I am still at the stage of trying to assimilate contravariant and covariant tensors, so my question probably has a simpler answer than I realize. A covariant tensor is like a gradient, as its units increase when the coordinate units do. A contravariant tensor's components decrease when the coordinates increase. In other words, covariant is inversely proportional to length; contravariant, proportional. This means that if I use the metric to transform a contravariant vector to a covariant tensor, I have changed the units by a factor of length^2. Yet, the metric is dimensionless, at least the Minkowski one is. What have I misunderstood? I suspect the units don't change, but then my idea of what makes a tensor co- or contravariant is all wrong.