What is Line element: Definition and 55 Discussions

In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc length, is a function of the metric tensor and is denoted by ds
Line elements are used in physics, especially in theories of gravitation (most notably general relativity) where spacetime is modelled as a curved Pseudo-Riemannian manifold with an appropriate metric tensor.

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  1. B

    How Do You Derive the Line Element on a 3-Sphere?

    Hello, I have been trying to figure out the line element on a 3-sphere for my GR class. Problem is that I am trying to go the traditional way of finding dx, dy, dz, and dw, then regroup and find the respective unit vectors and go from there. We are given that the answer as: ds^2 = a^2(d\chi^2...
  2. Mentz114

    Is this line element known to anyone ?

    This is based on the metric of the surface of a 3D sphere. A and B are constants with dimension (length)^2. Coordinates are x^0 = t, x^1 = \theta, x^2 = \phi ds^2 = -c^2AB^{-1}dt^2 + Bd\theta^2 + Bsin^2(\theta)d\phi^2 It satisfies the Einstein field equations with only one component of...
  3. P

    Us humans visualising the fourth dimensional line element

    Alright. So you guys already know that the first dimension is simply a line (according to our 3-space perspective). The second dimension extends a line at 90 degrees from the first, creating a plane. Adding yet another line 90 degrees from the first two lends a cube. You may spare no great...
  4. S

    General Orthogonal Coordinate System: Line Element Explained

    A general orthogonal coordinate system (u,v,w) will have a line elemet of the form: ds^2 = f^2 du^2 + g^2dv^2 + h^2dw^2 I have done a lot of vector calculus, but for some reason I can't figure out what this means! What is a line element? I know about the differential length element and its...
  5. B

    Line element in spherical coordinates

    Hi, I was just reading up on some astrophysics and I saw the line element (general relativity stuff) written in spherical coordinates as: ds^2 = dr^2 + r^2(d\theta^2 + \sin\theta\d\phi) I don't get this. dr is the distance from origo to the given point, so why isn't ds^2 = dr^2 without...
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