Line element Definition and 56 Threads

Hey folks, Does anyone know how to get the line element on a torus? The ds^2 term. I want to find the metric for toroidal geometry. Any help appreciated! Richard

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...

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...

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...

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...

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...