How Do Indices Function in Tensor Notation?

[Miloslav]

I was wondering how the indices of tensors work. I do not understand how the indices of tensors in can be used. For example, \eta _{\mu \nu }, the metric tensor, is like a matrix, and x^{u} is a contravector. How does this extend to notations such as T{_{a}}^{bc} and T{_{ab}}^{c}?

 

[Matterwave]

use wraps to make the latex show up.<br /> <br /> The index notation is used in 2 ways. 1, it is used to denote the components of a tensor in some (arbitrary) coordinate system. And 2, it is often used to denote the tensor itself. The second way is a shortcut, but, strictly speaking, is an abuse of notation. <br /> <br /> Lower indices indicate covariance while upper indices indicate contravariance. The tensors you wrote at the end are respectively once covariant twicce contravariant and twice covariant once contravariant.

 

[Miloslav]

Ok. Thanks for the tip about Latex. I was hoping to clarify that tensors are used in a certain context. Thank you for the information, I now understand how it can be used, particularly in the case of the Kronecker delta.

 

[HallsofIvy]

In flat, Euclidean, space, in which we can use the Kronecker \delta as metric tensor, there is no distinction between "covariant" and "contravariant" tensors.