In geometry, the line element or length element can most generally be thought of as the change in a position vector in an affine space expressing the change of the arc length. An easy way of visualizing this relationship is by parameterizing the given curve by Frenet–Serret formulas. As such, a line element is then naturally a function of the metric, and can be related to the curvature tensor. It is usually denoted by s or ℓ, and differentials of this are then written ds or dℓ.
Line elements are used in physics, especially in theories of gravitation (most notably general relativity) where spacetime is modelled as a curved manifold with a metric. For example, if a massive object causes some curvature in spacetime, the trajectory of an object with negligible mass over that curvature would follow the line element according to the geodesic equation.
Famous quotes containing the words line and/or element:
“Today, the notion of progress in a single line without goal or limit seems perhaps the most parochial notion of a very parochial century.”
—Lewis Mumford (1895–1990)
“Other sins only speak; murder shrieks out:
The element of water moistens the earth,
But blood flies upwards, and bedews the heavens.”
—John Webster (1580–1625)