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Concept
Line element
Summary
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 .
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.
The coordinate-independent definition of the square of the line element ds in an n-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement (in pseudo Riemannian manifolds possibly negative) whose square root should be used for computing curve length:
where g is the metric tensor, · denotes inner product, and dq an infinitesimal displacement on the (pseudo) Riemannian manifold. By parametrizing a curve , we can define the arc length of the curve length of the curve between , and as the integral:
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 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 and volume elements etc.
Since is arbitrary "square of the arc length" completely defines the metric, it is therefore usually best to consider the expression for as a definition of the metric tensor itself, written in a suggestive but non tensorial notation:
This identification of the square of arc length with the metric is even more easy to see in n-dimensional general curvilinear coordinates q = (q1, q2, q3, .
Official source
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