GR vs SR: Reconciling Contravariant & Covariant Vector Components

[nigelscott]

I am trying to reconcile the definition of contravariant and covariant
components of a vector between Special Relativity and General Relativity.

In GR I understand the difference is defined by the way that the vector
components transform under a change in coordinate systems.

In SR it seems that it is more of a notational thing that allows for the
derivation of the invariant interval.

How are the 2 things related?

 

[Nugatory]

Do you have a source for the different treatment in SR? It can be made to look like it's just a notational thing in SR by using Minkowski coordinates where $g_{\mu\rho}=g^{\mu\rho}$ so many of the differences between contravariant and covariant components disappear - but that's just taking advantage of the fact that flat spacetime is an especially nice special case (which is why we call it "special" relativity).

It's a good exercise to choose some perverse coordinate transform in which the metric acquires off-diagonal and non-constant components even in flat space-time, work a few otherwise-easy problems in those coordinates, just so that you can see the machinery working. You'll find that the contra/co distinction appears even in SR.

 

[nigelscott]

OK, soon after I posted I realized that the connection is through the metric tensor.