On Einstein notation with multiple indices On Einstein notation with multiple indices: For example, consider the expression:
$$a^{ij} b_{ij}.$$
Does the notation signify,
$$a^{00} b_{00} + a^{01} b_{01} + a^{02} b_{02} + ... $$
i.e. you sum over every combination of the indices? Or do you sum over the indices at the same time, i.e. they take on the same values:
$$ a^{00} b_{00} + a^{11} b_{11} +... ?$$
 A: Written explicitly, (assuming summation over indices from 0 to 3)
$$a^{ij}b_{ij} = \sum_{i=0}^3 \sum_{j=0}^3 a^{ij}b_{ij}$$
You can expand this to
$$a^{ij}b_{ij} = \sum_{i=0}^3 \left( a^{i0}b_{i0} + a^{i1}b_{i1} + a^{i2}b_{i2} + a^{i3}b_{i3} \right) $$
$$\implies a^{ij}b_{ij} = a^{00}b_{00} + a^{01}b_{01} + a^{02}b_{02} + a^{03}b_{03} + a^{10}b_{10} + a^{11}b_{11} + a^{12}b_{12} + a^{13}b_{13} + a^{20}b_{20} + a^{21}b_{21} + a^{22}b_{22} + a^{23}b_{23} + a^{30}b_{30} + a^{31}b_{31} + a^{32}b_{32} + a^{33}b_{33}$$
A: You would sum over every combination of indices that match. So the i should match, and the j should match. For instance, if each is from 1 to 3, you would get:
$a^{11}b_{11}+a^{12}b_{12}+a^{13}b_{13}+a^{21}b_{21}+a^{22}b_{22}+a^{23}b_{23}+a^{31}b_{31}+a^{31}b_{32}+a^{33}b_{33}$
A: Well, you do it one at a time:
$$a^{ij}b_{ij} = \sum_{j}a^{ij}b_{ij} = a^{i0}b_{i0}+a^{i1}b_{i1}+(\dots). $$
Then you expand on the other index
$$a^{ij}b_{ij} = a^{i0}b_{i0}+a^{i1}b_{i1}+\dots = (a^{00}b_{00}+b^{10}b_{10}+\dots)+(a^{01}b_{01}+a^{11}b_{11}+\dots)+(\dots).$$
If you write $a^{jj}b_{jj}$, then you will obtain the second sum you wrote, i.e., 
$$a^{jj}b_{jj}=a^{00}b_{00}+a^{11}b_{11}+(\dots).$$
