Line element

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 ds 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 ds 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 infinitessimal displacement d q {displaystyle dmathbf {q} } (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 parameterising a curve q ( λ ) {displaystyle q(lambda )} parametrised by a parameter λ {displaystyle lambda } , we can define the arc length of the curve length of the curve between q ( λ 1 ) {displaystyle q(lambda _{1})} , and q ( λ 2 ) {displaystyle q(lambda _{2})} is 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 − + + + {displaystyle -+++} 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 d q {displaystyle dmathbf {q} } is arbitrary 'square of the arc length' d s 2 {displaystyle ds^{2}} completely defines the metric, it is therefore usually best to consider the expression for d s 2 {displaystyle ds^{2}} as a definition of the metric tensor itself, written in a suggestive but non tensorial notation: This identification of the square of arc length d s 2 {displaystyle ds^{2}} with the metric is even more easy to see in n-dimensional general curvilinear coordinates q = (q1, q2, q3, ..., qn), where it is written as a symmetric rank 2 tensor coinciding with the metric tensor: Here the 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.