Understanding the Relationship between Line Element and Length in Four Vectors

[asdqwe]

What is the difference between the line element and length of a four vector? They both seem to have the same definition just with slightly different notation, so is the line element just the length of a specfic vector.

Also, if the magnitude of a four-vector is calculated to b -1 is this still a unit vector?

 

[WannabeNewton]

The line element, when you see it in the notation $ds^2$ in GR books, is usually written in the coordinate basis as $ds^2 = g(\partial_{\mu},\partial_{\nu})dx^{\mu}dx^{\nu}= g_{\mu\nu}dx^{\mu} dx^{\nu}$ where the $\partial_{\mu}$ are the coordinate vector fields and the $dx^{\mu}$ are the corresponding covector fields on $U\subseteq M$ for some space-time $(M,g)$. It is a notational way of conveying "infinitesimal arc-length" because the actual $g = g(\partial_{\mu},\partial_{\nu})dx^{\mu}\otimes dx^{\nu}$, still in the coordinate basis, doesn't really convey the same intuition. The metric tensor itself is just a map that assigns an inner product to each $T_{p}M$ (with some extra conditions e.g. the inner product must vary smoothly from tangent space to tangent space). All we have done is express this map in the coordinate basis.

On the other hand, for $v\in T_{p}M$, the length of this vector is simply, in the coordinate basis, $g(v,v) = g_{\mu\nu}v^{\mu}v^{\nu}$. And yes, if $g(v,v) = -1$ it is, at least in GR books, still called a vector of "unit" norm if the $g_{\mu\nu}v^{\mu}v^{\nu} < 0$ for time-like vectors convention is used anyways.