retrotransposon#LINEs}}

In [[geometry]], the '''line element''' or '''length element''' can be informally thought of as a line segment associated with an [[infinitesimal]] [[displacement vector]] in a [[metric space]]. The length of the line element, which may be thought of as a differential [[arc length]], is a function of the [[metric tensor]] and is denoted by ''<math>ds</math>''.

Line elements are used in [[physics]], especially in theories of [[gravitation]] (most notably [[general relativity]]) where [[spacetime]] is modelled as a curved [[Pseudo-Riemannian manifold]] with an appropriate [[metric tensor (general relativity)|metric tensor]].<ref name="WheelerMisnerThorne">Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, {{isbn|0-7167-0344-0}}</ref>

==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]].

==Line elements in Euclidean space==
{{main|Euclidean space}}

[[File:Line element.svg|thumb|Vector line element d'''r''' (green) in [[three-dimensional|3d]] Euclidean space, where λ is a [[parametric equation|parameter]] of the space curve (light green).]]

Following are examples of how the line elements are found from the metric.

===Cartesian coordinates===

The simplest line element is in [[Cartesian coordinates]] - in which case the metric is just the [[Kronecker delta]]:
<math display="block">g_{ij} = \delta_{ij}</math>
(here ''i, j'' = 1, 2, 3 for space) or in [[matrix (mathematics)|matrix]] form (''i'' denotes row, ''j'' denotes column):
<math display="block">[g_{ij}] = \begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{pmatrix}</math>

The general curvilinear coordinates reduce to Cartesian coordinates:
<math display="block">(q^1,q^2,q^3) = (x, y, z)\,\Rightarrow\,d\mathbf{r}=(dx,dy,dz)</math>
so
<math display="block"> ds^2 = g_{ij} dq^i dq^j = dx^2 +dy^2 +dz^2 </math>

===Orthogonal curvilinear coordinates===

For all [[orthogonal coordinates]] the metric is given by:<ref name="SpiegelLipschutzSpellman"/>
<math display="block">[g_{ij}] = \begin{pmatrix}
h_1^2 & 0 & 0\\
0 & h_2^2 & 0\\
0 & 0 & h_3^2
\end{pmatrix}</math>
where
<math display="block">h_i = \left|\frac{\partial\mathbf{r}}{\partial q^i}\right|</math>

for ''i'' = 1, 2, 3 are [[curvilinear coordinates#Orthogonal curvilinear coordinates in 3d|scale factor]]s, so the square of the line element is:
<math display="block">ds^2 = h_1^2(dq^1)^2 + h_2^2(dq^2)^2 + h_3^2(dq^3)^2 </math>

Some examples of line elements in these coordinates are below.<ref name="Kay"/>

{| class="wikitable"
|-
! Coordinate system
! {{math|(''q''<sup>1</sup>, ''q''<sup>2</sup>, ''q''<sup>3</sup>)}}
! Metric
! Line element
|-
|[[Cartesian coordinate system|Cartesian]]
|{{math|(''x'', ''y'', ''z'')}}
|<math>[g_{ij}] = \begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{pmatrix}</math>
|<math> ds^2 = dx^2 + dy^2 + dz^2 </math>
|-
|[[Polar coordinate system|Plane polars]]
|{{math|(''r'', ''θ'')}}
|<math>[g_{ij}] = \begin{pmatrix}
1 & 0 \\
0 & r^2 \\
\end{pmatrix}</math>
|<math> ds^2= dr^2 +r^2 d \theta^2</math>
|-
|[[Spherical coordinate system|Spherical polar]]s
|{{math|(''r'', ''θ'', ''φ'')}}
|<math>[g_{ij}] = \begin{pmatrix}
1 & 0 & 0 \\
0 & r^2 & 0 \\
0 & 0 & r^2\sin^2\theta \\
\end{pmatrix}</math>
|<math> ds^2=dr^2+r^2 d \theta\ ^2+ r^2 \sin^2 \theta d \varphi^2 </math>
|-
|[[Cylindrical polar coordinates|Cylindrical polar]]s
|{{math|(''r'', ''φ'', ''z'')}}
|<math>[g_{ij}] = \begin{pmatrix}
1 & 0 & 0 \\
0 & r^2 & 0 \\
0 & 0 & 1 \\
\end{pmatrix}</math>
|<math> ds^2=dr^2+ r^2 d \varphi^2 +dz^2 </math>
|}

