# Slight Subtlety in Einstein Index Notation

If we have an expression like $$a^ib_i + c^{ij}d_{ij} + e^{ijk} f_{ijk}$$ should it be interpreted as $$\sum_{i,j,k} a^ib_i + c^{ij}d_{ij} + e^{ijk} f_{ijk}$$ or $$\sum_i a^ib_i + \sum_{i,j} c^{ij}d_{ij} + \sum_{i,j,k} e^{ijk} f_{ijk}~?$$ I'm thinking the second one is the answer: we append different summations to different summands.

• Yes, second one. Dec 15, 2019 at 20:29
• I doubt it's possible to generate those sums using tensors since it implies adding tensors of different orders is allowed - which is not true. However, assuming it is possible, since addition is associative, I can relabel the sums - similar to the relabeling of the sum for the Christoffel symbol in the covariant derivative. I would argue $(2)$ doesn't make sense, and $(1)$ is equivalent to $(3)$. Dec 15, 2019 at 22:41
• @CinaedSimson You're right; in that case, what I mean is something like $a^{ij} b_{ik} + c^{im} d_{im}$ Dec 15, 2019 at 22:47
• @CinaedSimson There are many examples where this shows up. Think of the definition of the Ricci scalar. In the $\partial \Gamma$ terms three pairs of indices are contracted and in the $\Gamma\Gamma$ terms four pairs are contracted. Dec 16, 2019 at 0:31
• @MannyC: the OP examples were all contractions - for $3$ tensors each with a different order and summed as if they generated by the same object. Regarding the Ricci tensor, the scalar curvature $S$ is the contraction between the metric tensor and Ricci tensor, namely, $S=\sum g^{ij}R_{ij}$. Dec 16, 2019 at 3:22

## 1 Answer

It means the second thing. There are a couple of reasons why it needs to be this way.

In ordinary algebra, if $$A=B$$ and $$C=D$$, then $$A+C=B+D$$. Under your first interpretation, this would fail when you substituted sums for the symbols $$A$$...

The notation also involves a lot of implicit use of the metric to raise and lower indices. The metric is being multiplied by things, and this has higher priority than addition.