- #1

Wannabe Physicist

- 17

- 3

- 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.

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.