Line Element

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 , 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.

Other articles related to "line, line element, lines":

Technical Lettering - Dimensions of Letters
... Central Line is the imaginary line in the middle of each line or line element which is a constitutive part of a graphic character set ... If we consider d as the width of the line element and h as the height of the line element, then the two standard ratios for d/h are 1/14 and 1/10, which are feasible because they result in a minimum number ... Location of Central Lines- The nominal size (h) and the spacing between characters (a) shall be taken as the basis for defining the central line ...
Line Elements in 4d Spacetime - General Spacetime
... The coordinate-independent definition of the square of the line element ds in spacetime is In terms of coordinates where for this case the indices α and β run over 0, 1, 2, 3 for spacetime. ...
Born Coordinates - Radar Distance in The Small
... However, because the Langevin congruence is stationary, we can imagine replacing each world line in this congruence by a point ... To see this, consider the Born line element Setting ds2 = 0 and solving for dt we obtain The elapsed proper time for a roundtrip radar blip emitted by a Langevin observer is then Therefore ... To do this, we need only integrate our line element over the appropriate null geodesic track ...

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