==General curvilinear coordinates==

Given an arbitrary basis <math>\{\hat{b}_{i}\}</math> of a space of dimension <math> n </math>, the metric is defined as the inner product of the basis vectors.
<math display="block">g_{ij}=\langle\hat{b}_{i},\hat{b}_{j}\rangle</math>

Where <math>1\leq i,j\leq n</math> and the inner product is with respect to the ambient space (usually its <math>\delta_{ij}</math>)

In a coordinate basis <math>\hat{b}_{i} = \frac{\partial}{\partial x^{i}}</math>

The coordinate basis is a special type of basis that is regularly used in differential geometry.

==Line elements in 4d spacetime==

===Minkowski spacetime===

The [[Minkowski metric]] is:<ref>Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, {{isbn|0-07-145545-0}}</ref><ref name="WheelerMisnerThorne"/>
<math display="block">[g_{ij}] = \pm \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \\
\end{pmatrix}</math>
where one sign or the other is chosen, both conventions are used. This applies only for [[flat spacetime]]. The coordinates are given by the [[4-position]]:
<math display="block">\mathbf{x} = (x^0,x^1,x^2,x^3) = (ct,\mathbf{r}) \,\Rightarrow\, d\mathbf{x} = (c dt, d\mathbf{r})</math>

so the line element is:
<math display="block">ds^2 = \pm (c^2 dt^2 - d\mathbf{r} \cdot d\mathbf{r}) .</math>

===Schwarzschild coordinates===

In [[Schwarzschild coordinates]] coordinates are <math> \left(t, r, \theta, \phi \right)</math>, being the general metric of the form:
<math display="block">[g_{ij}] = \begin{pmatrix}
-a(r)^2 & 0 & 0 & 0 \\
0 & b(r)^2 & 0 & 0 \\
0 & 0 & r^2 & 0 \\
0 & 0 & 0 & r^2 \sin^2\theta \\
\end{pmatrix}</math>

(note the similitudes with the metric in 3D spherical polar coordinates).

so the line element is:
<math display="block">ds^2 = -a(r)^2 \, dt^2 + b(r)^2 \, dr^2 + r^2 \, d\theta^2 + r^2 \sin^2\theta \, d\phi^2 .</math>

===General spacetime===

The coordinate-independent definition of the square of the line element d''s'' in [[Spacetime#Spacetime intervals|spacetime]] is:<ref name="WheelerMisnerThorne"/>
<math display="block"> ds^2 = d\mathbf{x} \cdot d\mathbf{x} = g(d\mathbf{x},d\mathbf{x}) </math> 

In terms of coordinates:
<math display="block"> ds^2 = g_{\alpha\beta} dx^\alpha dx^\beta </math>
where for this case the indices {{math|''α''}} and {{math|''β''}} run over 0, 1, 2, 3 for spacetime.

This is the [[spacetime interval]] - the measure of separation between two arbitrarily close [[Event (relativity)|events]] in [[spacetime]]. In [[special relativity]] it is invariant under [[Lorentz transformation]]s. In [[general relativity]] it is invariant under arbitrary [[inverse function|invertible]] [[Differentiable function|differentiable]] [[coordinate transformations]].

==See also==

*[[Covariance and contravariance of vectors]]
*[[First fundamental form]]
*[[List of integration and measure theory topics]]
*[[Metric tensor]]
*[[Ricci calculus]]
*[[Raising and lowering indices]]
*[[Volume element]]

==References==

{{reflist}}

[[Category:Affine geometry]]
[[Category:Riemannian geometry]]
[[Category:Special relativity]]
[[Category:General relativity]]

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[[de:Linienelement]]