What Do Upper and Lower Indices in Tensor Notation Signify?

[taylrl3]

Hi,

I am very new to general relativity and have only just started to learn how to do some very basic manipulation of tensors. I can understand the methods I am using and have some idea of what a tensor is but am not sure what the difference between upper and lower indices signifies. I can identify that one is covariant and another contravariant but what is the difference between the two and what about when a tensor has both indices? I feel I need to clear this conceptual issue up before I can understand things further. Thanks :-)

Taylrl

 

[atyy]

Usually one of them is associated to "covectors" and the other to "vectors". Both are vectors in the linear algebra sense, but on a manifold, the former correspond to "forms" which are things that can be integrated even without a metric. In the presence of a metric, as in general relativity, there is one-to-one correspondence between covectors and vectors. http://www.math.ucla.edu/~tao/preprints/forms.pdf

A perhaps more physical explanation goes something like https://www.physicsforums.com/showpost.php?p=3361102&postcount=14.