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:
“We have not passed that subtle line between childhood and adulthood until we move from the passive voice to the active voice—that is, until we have stopped saying “It got lost,” and say, “I lost it.””
—Sydney J. Harris (b. 1917)
“I think it is a wise course for laborers to unite to defend their interests.... I think the employer who declines to deal with organized labor and to recognize it as a proper element in the settlement of wage controversies is behind the times.... Of course, when organized labor permits itself to sympathize with violent methods or undue duress, it is not entitled to our sympathy.”
—William Howard Taft (1857–1930)