Units if conversion between covariant/contravariant tensors

[joneall]

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.

 

[jvicens]

Thank you for your question. It is common for people to struggle with the concept of covariant and contravariant tensors, so do not feel discouraged.

First of all, it is important to understand that the terms "covariant" and "contravariant" refer to how the components of a tensor transform under a change of coordinates. In other words, it is a mathematical property of the tensor and not related to its physical units.

The confusion may arise because we often use the same notation for both covariant and contravariant tensors, such as using upper and lower indices. However, the position of the indices does not determine the type of tensor; it is the transformation rule that does.

Now, let's address your example of using the metric to transform a contravariant vector to a covariant tensor. The metric is indeed dimensionless, but it does not change the units of the tensor. The units are determined by the physical quantities that the tensor represents, such as length, time, or mass. The metric simply provides a way to relate these quantities in different coordinate systems.

In summary, the concept of covariant and contravariant tensors is not related to their physical units. It is a mathematical property that describes how the components of a tensor transform under a change of coordinates. I hope this helps clarify your understanding. Keep exploring and asking questions, and you will develop a better understanding of these important concepts in tensor analysis.