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:
“As for conforming outwardly, and living your own life inwardly, I do not think much of that. Let not your right hand know what your left hand does in that line of business. It will prove a failure.... It is a greater strain than any soul can long endure. When you get God to pulling one way, and the devil the other, each having his feet well braced,—to say nothing of the conscience sawing transversely,—almost any timber will give way.”
—Henry David Thoreau (1817–1862)
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—Joshua Meyrowitz, U.S. educator, media critic. “The Blurring of Public and Private Behaviors,” No Sense of Place: The Impact of Electronic Media on Social Behavior, Oxford University Press (1985)