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I am having trouble figuring out what is the development of the following equation due to its notation equation

Its a definition for the <> notation, and all that was previously stated was that $u^{(ab)} =\frac{1}{2}(u^{ab} + u^{ba}) $ is the symmetrization of the tensor $u$. Just to be clear, my problem is with the $h^{(a}_ch^{b)}_d$ part.

Any help would be appreciated, thank you.

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You symmetrize on the parenthesized indices, so

$$h^{(a}{}_c h^{b)}{}_d=\frac12(h^a{}_ch^b{}_d+h^b{}_ch^a{}_d).$$

It doesn’t matter that these indices are on different objects. You just think of the product of the two $h$’s as a four-index tensor (which it is).

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  • $\begingroup$ Thank you very much. Not sure how to select this as an answer to my post thought $\endgroup$
    – TheStressTensor
    Commented Nov 17, 2020 at 18:59

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