Why must contravariant tensors contract with covariant tensors?

[Master J]

Why is it that a contravariant tensor must be contracted with a covariant tensor, and vice versa? Why is this so?

 

[dextercioby]

You can't justify a definition, just read it carefully. So reread what a contracted tensor product is and you'll understand what exactly happens.

 

[Sina]

As said it is a part of the definition. If you time and want to learn the more geometric picture involving vector spaces and their duals then I suggest for instance as a start

http://www.strw.leidenuniv.nl/~yuri/GR/handout1.pdf

This type of perspective will certainly make more sense and you will understand why they are defined by their transformation properties.

ps: what you can do with two contravariant tensors is for instance to take their tensor product to obtain another contravariant tensor of higher rank (that is with more indices). Similiarly you may take tensor product of two tensors of any type and possibly obtain tensors of mixed type too.