 #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.