What does this notation mean? (suffix/prefix on tensors?)

• I
• sa1988
In summary, the conversation discusses the use of upper and lower indices in tensor analysis. Lower indices are covariant and upper indices are contravariant, and they are used to distinguish between the two. This allows for a methodical way of organizing units and per unit measurements. The confusion about these indices was encountered in the context of relativistic kinematics, where the raised index represents spatial coordinates and the lower index represents time. This is because the raised index changes contrary to the basis change, while the lower index remains on the time axis in a Minkowski diagram.
sa1988
Looking at relativistic transformations and suddenly we have this transformation matrix with an upper and lower index. See below:

A bit of googling tells me the upper index means a co-ordinate. However I'm not sure what the lower index is. Overall I have no idea what makes it so special, or how to perform the operation in any way different from the sort of index notation I've already come across in fluid dynamics, wherein we have things like this for the divergence of a tensor:

$$\nabla \cdot A = \frac{\partial}{\partial x_i}A_{ij}$$

I simply don't get what's going on with the sudden separation into upper and lower indices here. Any advice would be appreciated, thanks.

In tensor analysis, there are covariant and contravariant indices. The lower and upper indices are used to distinguish between the two. Lower indices are covariant and upper indices are contravariant. With a contravariant index, when a unit of measure is converted to a smaller one, the associated number of units gets larger (e.g. conversion of 1 hour to 3600 seconds). With a covariant index, when a unit of measure is converted to a smaller one , the associated number gets smaller ( e.g. 1 mile/hour conversion to 1/3600 miles/sec ). You can find a more formal explanation, which still attempts to keep some intuitive motivation, in http://www.ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf

When you are dealing with a mixture of "units" and "per unit", it is helpful to have a methodical way of keeping things organized.

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sa1988
FactChecker said:
In tensor analysis, there are covariant and contravariant indices. The lower and upper indices are used to distinguish between the two. Lower indices are covariant and upper indices are contravariant. With a contravariant index, when a unit of measure is converted to a smaller one, the associated number of units gets larger (e.g. conversion of 1 hour to 3600 seconds). With a covariant index, when a unit of measure is converted to a smaller one , the associated number gets smaller ( e.g. 1 mile/hour conversion to 1/3600 miles/sec ). You can find a more formal explanation, which still attempts to keep some intuitive motivation, in http://www.ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf

When you are dealing with a mixture of "units" and "per unit", it is helpful to have a methodical way of keeping things organized.

Great stuff, thanks for that. Extremely helpful little document!

EDIT: So just to make sure I'm on the right track: Raised indices are contravariant vectors, i.e. if you change the basis, the vectors change 'contrary' to that because they essentially have to be changed in the opposite direction of the basis change. Covariant vectors are ones that change with the basis.

The place I bumped into this confusion was in the relativistic kinematics section of the advanced mechanics module I'm doing. I hope I'd be right in thinking that the raised index in this context is spatial co-ordinates as they will surely be contravariant with respect to lorentz transforms which essentially alter the bases from which the situation is being looked at, whereas the lower index is time, which sticks to the time axis on, for example, a Minkowski diagram.

So, in a space-time vector (ct, x1, x2, x3) , we have (covariant, contravariant, contravariant, contravariant)

Is this right?

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1. What is a tensor?

A tensor is a mathematical object that represents a multi-dimensional array of numbers. It is used to describe physical quantities and their relationships in a coordinate-independent manner.

2. What is a suffix on a tensor?

A suffix on a tensor is a numerical index that represents the dimension of the tensor in a specific coordinate system. It is typically represented by a lower-case Latin letter, such as i, j, or k.

3. What is a prefix on a tensor?

A prefix on a tensor is a numerical index that represents the order of the tensor. It is typically represented by an upper-case Latin letter, such as A, B, or C.

4. How do I read a tensor notation?

A tensor notation is read from left to right, with the suffixes placed to the right of the prefix. For example, Aij represents the element of the tensor A at the ith row and jth column.

5. Why are tensors used in science?

Tensors are used in science because they provide a way to describe physical quantities and their relationships in a coordinate-independent manner. This makes them applicable in various fields such as physics, engineering, and computer science.

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