# 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:

$$a^{00} b_{00} + a^{11} b_{11} +... ?$$

• The first case. Sum over each index independently.
– Will
Jun 6, 2013 at 17:40

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

$$a^{ij}b_{ij} = \sum_{i=0}^3 \left( a^{i0}b_{i0} + a^{i1}b_{i1} + a^{i2}b_{i2} + a^{i3}b_{i3} \right)$$ $$\implies a^{ij}b_{ij} = a^{00}b_{00} + a^{01}b_{01} + a^{02}b_{02} + a^{03}b_{03} + a^{10}b_{10} + a^{11}b_{11} + a^{12}b_{12} + a^{13}b_{13} + a^{20}b_{20} + a^{21}b_{21} + a^{22}b_{22} + a^{23}b_{23} + a^{30}b_{30} + a^{31}b_{31} + a^{32}b_{32} + a^{33}b_{33}$$

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:

$a^{11}b_{11}+a^{12}b_{12}+a^{13}b_{13}+a^{21}b_{21}+a^{22}b_{22}+a^{23}b_{23}+a^{31}b_{31}+a^{31}b_{32}+a^{33}b_{33}$

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.,

$$a^{jj}b_{jj}=a^{00}b_{00}+a^{11}b_{11}+(\dots).$$

• Ack! $a^{jj}b_{jj}$ is not right, you can't repeat an index more than twice in a term! Jun 6, 2013 at 17:46
• I think the answer was just suggesting what it could mean if you wrote that expression, so not really deserving of a downvote imo. Jun 6, 2013 at 19:51
• An answer about correct Einstein summation notation should contain within it correct Einstein summation notation. I have no problem removing the downvote when either the notation is corrected or it's removed as it's tangential to the question. Once it's done, ping me and I'll remove the vote but I don't remember off hand what the correct notation is to edit it myself. I'd rather see it corrected than removed as it's still interesting. Jun 6, 2013 at 23:20
• @AlexNelson Twice is not the problem, 4 times is the problem. Going through all my books, I have yet to see any that say repeating an index more than 2 times in a term is okay. Jun 7, 2013 at 1:28
• You may want to specify that $a^{jj}b_{jj}$ is not valid Einstein summation notation, and it uses the Euclidean convention. Jun 7, 2013 at 4:37