Editing Line element (section)

==Line elements in 4d spacetime==

===Minkowski spacetime===

The [[Minkowski metric]] is:<ref>Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, {{isbn|0-07-145545-0}}</ref><ref name="WheelerMisnerThorne"/>
<math display="block">[g_{ij}] = \pm \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \\
\end{pmatrix}</math>
where one sign or the other is chosen, both conventions are used. This applies only for [[flat spacetime]]. The coordinates are given by the [[4-position]]:
<math display="block">\mathbf{x} = (x^0,x^1,x^2,x^3) = (ct,\mathbf{r}) \,\Rightarrow\, d\mathbf{x} = (c dt, d\mathbf{r})</math>

so the line element is:
<math display="block">ds^2 = \pm (c^2 dt^2 - d\mathbf{r} \cdot d\mathbf{r}) .</math>

===Schwarzschild coordinates===

In [[Schwarzschild coordinates]] coordinates are <math> \left(t, r, \theta, \phi \right)</math>, being the general metric of the form:
<math display="block">[g_{ij}] = \begin{pmatrix}
-a(r)^2 & 0 & 0 & 0 \\
0 & b(r)^2 & 0 & 0 \\
0 & 0 & r^2 & 0 \\
0 & 0 & 0 & r^2 \sin^2\theta \\
\end{pmatrix}</math>

(note the similitudes with the metric in 3D spherical polar coordinates).

so the line element is:
<math display="block">ds^2 = -a(r)^2 \, dt^2 + b(r)^2 \, dr^2 + r^2 \, d\theta^2 + r^2 \sin^2\theta \, d\phi^2 .</math>

===General spacetime===

The coordinate-independent definition of the square of the line element d''s'' in [[Spacetime#Spacetime intervals|spacetime]] is:<ref name="WheelerMisnerThorne"/>
<math display="block"> ds^2 = d\mathbf{x} \cdot d\mathbf{x} = g(d\mathbf{x},d\mathbf{x}) </math> 

In terms of coordinates:
<math display="block"> ds^2 = g_{\alpha\beta} dx^\alpha dx^\beta </math>
where for this case the indices {{math|''α''}} and {{math|''β''}} run over 0, 1, 2, 3 for spacetime.

This is the [[spacetime interval]] - the measure of separation between two arbitrarily close [[Event (relativity)|events]] in [[spacetime]]. In [[special relativity]] it is invariant under [[Lorentz transformation]]s. In [[general relativity]] it is invariant under arbitrary [[inverse function|invertible]] [[Differentiable function|differentiable]] [[coordinate transformations]].