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 [tex] wraps to make the latex show up.

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.

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 [itex]\delta[/itex] as metric tensor, there is no distinction between "covariant" and "contravariant" tensors.

 

[blue_raver22]

The indices of tensors play a crucial role in denoting the components of a tensor in a specific coordinate system. In the example provided, \eta _{\mu \nu } is the metric tensor, which is used to measure distances and angles in a given space. The indices \mu and \nu represent the rows and columns of the matrix, respectively, and indicate the specific components of the tensor.

The notation x^{u} refers to a contravariant vector, where the superscript u indicates the component of the vector in a given direction. This notation is commonly used in relativity and other areas of physics to represent the components of a vector in a specific coordinate system.

The notation T{_{a}}^{bc} and T{_{ab}}^{c} extend the use of indices to higher order tensors. In these cases, the indices a, b, and c represent the different dimensions of the tensor, and the superscripts and subscripts indicate the specific components within those dimensions. This notation is essential in performing tensor operations and transformations, as it allows us to keep track of the components and their transformations between different coordinate systems.

In summary, indices in tensor notation are a powerful tool for denoting the components of tensors in a specific coordinate system and performing tensor operations. They allow us to work with tensors of different dimensions and keep track of their transformations, making them essential in various areas of physics and mathematics.