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:
“There’s something like a line of gold thread running through a man’s words when he talks to his daughter, and gradually over the years it gets to be long enough for you pick up in your hands and weave into a cloth that feels like love itself. It’s another thing, though, to hold up that cloth for inspection.”
—John Gregory Brown (20th century)
“Justice has its anger, my lord Bishop, and the wrath of justice is an element of progress. Whatever else may be said of it, the French Revolution was the greatest step forward by mankind since the coming of Christ. It was unfinished, I agree, but still it was sublime. It released the untapped springs of society; it softened hearts, appeased, tranquilized, enlightened, and set flowing through the world the tides of civilization. It was good. The French Revolution was the anointing of humanity.”
—Victor Hugo (1802–1885)