# On the Validity of Swapping Dummy Indices in Tensor Manipulation

• Wannabe Physicist
In summary, property (a) states that a second rank tensor that vanishes in one frame will also vanish in all frames related by rotations. Using this property, it can be shown that a symmetric tensor remains symmetric under all rotations. This is done by considering the transformation law under rotation and the definition of a symmetric tensor. The statement of swapping indices is valid, as shown in the equations.
Wannabe Physicist
Homework Statement
Using property (a), show that a symmetric tensor ##T_{i_1 i_2 }## remains symmetric under all rotations.
Relevant Equations
(1) Transformation law under rotation: ##T_{i_1 i_2 }' = r_{i_1 j_1} r_{i_2 j_2} T_{j_1 j_2}##
(2) Definition of symmetric tensor: ##T_{i_1 i_2} - T_{i_2 i_1} = 0##
Property (a) simply states that a second rank tensor that vanishes in one frame vanishes in all frames related by rotations.

I am supposed to prove: ##T_{i_1 i_2} - T_{i_2 i_1} = 0 \implies T_{i_1 i_2}' - T_{i_2 i_1}' = 0##

Here's my solution. Consider,

$$T_{i_1 i_2}' - T_{i_2 i_1}' = r_{i_1 j_1} r_{i_2 j_2} T_{j_1 j_2} - r_{i_2 j_1} r_{i_1 j_2} T_{j_1 j_2}$$

**Now consider this statement:** Because ##j_1## and ##j_2## are dummy indices and both are summed from 1 to 3, we can swap these indices exclusively for the second term in the above expression.

If I assume the above statement it is easy to obtain

$$T_{i_1 i_2}' - T_{i_2 i_1}' = r_{i_1 j_1} r_{i_2 j_2} T_{j_1 j_2} - r_{i_2 j_2} r_{i_1 j_1} T_{j_2 j_1}$$
$$T_{i_1 i_2}' - T_{i_2 i_1}' = r_{i_1 j_1} r_{i_2 j_2} [T_{j_1 j_2} - T_{j_2 j_1}]$$And then using property (a), I can prove the required statement.

But I am not sure if the statement of swapping indices is valid.

vanhees71
Wannabe Physicist said:
Homework Statement:: Using property (a), show that a symmetric tensor ##T_{i_1 i_2 }## remains symmetric under all rotations.
Relevant Equations:: (1) Transformation law under rotation: ##T_{i_1 i_2 }' = r_{i_1 j_1} r_{i_2 j_2} T_{j_1 j_2}##
(2) Definition of symmetric tensor: ##T_{i_1 i_2} - T_{i_2 i_1} = 0##

But I am not sure if the statement of swapping indices is valid.
It simply says obviously
$$\sum_{i=1}^n A_i B_i=\sum_{j=1}^n A_j B_j=\sum_{\gamma=1}^n A_\gamma B_\gamma$$
where ##\gamma=\{a,b,c,...,\alpha,\beta,...,\xi,\eta,\zeta,...\}## any symbol you like.

vanhees71
Oh right! Thanks a lot!

## 1. What is the purpose of swapping dummy indices in tensor manipulation?

The purpose of swapping dummy indices in tensor manipulation is to simplify and streamline calculations involving tensors. By swapping dummy indices, we can rearrange the order of operations and perform calculations more efficiently.

## 2. How do we determine the validity of swapping dummy indices?

The validity of swapping dummy indices can be determined by checking if the resulting equation is equivalent to the original equation. If the two equations yield the same result, then swapping dummy indices is valid.

## 3. Are there any limitations to swapping dummy indices in tensor manipulation?

Yes, there are limitations to swapping dummy indices in tensor manipulation. Swapping indices may not be valid if the tensor has certain symmetries or if the indices are repeated multiple times in the same term.

## 4. Can swapping dummy indices be applied to all types of tensors?

Swapping dummy indices can be applied to all types of tensors as long as the indices are in the same position and have the same range of values. However, it may not always be valid and should be checked for each specific case.

## 5. Are there any other methods for simplifying tensor calculations besides swapping dummy indices?

Yes, there are other methods for simplifying tensor calculations, such as using index notation, Einstein notation, and the contraction rule. These methods can also help to streamline calculations involving tensors.

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