In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimaldisplacement 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 .
The coordinate-independent definition of the square of the line element ds in an n-dimensionalRiemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement [2] (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 infinitesimaldisplacement 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:[3]
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.
Identification of the square of the line element with the metric tensor
Since is an arbitrary "square of the arc length", completely defines the metric, and 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 coordinatesq = (q1, q2, q3, ..., qn), where it is written as a symmetric rank 2 tensor[3][4] coinciding with the metric tensor:
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:(here i, j = 1, 2, 3 for space) or in matrix form (i denotes row, j denotes column):
The general curvilinear coordinates reduce to Cartesian coordinates:so
for i = 1, 2, 3 are scale factors, so the square of the line element is:
Some examples of line elements in these coordinates are below.[2]
General curvilinear coordinates
Given an arbitrary basis of a space of dimension , the metric is defined as the inner product of the basis vectors.
Where and the inner product is with respect to the ambient space (usually its )
In a coordinate basis
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:[5][1]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: